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An Introduction to the Navier-Stokes Initial-Boundary Value Problem
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1 An Introduction to the Navier-Stokes Initial-Boundary Value Problem Giovanni P. Galdi Department of Mechanical Engineering University of Pittsburgh, USA Rechts auf zwei hohen Felsen befinden sich Schlösser, unten breitet sich die Stadt J.W. GOETHE Introduction The equations of motion of an incompressible, Newtonian fluid usually called Navier-Stokes equations have been written almost one hundred eighty years ago. In fact, they were proposed in 1822 by the French engineer C. M. L. H. Navier upon the basis of a suitable molecular model. It is interesting to observe, however, that the law of interaction between the molecules postulated by Navier were shortly recognized to be totally inconsistent from the physical point of view for several materials and, in particular, for liquids. It was only more than twenty years later that the same equations were rederived by the twenty-six year old G. H. Stokes (1845) in a quite general way, by means of the theory of continua. In the case where the fluid is subject to the action of a body force, the Navier-Stokes equations can be written as follows div (0.1)
2 2 Navier-Stokes Initial-Boundary Value Problem. where is the velocity field evaluated at the point and at time, is the pressure field, is the constant density of the fluid, and is the coefficient of kinematical viscosity. Finally, denotes the relevant geometrical domain where the spatial variables are ranging. Therefore, it will coincide with the region of flow for three-dimensional motions (, ), while it will coincide with a two-dimensional region, in case of plane flows ( ). To the equations (0.1) we append the initial condition: 1 and the boundary condition (0.2) 2 (0.3) In the case where extends to infinity, we should impose also convergence conditions on (and/or, possibly, on ) when. Several mathematical properties for system (0.1) have been deeply investigated over the years and are still the object of profound researches. However, after more than one hundred seventy years from their formulation, the Fundamental Problem related to them remains still unsolved, that is: Given the body force and the initial distribution of velocity (no matter how smooth), to determine a corresponding unique regular solution to for all times. So far, this problem is only partially solved, despite numerous efforts by mathematicians and despite being viewed as an obvious truth by engineers. All this adds more weight to the following profound consideration due to Sir Cyril Hinshelwood, see Lighthill (1956, p. 343) Fluid dynamicists were divided into hydraulic engineers who observe what cannot be explained and mathematicians who explain things that cannot be observed One of the aims of this article is to furnish an elementary presentation of some of the basic results so far known for. In Section 1, we shall discuss the main features of system (0.1) and describe the main difficulties related to 1 Without loss of generality, we can takeas initial time. 2 For simplicity, we shall consider the case of homogeneous no-slip conditions.
3 1. Some Considerations on the Structure of the Navier-Stokes Equations. 3 it. Successively, following the classical methods of Leray (1934a, 1934b) and Hopf (1951/1952), we introduce the definition of weak solution to (0.1)-(0.3) and study some of the related properties (Section 2). These solutions play a major role in the mathematical theory of Navier-Stokes equations, in that they are the only solutions, so far known, which exist for all times and without restrictions on the size of the data. In Section 3 we shall show the existence of a weak solution for all times. Uniqueness and regularity of Leray-Hopf solutions will be presented in Sections 4 and 5, respectively. Due to the particular form of the nonlinearity involved in the Navier-Stokes equations, this study will naturally lead to the functional class,,, 3 such that any weak solution belonging to is unique and regular. In view of this result, we shall see that every weak solution in dimension two is unique within its class, and that it possesses as much space-time regularity as allowed by the data. Since it is not known if a weak solution in dimension three is in, it is not known if these properties continue to hold for threedimensional flows. However, partial regularity results are available. To show some of these latter, we begin to prove the existence of more regular solutions in Sections 6. This existence theory will lead to the celebrated théorème de structure of Leray, which, roughly speaking, states that every weak solution is regular in space and time, with the possible exception of a set of times of zero 1/2-dimensional Hausdorff measure. Moreover, defining a finite time an epoch of irregularity for a weak solution, if is regular in a leftneighborhood of but it can not be extended to a regular solution after, we shall give blow-up estimates for the Dirichlet norm of at any (possible) epoch of irregularity. In view of the relevance of the functional class, in Section 7 we will investigate the existence of weak solutions in such a class. Specifically, we shall prove the existence of weak solutions in, at least for small times, provided the initial data are given in Lebesgue spaces, for a suitable. To avoid technical difficulties, this study will be performed for the case (Cauchy problem). As a consequence of these results, we shall enlarge the class of uniqueness of weak solutions, to include the case. In addition, we shall give partial regularity results of a weak solution belonging to. The important question of whether a weak solution in is regular, is left open. 3 denotes the space dimension.
4 4 Navier-Stokes Initial-Boundary Value Problem. 1 Some Considerations on the Structure of the Navier- Stokes Equations. Before getting involved with weak solutions à la Leray-Hopf and with their regularity, we wish to emphasize the main mathematical difficulties relating to. First of all, we should notice that the unknowns do not appear in (0.1) in a symmetric way. In other words, the equation of conservation of mass is not of the following form This is due to the fact that, from the mechanical point of view, the pressure plays the role of reaction force (Lagrange multiplier) associated with the isochoricity constraint div. In these regards, it is worth noticing that, in a perfect analogy with problems of motion of constrained rigid bodies, the pressure field must be generally deduced in terms of the velocity field, once this latter has been determined. In particular, we recall that the field can be formally obtained by operating with div on both sides of as a solution of the following Neumann problem div in (1.1) at wheredenotes the unit outer normal to 4. Because of the mentioned lack of symmetry in e, the system (0.1) does not fall in any of the classical categories of equations, even though, in a sense, it could be considered close to a quasi-linear parabolic system. Nevertheless, the basic difficulty related to problem (0.1) (0.3) does not arise from the lack of such a symmetry but, rather, from the coupled effect of the lack of symmetry and of the presence of the nonlinear term. In fact, the formulated for any of the following systems div (0.1 ) 4 From this it is clear that to prescribe the values of the pressure at the bounding walls or at the initial time independently of, could be incompatible with (1.1) and, therefore, could render the problem ill-posed.
5 2. Leray-Hopf Weak Solutions and Related Properties. 5 (0.1 ) obtained from (0.1) by disregarding either the nonlinear term [ ] or the isochoricity condition [ )] can be completely solved. While for this solvability will be clear when we shall consider the solvability of for (0.1), the solvability of is a consequence of an interesting a priori estimate discovered by Kiselev and Ladyzhenskaya (1957) and based on a maximum principle that we would like to mention here. Setting from we obtain (1.2) Consider a point of the cylinder where assumes its maximum. If such a point lies either on the bottom face of the cylinder (, at ) or on its lateral surface (, at ) we have (1.3) If, on the contrary, is an interior point of the cylinder or lies on its top face we find evaluated at. Therefore, from (1.2) we deduce (1.4) As a consequence, from (1.3), (1.4) we prove the following a priori estimate holding for all (sufficiently regular) solutions to system (1.5) Notice that (1.5) is independent of the spatial dimension. Unfortunately, nothing similar to (1.5) is so far known for system (0.1) in dimension 3. Nevertheless, as we shall show later on, in dimension 2 the global (, for all times) estimates that we are able to derive will suffice to ensure the existence and uniqueness of a regular solution for (0.1).
6 6 Navier-Stokes Initial-Boundary Value Problem. 2 The Leray-Hopf Weak Solutions and Related Properties. We shall begin by giving the definition of weak solution in the sense of Leray- Hopf. To this end, we need to introduce some notation. By and,,, we denote the usual Lebesgue and Sobolev spaces, respectively. The norm in is indicated by. For, it is and we set. Whenever we need to specify the domain on which these norms are evaluated, we shall write. We denote by the completion in the norm of the space constituted by all infinitely differentiable functions with compact support in. The dual space of will be denoted by. Let 5 div in We define as the completion of in the Lebesgue space. Moreover, we denote by the completion of in the Sobolev space. For, we shall simply write and, respectively. It is well known, see Galdi (1994, Chapter III), that if has a bounded boundary which is locally lipschitzian, or if is a half-space, the following characterizations for and hold, for : div div where the values at the boundary have to be understood in the trace sense. Furthermore, if is of class, the following Helmholtz-Weyl decomposition holds (2.1) where for some with (we set ). The projection of onto is denoted by (, for ). In the case, and are orthogonal subspaces of, and (2.1) holds for any open set. 5 If is a space of scalar functions, we denote by the space constituted by vector or tensor valued functions having components in.
7 2. Leray-Hopf Weak Solutions and Related Properties. 7 For we set and define div in Notice that for, need not be zero. For vector functions in we put If we need to specify the domain on which these quantities are evaluated, we shall write Moreover, for a given Banach space!, with associated norm, and a real interval, we denote by! the linear space of (equivalence classes of) functions "! such that the functional " " if " if is finite. It is known that this functional defines a norm with respect to which! becomes a Banach space (Hille and Phillips 1957, Chapter III). Likewise, for a non-negative integer and a real interval, we denote by! the class of continuous functions from to!, which are differentiable in up to the order included. Finally, if is open or semi-open, by $!, we denote the subspace of! such that %. Depending on!, these spaces might share several properties with the usual Lebesgue spaces and spaces, and we refer to the monograph of Hille and Phillips for further information. Assume now is a classical solution to (0.1)-(0.3). 6 Then, 6 For instance,is one time differentiable inand twice differentiable in, whileis one time differentiable in. Moreover, assumes continuously the initial and boundary data.
8 8 Navier-Stokes Initial-Boundary Value Problem. multiplying by and integrating over we find for all (2.2) Conversely, if is a vector field satisfying (2.2), and having enough smoothness as to allow for integration by parts over in some sense, 7 we easily obtain & for all & and. Therefore, for every such and by a well known result of Hopf (1950/1951), see Galdi (1994, Lemma III.1.1), we conclude the validity of (0.1) for some pressure field. However, it is clear that if is a solenoidal vector field that satisfies (2.2) but is not sufficiently differentiable, we can not go from (2.2) to and it is precisely in this sense that (2.2) has to be considered as the weak formulation of Remark 2.1 It is simple to give examples of solenoidal vector fields which satisfy (2.2) but which do not have enough smoothness as to verify, no matter how smooth and are. Take, for instance ' ' in (2.3) where has no more regularity than the local integrability in with finite. Since ' and div, we deduce that is a non-smooth solenoidal vector field satisfying (2.2) with and (. Notice that, since ' is harmonic, is analytic. 7 For instance, in the sense of generalized differentiation.
9 2. Leray-Hopf Weak Solutions and Related Properties. 9 We wish to give a generalized meaning to the solenoidality condition and to the boundary condition (0.4). This will be accomplished, for instance, if we require that, for almost all times, belongs to. Moreover, to ensure that all integrals in (2.2) are meaningful, we may require. These considerations then lead to the following definition of weak solution, due to Leray (1934a, 1934b) and Hopf (1951/1952). Definition 2.1 Let,. A measurable function,, 8 is said a weak solution of the problem (0.1)-(0.4) in if a) ) ; b) verifies (2.2). If for all,will be called a global weak solution if it is a weak solution in for all. Remark 2.2 In a) we have included the condition that which, a priori, does not seem to be strictly necessary. However, on one hand, this condition ensures that the kinetic energy of a weak solution is essentially bounded in the time interval, and this is a natural request from the physical point of view. On the other hand, excluding such a condition would result in a definition of weak solution too poor to allow for the development of any further relevant property. And last, but not least, we shall prove that the class of weak solutions is not empty, see Theorem 3.1. Remark 2.3 Definition 2.1 is apparently silent about the pressure field. Later on (Theorem 2.1) we shall see that to every weak solution we can always associate a corresponding pressure field. Our next objective is to collect a certain number of properties of weak solutions which will eventually lead, among other things, to a definition equivalent to Definition 1.1. The following result is due to Hopf (1951/1952, Satz 2.1); see also Prodi (1959, Lemma 1) and Serrin (1963, Theorem 4). Lemma 2.1 Let be a weak solution in. Then can be redefined on a set of zero Lebesgue measure in such a way that for all 8 Of course, the definition of weak solution can be given for any spatial dimension, but we shall be mainly interested in the physical interesting cases of two and three dimensions.
10 10 Navier-Stokes Initial-Boundary Value Problem. and satisfies the identity * * (2.4) for all,, and all. Proof. It is clear that to show (2.4) for arbitrary, it is enough to prove it for. We begin to show that (2.4) holds for and almost every. Let + be a monotonic, non-negative function such that +, if, if, For a fixed and & with & we set Notice that + * + + * * & & & + * * Choosing in (2.2) as +, we obtain + * * (2.5) + * * * (2.6) Letting & in this relation and recalling Definition 2.1, we easily deduce that the integral on the left-hand side of this relation tends to *
11 2. Leray-Hopf Weak Solutions and Related Properties. 11 Let now investigate the behavior of the integral at the right-hand side of (2.6). In view of (2.5) and of a) of Definition 2.1, we have for each fixed -& + * * + * * * * * * * * * & * & * * & * *. * Denote by the set of all those for which * * & * * As is well known from the theory of Lebesgue integration (Titchmarsh, 1964, 11.6, Hille and Phillips, 1957, Theorem 38.5), can differ from only by a set of zero Lebesgue measure. Therefore, since we obtain * -& for all and so identity (2.4) follows for and all. We set /. Moreover, by a) of Definition 2.1, there exists a constant. and a set / of zero Lebesgue measure such that. for all / (2.7) Put / / / and pick /. Then, there exists a sequence / converging to as 0. By (2.7),. and so, by the weak
12 12 Navier-Stokes Initial-Boundary Value Problem. compactness of the spaces we find such that for all Define if / if / (Notice that.) Clearly,, for all. Furthermore, evaluating (2.4) along the sequence associated to and letting 0 it is easy to verify the validity of the following statements: 1) satisfies (2.4) for all ; 2) does not depend on the sequence. The lemma is then completely proved. As a corollary to this result, we have Lemma 2.2 Letbe a weak solution in. Then can be redefined on a set of zero Lebesgue measure in such a way that it satisfies the identity (2.8) for all and all. Furthermore, is weakly continuous, that is, for all and all. Proof. We put in (2.4) and choose +, where + is the function introduced in the proof of the previous lemma and. Noticing that in, (2.8) follows at once. To show the weak continuity, we observe that for any fixed from (2.8) it follows that
13 2. Leray-Hopf Weak Solutions and Related Properties. 13 for all. It is clear that this property continues to hold (by density) for all. Let now. By the Helmholtz-Weyl decomposition (2.1) we may write and so, since, we have and the lemma follows. Remark 2.4 Lemma 2.2 tells us, in particular, the way in which a weak solution assumes the initial data, namely, in the sense of the weak convergence. Throughout the rest of these notes, we shall assume that every weak solution has been modified on a set of zero Lebesgue measure in such a way that it verifies the assertions of Lemma 2.1 and Lemma 2.2 Our next concern is to investigate if and in which sense, we can associate a pressure field to a weak solution. Let us first assume that is a classical solution to (0.1)-(0.4). Then, multiplying by and integrating by parts over we formally obtain div (2.9) In the next theorem, we shall show that to any weak solution in we can associate a function 3,, 3, such that div (2.10) for all. We wish to emphasize that, using only (2.8) and the local regularity property of weak solution, in general, we can not write for some (2.11)
14 14 Navier-Stokes Initial-Boundary Value Problem. due to the fact that a weak solution has a priori only a mild degree of regularity in time. To see this, let us consider the vector field given in (2.3) and choose but. By a straightforward calculation we find that and therefore (2.10) is satisfied with and ' div' div ' ' Since, does not verify (2.11); see also Remark 2.6. Theorem 2.1 Let be a weak solution in. Then, there exists a scalar field with 3 for all and 3 verifying (2.10) for all and all. Moreover, if 3 satisfies the cone condition, there exist 3 and 3 such that.. for all, where. and 4 if if if if.
15 2. Leray-Hopf Weak Solutions and Related Properties. 15 Proof. Let us consider a sequence of bounded invading domains, that is, is bounded for each 0, and Without loss, we may assume that satisfies the cone condition for each 0. For fixed, consider the functional It is clear that is linear functional on. Moreover, using the Schwarz inequality and the following ones (see, 5, Galdi, 1994, Chapter II) it is easy to see that % % % % % % 6.. (2.12) (2.13) where. As a consequence, is a continuous linear functional on which, by Lemma 2.2, vanishes on. Thus, since is bounded for all 0, by known results (Galdi, 1994, Corollary III.5.1) there exists such that div for all. Likewise, we show that there exists such that div for all Since, for, we have 6, 6, we can modify by a function of time so that in. By means of
16 16 Navier-Stokes Initial-Boundary Value Problem. an induction argument, we then prove the existence of a function with, for all 0. 9 Furthermore, div for all. and, again by Galdi (1994), Corollary III.5.1, and (2.13), we have.. which proves the theorem. Remark 2.5 If has a bounded boundary satisfying the cone condition, one can show that the field introduced in the previous theorem can be chosen to belong to. In fact, in such a case, assuming some more regularity on one shows that relation (2.11) holds, see Sohr and von Wahl (1986). Remark 2.6 In a recent paper, J. Simon (1999) has shown that, when is bounded, there exists at least one weak solution satisfying (2.9), with corresponding, if has no regularity, and with., if is locally lipschitzian. For this result to hold it is sufficient to assume. We wish now to prove a converse of Lemma 2.2, that is, any function ) (see Definition 2.1) which satisfies (2.8) for all and all must also satisfy (2.2). This will lead to an equivalent formulation of weak solution involving identity (2.8) instead of (2.2). We begin to show that if ) solves (2.8) for all and all, then it also satisfies (2.2) with 7 (2.14) where 7. By the linearity of (2.8) in, it is enough to show this statement for 8. Now, (2.2) with 7 and (2.8) can be written in the following forms denotes the set of all positive integers (2.2 )
17 2. Leray-Hopf Weak Solutions and Related Properties. 17 and respectively, where 5 5 (2.8 ) 5 From Lemma 2.2 we already know that implies. Conversely, from classical results on Lebesgue integration (see, 5, Titchmarsh, 1964, 11), one shows that implies. 10 To complete the equivalence of the two formulations, it remains to show that every together with their first and second spatial derivatives and first time derivatives, can be approximated in by functions of the type (2.14). This is the objective of the following lemma. Lemma 2.3 Let be an arbitrary domain in,, and let. Then, there exists a sequence of functions with the following properties. Given and 1 there are functions 7, 0 8 such that 1 with given in (2.14). Moreover, can be chosen to be an orthonormal basis in. Proof. Let be the completion of in the norm of the Sobolev space and let be a basis of constituted by elements of. 11 For arbitrary 9, let be a partition of such that 9 (2.15) Denoting by the scalar product in and setting 10 If, equation is obtained from after multiplying this latter by and integrating by parts. 11 This is always possible, owing to the separability of.
18 18 Navier-Stokes Initial-Boundary Value Problem. we have for all (2.16) and so, by the Schwarz inequality, we find for all 9 (2.17) Thus, from (2.15)-(2.17), for and sufficiently large : we find 9 Choosing, by the Sobolev embedding theorem we conclude 9 with. Moreover, for all, it is and so, by the same kind of argument used before, we show To the set we can apply the Schmidt orthogonalization procedure in, thus obtaining another system, whose generic element is a linear combination of, - -. Since is dense in, it is easy to show that satisfies all requirements stated in the lemma which, consequently, is proved. From what we have shown, we deduce the following result. Lemma 2.4 A measurable function is a weak solution of the problem (0.1)-(0.4) in if and only if a) ) ; b) verifies (2.8), for all and all.
19 2. Leray-Hopf Weak Solutions and Related Properties. 19 Remark 2.7 Differentiating (2.8) with respect to and recalling that ), we find (2.18) for a.a. and all. It is easily seen that the right-hand side of (2.18) defines a linear, bounded functional in. In fact, denoting by such a functional, by the Schwarz inequality and by (2.12) we have, 6 (2.19) where 4 if 4 if Thus, denoting by the dual space of, for almost all, there exists such that where denotes the duality pairing between and. Notice that is in but not necessarily in. Moreover, by (2.18), (2.19) and the assumption ), we find where ' if and ' if. Remark 2.8 The method of proof used for Lemma 2.3 enables us to give a density result which will be used several times in the next sections. To this end, we recall standard facts concerning the theory of approximation of functions. Let ;!, For &, the mollifier ; (in the sense of Friederichs) of ; is defined by ; < ; (2.20) where < is an even, positive, infinitely differentiable function with support in &&, and <. We have (see, 5, Hille and Phillips, 1957)
20 20 Navier-Stokes Initial-Boundary Value Problem. Lemma 2.5 Let ;!,. Then ;!, for all 0. Moreover ; ; Finally, if;! converges to ; in the norm of! then ; ; We also have. Lemma 2.6 is dense in. Proof. Let be a basis of and let!. Denoting by the scalar product in, and setting we have for all and & (2.21) Clearly,. By Lemma 2.5, for a given 1, there is & 1 On the other hand, from (2.21) and the Lebesgue dominated convergence theorem, we have for all fixed & Thus the result follows from the last two displayed relations and the triangle inequality. 3 Existence of Weak Solutions. The aim of this section is to prove the following existence theorem of weak solutions.
21 3. Existence of Weak Solutions 21 Theorem 3.1 Let be any domain in and let. Then, for any given there exists at least one weak solution to (0.1)-(0.3) in. This solution verifies, in addition, the following properties i) The energy inequality: ii). * * *** Proof. We shall use the so called Faedo-Galerkin method. Let be the basis of given in Lemma 2.3. We shall look for approximating solutions of the form (EI) 6 0 (3.1) where the coefficients 6 are required to satisfy the following system of ordinary differential equations 6 with the initial condition " 0 (3.2) 6 0 (3.3) where " Since " for all 0, from the elementary theory of ordinary differential equations, we know that problem (3.1) (3.3) admits a unique solution 6, 0, where. Multiplying (3.2) by 6, summing over and employing the orthonormality conditions on along with the identity for all
22 22 Navier-Stokes Initial-Boundary Value Problem. we obtain for all * * *** (3.4) with. Since, Using in (3.4) the Schwarz inequality along with Gronwall s lemma, we easily deduce the following bound * *. for all (3.5) with. independent of and 0. From this inequality it follows, in particular, that 6. for all 0 which in turn, by standard results on ordinary differential equations, implies, for all 0. We shall now investigate the properties of convergence of the sequence when 0. To this end, we begin to show that, for any fixed, the sequence of functions is uniformly bounded and uniformly continuous in. The uniform boundedness follows at once from (3.5). To show the uniform continuity, we observe that from (3.2), (3.5), with the help of the Schwarz inequality it easily follows that = * * * =. * * (3.6) where = = Thus, using the Schwarz inequality into (3.6) and recalling (3.5), we readily show the equicontinuity of. By the Ascoli-Arzelà theorem, from the sequence we may then select a subsequence which we continue to denote by uniformly converging to a continuous function. The selected sequence may depend on. However, using the classical Cantor diagonalization method, we end up with a sequence again denoted by converging to, for all, uniformly in t. This
23 3. Existence of Weak Solutions 23 information, together with (3.5) and the weak compactness of the space, allows us to infer the existence of such that uniformly in and for all (3.7) Let us now prove that converges weakly in to, uniformly in, that is, uniformly in and for all (3.8) By the Helmholtz-Weyl orthogonal decomposition (2.1), it is enough to show (3.8) for. To this end, writing % % and using the Schwarz inequality together with (3.5), we find For 1, we choose 8 so large that %. 1 Further, by (3.6) we can pick 0 01 so that 1 and (3.8) follows from (3.7) and the last two displayed inequalities. In view of (3.5) we clearly have. Again from (3.5), by the weak compactness of the space it follows the existence of such that for (with ) for all.
24 24 Navier-Stokes Initial-Boundary Value Problem. Choosing in this inequality and using (3.8), it is easy to show that. Thus, in particular, we find for all,. (3.9) We wish now to prove that (3.8) and (3.9) imply the strong convergence of to in 3, for all 3. To show this, we need the following Friederichs inequality, see, 5, Galdi (1994, Lemma II.4.2): Let be a cube in, then for any 9, there exist >9 functions, > such that! 9 If we apply this inequality with and use (3.5), (3.7) we find (3.10) With the help of (3.8)-(3.10), we shall now show that is a weak solution to (0.1)-(0.3). Since we already proved that ), by Lemma 2.4, it remains to show that satisfies (2.8). Integrating (3.1) from to we find (3.11) From (3.8), (3.9) we at once obtain (3.12) Furthermore, denoting by a cube containing the support of, we have (3.13)
25 3. Existence of Weak Solutions 25 Setting =, by (3.5) we also have = and so, using (3.10), we find Furthermore, we have =. (3.14) and since?, from (3.9) we deduce? (3.15) Therefore, passing into the limit 0 in (3.11), from (3.12)-(3.15) we conclude (3.16) However, from Lemma 2.3 we know that every function can be uniformly approximated in by functions of the form So, writing (3.16) with in place of, we may pass to the limit 8 in this new relation and use the fact that ) to show the validity of (2.8) for all. We shall now prove the energy inequality (EI). To this end, we
26 26 Navier-Stokes Initial-Boundary Value Problem. shall take the as 0 of both sides of (3.4). By the definition of, the properties of, and (3.8) we deduce *** *** Moreover, by (3.8), (3.9) and a classical property of weak limits, we find that * * * * (3.17) and (EI) follows from (3.4) and the last two displayed relations. From (EI) we deduce at once On the other hand, is weakly continuous in (see Lemma 2.2), and so we have which implies This relation together with the weak continuity of allows us to conclude and the theorem is thus proved. Remark 3.1 In the literature, one may find many different definitions of weak solution (see, 5, Lions, 1969; Masuda, 1984; von Wahl (1985)). The one chosen here is due to Leray and Hopf. Likewise, there are many different constructive procedures of weak solutions (see Leray 1934a, 1934b; Kiselev and Ladyzhenskaya, 1957; Shinbrot, 1973). Since, as we shall see in the next section, there is no uniqueness guaranteed for weak solutions in dimension 3 (or higher), these procedures may conceivably lead to different solutions. 4 The Energy Equality and Uniqueness of Weak Solutions. An interesting feature of weak solutions that should be emphasized, is that they obey only an energy inequality rather than the energy equality (that is (EI) with
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text:
Manga by Initial Alphabetic:#
19 DaysChapter 269
The cute and funny adventures of a boy and his best friend. Also contains insert art of the characters by the mangaka. Read more
12-ji Kara HajimaruChapter 7
Shoyu operates a cozy pub called "Tamaya's" in a sleepy part of town. In the evening all kinds of tired office workers gather in his shop to drink away there daily worries. Shoyu has more or less settled down in his lifestyle as a gay bar owner, even though his love life has been rather lukewarm since the incident 12 years ago. What will happen when his past unexpectedly catches up with him? Read more
12 BeastChapter 25
From Seven Seas: Three things you should know about Touga Eita, second-year high school student and heir to the Touga-style ninjutsu dojo: First, he really, really loves video games. Second, when given a choice between heroics and saving his own skin, he will always choose the latter. Third, he's never so much as kissed a girl. All this changes when a voluptuous girl with wings and taloned feet named Aero appears and calls on Eita to help save her people-the harpies of Re-Verse-from the merciless onslaught of giant robot monsters known as Gigas. Eita will follow his newfound harpy friend into a... Read more
29 to JKChapter 15
Yariba Eiji (29 years old with intimidating eyes), a corporate slave whose only hobbies are video games and manga, gets through life by spending his time off relaxing at net cafes. One day, he happens upon Minamisato Karen (a 15 year old high school girl) and ends up scolding her. This leads to her warming up to the 29 year old Eiji...! Not only that, Eiji's little sister (a lazy middle schooler), childhood friend (a nosy ex-girlfriend), and his subordinate (a cool and serious girl) are also becoming interested in him. Love is coming at him from every direction! Juggling... Read more
/Blush-DC.Chapter 35
Akiyoshi Wataru is your average student. He attends cram school, hangs out with his friends and does what boys his age do: Go after girls. One day a particular girl, Iwashita Ayane, catches his attention in cram school, because of her red fingernail. Later in town he sees a familiar girl and follows her. But the girl enters an image club. A brothel that specializes on cosplay. What's more, one of the girls working there goes by the name of "Ayane". What does that mean? And what will Wataru do? Read more
1/8 KanojoChapter 40
Asuha wakes up to find herself naked in her childhood friend's room. On top of that, she's the size of a doll! Read more
81 DiverChapter 375
Kentarō's former shogi teacher taught him to (metaphorically) dive into the 81 squares of the shogi board. Thus, his nickname is "81diver". He has failed to enter the professional league of shogi competition. However, he continues to play the game for a living by gambling at amateur shogi clubs. Read more
1 x 1/2Chapter 11.1
Asuka Mori is infactuated with a certain someone. That certain someone being the person who makes up half of her DNA; her mother, Ayako. Contains yuri and incest themes. Read more
16-sai, Shinkon Hajimemashita.Chapter 5
Collection of stories: 1-2. 16-sai, Shinkon Hajimemashita Getting married at16th birthday? Swayed by the husband's unreasoning love? When she was six, Sou-chan promised Wakaba that he would marry her when she turns 16. Ten years later, does Sou-chan still remember the promise he made? Can Wakaba be honest with her own feelings? What will happen when Sou-chan's father's company goes bankrupt? 3. Akuma no Yuuwaku ni, Amaku Torokeru Matsutani Mika wants to protect the pastry shop her father left her but lately no customer visits her shop. Can Katsuya help her? 4. Idol-sama wa Watashi ni Muchuu?! An aspiring idol group 'Infinity three' and its manager are... Read more
+ Tic NeesanChapter 151
Gag driven, short episodes built around a third-year high-school girl (Nee-san), who likes building plastic models in a model-making club along with fellow club members Makimaki and Okappa. Read more
3-manen no KareshiChapter 6
From Yaoi Is Life: Takagi is like the prince of his office every girl admires. But in his private life, he just broke up with his lover, and his body is sexually deprived... At the time, he learned about "rental boyfriend" by chance. He tried, booked someone, and the one who came was a really cool young man named Daichi. His body is perfect, and Takagi wasted no time seducing him to bed, but the service strictly prohibits sexual activities?! Takagi lost his interest in Daichi when...? Read more
30-sai, Shojo OLChapter 0
From Antisense Scanslations: Hinako, a 30-year-old virgin, has always had relationship problems. Due to her phobia of dirty things, she punches any guy who touches her. Desperate to find someone to love and help her graduate from being a virgin, she takes her much younger co-worker Kai up on his offer to date his friend Machida. Will she finally be free from her virginity streak and find someone to love her despite her phobia? Read more
1/2 PrinceChapter 78
It is the year 2100A.D. and humans have invented a virtual reality game called "Second Life". Feng Lan, due to a bet with her brother, vows to beat him in the game. Thus, she becomes the first transgendered character in "Second Life", in other words, she is the only person who is the opposite gender in the game from her real life. As her popularity rises because of her looks, skills and team mates, can she keep her identity a secret from her loved ones and friends to prevent trouble? Note: This manga is actually Chinese, so most names are different... Read more
8 Tales of the ZQNChapter 8
An anthology manga containing 8 short stories based around the theme "ZQN" (From "I Am a Hero") Read more
5 Ji Kara 9 Ji MadeChapter 21
Sakuraba Junko is turning 27 years old soon, but even so, she is without a boyfriend. But who cares?! Junko has a stable job as an English teacher and she is fine with spending her upcoming birthday alone. Well, until she is set up with on an omiai with a Buddhist monk... which, of course, she rejects. But then he shows up at her school where he has enrolled in private lessons... what will become of our Junko-sensei? Read more
15-sai (ASAGI Ryuu)Chapter 6
Third year middle school student Anna has begun living with fourth year college student Ema, and is swung about by the tactless Ema. Drawn in by the romantic designs of Ema, the once perfectly ordinary Ema, is brought into the adult world. And then, she is shown the scene of a kiss between two girls... Read more
18 RinChapter 3
Follows the story of Rinko Ikeuchi, the daughter of wealthy parents who's on the fast track to enter medical school, but when her family falls on hard times Rinko is forced to take a job, any job, to make ends meet. When the handsome Sawakura shows up offering Rinko gainful employment she can't really say no, even after she realizes that it involves working as a production assistant in the world of hardcore adult videos. Read more
...CurtainChapter 5
Contains 5 stories: 1. ...Curtain. Living purely together with Sensei: Hiyori Okamura is a new mathematics teacher at a high school she graduated from. She still loves Shibasaki, a chemistry teacher who she loved as a student. One day, it happened that Hiyori couldn't use her room, so she moved to Shibasaki's room in the same apartment house, and began to live there. A curtain divided the room in two, and Hiyori and Shibasaki...!? 2. Love Duet: Honda Ema works with and is attracted towards Sawaki Koutaro. Honda plays the Piano and is taught by Sawakis little sister. But due to an accident... Read more
3-Second StripChapter 503
Just spend three seconds of your day on this three-panel comic! Read more
1 x ½Chapter 11.1
Mother × daughter. It is Yuri manga of real parent and child objects. Read more
-6mm no TabooChapter 6
Collection of oneshots: 1) Half Moon Tsukito is known as the "Angel of Death" due to the misfortune that falls to anyone that associates with him. A yakuza intrusted with the mission to kill a trader, is looking for a one sweet night before the deadly mission. But perhaps the night will become more than just one night. "remember me in night of the half moon" 2) Private Gallery What is fake and what is real - in art and love? 3) Lover's Return Akimi Kenishi was kidnapped as a little boy in a foreign country and became a pedophiles toy in human trafficking business. His... Read more
+a no TachiichiChapter 13
Satoki Houma is an imaginative girl who enjoys shoujo manga. She has long comes to terms with the fact she will never be the type of girl that is the heroine of a manga — purely because she isn't pretty, and she has decided she prefers to watch events unfold rather that get involved in any case. However, in the perfect shoujo manga twist, one day she gets dragged to an alternate world where the people have summoned a Holy Maiden to save them…! Except so did her classmate, Haruka Toudou, who fits the exact model of a Shoujo Protagonist. Naturally,... Read more
17 Sai no Hisoka na YokujouChapter 6
"It's fine if our hearts aren't connected", that is what Takamura sensei thought when he confessed to his student, Rin. But after a few times of bodily contact, Takamura started to desire Rin's heart. Will Rin open himself up and accept Takamura? [ From Attractive Fascinante ] Also contains a doujinshi (Takamura X Rin). Read more
17-sai, Kiss to DilemmaChapter 20.3
From Chibi Manga: At 17 years old, I didn't know anything about life or love. And I wasn't seeing my cousin as a "man"... I'd like to not know anything at all but love changed everything... Read more
17-sai Hajimete no HChapter 7
From ShoujoMagic & Nekohana: A collection of short stories revolving around a girl's first time! Story 1: A Boy's Suggestion, by SHINJO Mayu Kana is on a mission of love. How pathetic would it be if a senior like her completed high school without ever having had a boyfriend? That's when Takuya, a freshman with a reputation as someone you don't want to mess with, confidently asks Kana to go out with him. She has misgivings about him being younger than herself, but can she be adult enough to overlook it? Story 2: Virgin Beast, by MINAMI Kanan See Ren-ai Shijou Shugi, volume... Read more
"Aishiteru", Uso Dakedo.Chapter 1
From Chibi Manga: He's a liar. He's a player. He's the worst. Even if I knew that...why did I fall in love with him? This is the story about the encounter between a charismatic handsome boy and a serious girl... Read more
#shin5 - Kekkonshite mo KoishiteruChapter 15
Collection of short stories of a family's daily adventures Read more
100% Gokuama Kareshi!Chapter 6
From Midnight Scans: A sweet love story that will melt your heart! A gathering of the sweetest boyfriends. "No matter where you are or the time of day, I will protect you." Story 1 Kodomo Jya, Naino (Yoshioka Ririko) Nono and Ryou are childhood friends. When they were younger, Nono always saved Ryou, but...?! Story 2 Natsu ga Chikai (Kimura Ayako) I like Tsuji Hideto, and in order to be able to speak with him every day, I purposely violate school regulations. Story 3 HOLE - Kirameku Sekai (Takahashi Rie) I never confessed my feelings to my unrequited love. In order to change a life of regret now, she... Read more
...Junai no SeinenChapter 4
Kirishima Kaoru is the representative of the Fujido Groups while Daigo is a young executive of an American corporation. They are business colleagues by day and lovers by night. Although, Kaoru wishes that he could give up his job and indulge himself in sex with Daigo day and night! Unfortunately due to his deep-rooted insecurities, Kaoru continuously tests Daigo's affection through manipulations. And now, Kaoru's previous boss and lover, Ian, wants Kaoru back in his bed. Who will win in the struggle for power and love? The series goes like this: 1. Happy Yarou Wedding 2. Utsukushiki Kemonotachi. 3. ...Virgin Love. 4. Toriko ni Natta... Read more
- All Manga
- 4 koma
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subset: commoncrawl


text:
Segovia Classical Guitar Series: Thiabaut Garcia
Evanston | Website
This event occurred in the past.
Map | Find Parking
French guitarist Thibaut Garcia, was the 2015 first prize winner of the prestigious Guitar Foundation of America International Competition. Garcia is in the midst of a 50-concert tour of the United States and Canada.


text:
+919816286512,9197361 83064
Before going to fly you must make sure with your pilot that para glider is equipped with rescue parachute , updated equipment like harness, helmet etc.
There are some category list which cannot go for a flight :-
Last but not least , you are advised to check all the details before flying with anyone just you are getting a great discount. It may cost a life flying without proper equipment .
Contact Abparagliding:
Manjeet Kumar
Owner at Ab Paragliding
Call Us:
Adventure Billing Paragliding © 2014
Website Designed By: Bhagwati Web Solutions


text:
- Sleeps 4
- 2 Bedrooms
- 1 Bathroom
- 1 Pet allowed
- Garden/courtyard
- Private parking
- WiFi available
With plenty of character features throughout, Box Cottage is thought to date back to around the 18th century. This cosy pet-friendly cottage is nestled within the popular village of Snape close by to the River Alde.
Having recently been renovated, Box Cottage has a lovely modern kitchen and diner which is flooded with natural light and is the perfect space for guests to gather together and enjoy meals. The snug sitting room has comfortable seating surrounding a cream-coloured woodburner encased in a brick fireplace.
Upstairs are two comfortable bedrooms and a modern family bathroom with the added benefit of a bath with an over-bath power shower.
The pretty front garden of the cottage looks towards the Snape Maltings Concert Hall as well as the River Alde and there is also a delightful rear garden which has pretty plants and shrubs and a seating area with garden furniture. From the rear garden gate you can access the parking area which is spacious enough for two cars.
Box Cottage is a fantastic holiday choice for nature lovers, concert goers, walkers and cyclists, with the property perched on an area of Outstanding Natural Beauty. Box Cottage is also within easy access of Snape Maltings which a small complex of shops, galleries, cafes and restaurants and also home to various concerts throughout the year. Shop: 300 yards. Pub: 100 yards.
Sitting Room: TV, DVD, brick fireplace with woodburner, painted ceiling beams.
Kitchen/Dining Area: Electric oven and ceramic hob, fridge with freezer compartment, microwave, dishwasher, washing machine, dining table and chairs.
Cloakroom: Washbasin and wc.
Bedroom 1: King size bed.
Bedroom 2: Twin beds.
Bathroom: Bath with over-bath shower, washbasin and wc.
Garden: Lovely enclosed front garden as well as an enclosed rear garden with plants, shrubs and seating area with garden furniture.
Parking: Off-road parking for two cars on a private drive to the rear.
Heating: Electric heating.
Pets: One well-behaved dog welcome.
Notes: Wi-Fi and towels provided.
Snape is best known as the venue for the Aldeburgh Festival and for other events throughout the year held in the Maltings Concert Hall. However, the pretty village, at the head of the tidal estuary, is a popular holiday venue in its own right and central for exploring the unique area of the Suffolk Heritage Coast – the tidal estuary and network of footpaths are one of the most interesting areas of Suffolk for exploring on foot. There is easy access to the fascinating coastal town of Aldeburgh, plus castles and much more to explore at Framlingham and Orford. The burial treasures at Sutton Hoo, near Woodbridge, should not be missed also.
- Sleeps 4
- 2 Bedrooms
- 1 Bathroom
- 1 Pet allowed
- Garden/courtyard
- Private parking
- 2 Single beds
- 1 King size/super king size bed
- WiFi available
- Equipped with a highchair
- Travel cot available
- Open fire/woodburner
- Bath towels included
- Bed linen included
- Equipped with a dishwasher
- Equipped with a washing machine
Lovely comfortable cottage. Felt completely at home right away.Mr TERENCE HEATHCOTE - 2nd October 2018
Overall, an excellent property in the perfect location. We were touched by the beautiful selection of local products in the welcome hamper.Ms Haruka Kawaguchi - 12th July 2018
Comfortable cottage in a good location.Chris C, Birmingham - 13th August 2016
Wonderful place and would definitely go back!Ruth S, London - 16th July 2015
Just to say how much we enjoyed our week at Box Cottage in Snape. The cottage was really lovely and so cosy and comfortable. We didn't want to leave and hope to return before too long.Vivian C. - 11th May 2015
Holiday Extras
Holiday Extras
We've hand-picked a selection of Holiday Extras from trusted partners to make your holiday extra special. When you make your booking, you'll see all the Extras on offer at your chosen property and be able to add them to your holiday. Booking them in advance gives you more time to relax when you arrive, and remember - many of these offers are exclusive so you won't find them anywhere else.
Discover Newmarket Discover Newmarket Tours
Once the sporting playground of royalty, the historic home of horseracing celebrates a heritage dating back over 350 years. The area has loads to offer with Dis...
Nomad Sea Kayaking Nomad Sea Kayaking Adventure
Get active with an adventure sea kayaking experience. See a different view of Suffolk's waterside landscape and wildlife, on one of these truly memorable tri...
Big Sky Yoga Private Yoga Session
Relax into your Suffolk holiday with a private yoga session in the comfort of your holiday cottage or garden from leading yoga instructor Angie Lee-Foster.
Ufford Park Health & Spa Two for One Fire and Ice Spa Day at Ufford Park Spa
Treat yourself and a partner to the ultimate indulgent day of relaxation at The Thermal Suite, Ufford Park Spa, plus enjoy 25% discount on treatments.
Cretingham Golf Club Unlimited Golf Packages - Cretingham Golf Club
We are delighted to be able to bring you this exclusive offer from one of the county's most picturesque golf courses. The 18-hole parkland course is...
Nick Catling Photography 10% Discount on Beach Photography Workshop
Capture beautiful images on film to help keep the memories of your Suffolk cottage holiday alive with this full day photography workshop with Nick Catling.
Virgin Wines 12 Bottle Classic Wine Selection
This classic case oozes class and is perfect to give you various flavours from across the globe! Enjoy 2 bottles of each wine to encourage sharing or indulgence...
- Normal price
- £115.87
- Price
- £95.50
Virgin Wines 12 Bottle Luxury Wine Selection
This 12 bottle case is packed full of wines to blow you away with flavours and complexities all round – a case guaranteed to impress all who get the pleas...
- Normal price
- £175.87
- Price
- £131.90
Aldeburgh Yacht Club 2 Hour Sailing Experience at Aldeburgh Yacht Club
Take to the water with and RYA instructor at Aldeburgh Yacht Club for a private sailing lesson.
Spicers of Hythe Ltd 20% discount on Hampers of Distinction
A hamper of distinction from Spicers of Hythe adds a little luxury to any occasion. We offer 20% off a beautifully presented hamper of gourmet delights.


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Neighbors, join Halcyon Neighborhood Association's Facebook group!
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Halcyon Neighborhood Association E-News September 2016
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Items appearing in the HNA E-News are deemed to be of general interest to neighbors but do not necessarily reflect the views of Halcyon Neighborhood Association (HNA), its Steering Committee, or the Editor. The Editor's introductory comments express her personal viewpoint.
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1. Celebration of the 20th Anniversary of Halcyon Commons, Sun. Oct. 9, 3-6 p.m., Volunteers Needed!
2. Trees! Trees! Trees! HNA Launches Street Tree Project
3. Crime-Watch Update
4. Whole Foods Market News: Next Drop-in Progress Meeting Fri., Sept. 9, 8:15-9:00 a.m.
5. CBCB News: Link to Article; Message from Director
6. Halcyon Neighborhood Association's Guiding Principles
Editor's Introductory Note:
Thanks to the crew who helped make our fun ice cream social for National Night Out possible, especially to volunteer coordinators Mary Stoker and Lynn Richards, and to the volunteers who made fresh, homemade ice cream and nondairy frozen items: Bruce Wicinas (an ongoing hero for the hand-cranking and the handpicked blackberries and peaches that went in the ice cream), Susan Hunter, and Mikel Delgado. Whole Foods Market once again helped make the event extra tasty with donations of frozen extras and delicious cake and appetizers. Thanks as well to all who helped with setup, serving, and cleanup. Over eighty neighbors stopped by during the course of the evening, as well as Councilmember Lori Droste and various City officials and police officers. We really appreciate all the neighbors who turned out! It makes a difference.
Thanks to Susan Hunter, for once again coordinating a successful multiblock yard sale; this year fifteen households participated, and it was great to see so many neighbors perusing the sales and getting to know one another better.
We have another great community-building event coming up in October – a celebration of the 20th anniversary of the completion and dedication of Halcyon Commons, the community-created park where there was once a parking lot (see item 1). Please get this on your calendars now so you don't miss this special celebration with live music and fun for all ages!
Happy September and Fall Equinox!
—Nancy Carleton, HNA Co-Chair and E-News Editor, email@example.com
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1. Celebration of the
20th Anniversary of Halcyon Commons,
Sun. Oct. 9, 3-6 p.m., Volunteers Needed!
Editor's note: Preliminary event description from HNA Co-Chair John Steere, who's leading the committee organizing this fun, celebratory event, with intro from neighbor Zoe Lake. Let us know if you can lend a hand!
Mark your calendar! The 20th anniversary celebration of our neighborhood park, Halcyon Commons, is coming up on Sunday, October 9, 3-6 p.m. Join your friends and neighbors and bring the whole family as we gather to enjoy this neighborhood jewel. Look forward to a dedication of the halcyon bird sculpture, live music, and displays of the park's evolution. Hear from community leaders. Activities for kids and refreshments will be plentiful. Hope to see you there!
Halcyon Commons 20th anniversary Celebration
Sunday, October 9, 3 to 6 p.m.
In the park on Halcyon Court between Webster and Prince Streets
Help us celebrate the 20th birthday of Halcyon Commons, the park that was designed by its namesake neighborhood, the park that has helped make the Halcyon neighborhood the place that it has become since then.
Please join us for a happy community gathering to appreciate the park and the neighborhood that created it and continues to take care of it.
With Live Music provided by local groups and a Brazilian Bateria, the dedication of the Halcyon bird sculpture in the park, a proclamation by city officials, activities for kids, and more.
Delicious food, cake, and drink made possible by Whole Foods Market and residents.
Volunteers Needed!
Please email email@example.com if you're able to help with setup, stocking the food and beverage table during the event, or cleanup, or are willing to help with pre-event tasks.
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2. Trees! Trees! Trees! HNA Launches Street Tree Project
Editor's note: An important part of Berkeley's Climate Action Plan is getting more trees planted in the city. HNA Steering Committee member Hank Abrons and neighbor Janice Greenberg have formed a committee to get more street trees planted in the Halcyon Neighborhood, and to better tend to the trees we already have. They'd welcome an additional volunteer or two (or more!); if you care about trees, this is a great way to make a visible difference where you live! Here's the announcement
Street Tree Project
In keeping with our celebration of the 20th anniversary of Halcyon Commons, HNA will undertake a project to plant and nurture more street trees. The project will unfold over the next year or so, and a small Tree Planning Team is being formed to guide it. The required time and work commitment will be modest. If you're interested in joining the Tree Planning Team, or have a friend/neighbor to recommend, please contact Hank Abrons (email: firstname.lastname@example.com). Thank you!
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3. Crime Watch Update
Editor's note: Our HNA Facebook group at http://www.facebook.com/#!/group.php?gid=292160762499 often shares breaking crime reports in between E-News editions (your home and email addresses plus phone are required to participate so we can be sure the group is kept safe for neighbors, and we require this information prior to accepting your request to join). Remember, we rely on neighbors to fill in the details of crimes in our neighborhood, so, in addition to filing a police report (which can now be done online through the City of Berkeley website for some kinds of crimes: http://www.ci.berkeley.ca.us/onlinereport/), please email firstname.lastname@example.com to pass the word on to your neighbors, or share it on our Facebook page, so we can all stay aware and vigilant. We don't want to engage in profiling, so if you include a description, please make it as detailed as possible and avoid generalities.
Photos of our well-attended Ice Cream Social for National Night Out were included in this gallery at Berkeleyside.com
BPD Advisory: Three Recent Robberies Related to Pokémon Go
The City of Berkeley has become the latest community to discover a growing crime trend related to the highly popular Pokémon Go smartphone game. In the last month there have been two completed and one attempted pedestrian robberies where the victims were all distracted by the game. One robbery resulted in the arrest of two adults.
On 7/12/16 at 9:15 p.m., Berkeley Dispatch received calls regarding a screaming woman in the 2300 block of Dana Street. Officers responded and made contact with the victim, who said she was walking eastbound on Durant Avenue when she had just finished playing the game. She said she cut through a parking lot on the south side of the street when she noticed she was being followed by two suspects. One of the two knocked her to the ground while he attempted to take her phone. The other suspect pulled her purse off of her shoulder. The suspects then fled northbound through the parking lot and got into a waiting SUV, which fled southbound on Dana Street. The victim described the suspects as two young "baby-faced" black male adults, slender build, wearing dark hooded sweatshirts.
On 7/31/16 at approximately 2:46 p.m., we responded to a report of a robbery that had just occurred in the area of Parker and Warring Streets. The suspect jumped out of a waiting vehicle, punched the victim, and then took his phone from his hand. The victim described the suspect as a black male juvenile, 16 years old, wearing a blue hooded sweatshirt. Following the robbery the suspect ran to a waiting sedan and fled southbound on Warring Street. The suspect vehicle was stopped a few miles from the crime scene, and two arrests were subsequently made.
The attempted robbery occurred most recently on 8/8/16 at around 5:30 p.m. in Civic Center Park, 2151 Martin Luther King Jr. Way. The intended male victim was in the park playing the game when someone tried to grab the phone from his hand. The victim struggled with the suspect and successfully retained his phone. The suspect was described by the victim as a black male juvenile, 15-16 years old, wearing a gray hooded sweatshirt with black and white Air Jordan sneakers. The suspect fled eastbound on Allston Way.
Like many other law enforcement agencies, the Berkeley Police Department encourages members of the community to follow these simple safety tips:
* Leave your smart phones in a pocket until you get where you are going.
* If you intend on playing the game at night, consider doing so with a group of friends.
If members of the community have information regarding the above cases they are encouraged to contact the Berkeley Police Department Robbery Detail at (510) 981-5900. For all emergencies or in progress calls please dial 9-1-1 from your landline or (510) 981-5911 from your cell phones.
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4. Whole Foods Market News: Next Drop-in Progress Meeting Fri., Sept. 9, 8:15-9:00 a.m.
Editor's note: News from Whole Foods Market, courtesy of Store Team Leader Melody Mestermacher.
Next Drop-in Progress Meeting, Friday, Sept. 9, 8:15-9:00 a.m.
Whole Foods Market (check in at Customer Service)
These meetings are held at the Community Table at the front of the store, and the purpose is to share our plans for ongoing changes in the store, street litter control, and progress on any of the ongoing issues we discuss [with HNA reps and nearby neighbors]. The Store Team Leader is usually present, and anyone is welcome to join us. Of course there will be complimentary coffee or tea. Our goal is to follow up immediately on any issues that we hear about, and to hear how the store can be a better neighbor.
As always, if there are ever any urgent issues, please call the store at 510-649-1333 and ask for the shift manager on duty. We have new shift leaders being trained, so please send an email to the group below so we can follow up appropriately. Always feel free to swing by too, if you'd like to have a word with us.
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5. CBCB News: Link to Article; Message from Director
Editor's note: HNA's Guiding Principles preclude us from getting involved in adversarial land-use matters when there are a range of views among neighbors, and call on us to seek partnership relationships with local businesses wherever possible. As the article at the link suggests, there are neighbors near the CBCB (Cannabis Buyers Club of Berkeley) who have expressed frustration about impacts of the business, as covered in a recent article at Berkeleyside.com (see link). Around the time the article came out, CBCB director Aundre Speciale contacted HNA's co-chairs seeking positive ways to reach out to the neighborhood. We let her know that HNA doesn't typically take positions on land-use matters, but that we encouraged a partnership approach with neighbors; we described the good results that have come from Whole Foods Market being willing to hold regular meetings with neighbors to hear complaints and address them with action goals and follow-up tracked via Excel spreadsheet. We encouraged CBCB to consider something similar. In that spirit, I'm passing on the message received from Aundre Speciale, which she asked me to include in this month's E-News. She also informed me that a meeting for neighbors to air concerns will be held soon; when I receive word of it, I'll post to our HNA Facebook group (and in an E-News edition depending on timing). If a neighbor or a couple of neighbors living near the CBCB wants to join the HNA Steering Committee to serve as a liaison to any meetings held by the CBCB and monitor such meetings for HNA in accordance with HNA's Guiding Principles (see item 6), please email email@example.com. The model HNA has encouraged with Whole Foods Market has been very effective at dealing with various complaints of immediate neighbors over the years and has contributed to a positive partnership; we hope something similar might prove possible in this case.
Berkeleyside article: "Neighbors Express Frustration with Berkeley Cannabis Club"
Message from CBCB Director Aundre Speciale:
Hi neighbors,
We at CBCB are proud to be a part of this wonderful and vibrant neighborhood, and strive to operate in a way that is an asset to the community.
We want to make sure that you have contact information for CBCB's Director, Aundre Speciale and General Manager, Deborah Sadler. In addition, CBCB employs two other guards to patrol the neighborhood during hours of operation, 9-9, and we are including their number for immediate response. Please feel free to contact us with any questions or concerns you may have.
Aundre Speciale
Patrolling guard: 510-631-2407
Aundre Speciale: 510-827-0665
Deborah Sadler: 619-851-2410
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6. Halcyon Neighborhood Association's Guiding Principles
Editor's note: Given the mention of HNA's Guiding Principles in item 5, this seems like a good time to repost them.
HNA is a community group dedicated to stewardship of the Halcyon Neighborhood in South Berkeley (bounded by Telegraph, Ashby, Adeline, and Woolsey). We encourage positive, proactive, partnership-oriented approaches to improving the well-being of our neighborhood, with an emphasis on the following goals:
* Community building (such as regular potlucks, special events in the park, mutual support among neighbors, and multiblock yard sales);
* Ongoing care of Halcyon Commons (a park conceived of and created by the neighbors who founded HNA) under the nonprofit umbrella provided by Berkeley Partners for Parks and in partnership with the City of Berkeley;
* Continued greening and care of the neighborhood (planting trees, cleaning litter off streets, graffiti removal, and helping maintain public landscaped features);
* Strengthening neighborhood watch (crime watch, community safety walks, emergency preparedness, and disaster supply cache);
* Networking with the larger Berkeley community (nearby neighborhood groups, neighborhood businesses, City staff, and elected officials);
* Sharing information and empowering residents to become proactive in addressing neighborhood needs and in expressing individual viewpoints regarding civic affairs (spreading news through meetings, flyers, and the HNA E-News, and providing contact information).
To fulfill these goals, HNA provides a sanctuary from partisan politics so that neighbors with diverse viewpoints feel welcome to participate. Thus, HNA only takes stands on larger issues when there is near-unanimity among neighbors. By focusing on immediate local concerns, we find we can have a greater impact and get better results from the time we invest. Guided by these principles, HNA invites neighbors who are willing to work together in a spirit of partnership to participate at whatever level makes sense for them. Opportunities for serving on our volunteer Steering Committee and project-oriented committees are available to neighbors who are willing to roll up their sleeves and work together in a nonpartisan spirit. Neighbors are invited to step forward into leadership positions defined by the work they do in accordance with HNA's primary goals and guiding principles.
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text: Annette Bening, who was watching at home when her husband, Warren Beatty, was onstage reading the wrong Best Picture envelope at the Oscars last year, has a few questions for Jimmy Kimmel, for instance, "What happened? Where were you when the whole debacle happened?" Then, the two people at the periphery of the whole mess try to get to the bottom of it, and in the process realize Warren Beatty might still have the La La Land envelope he read by accident hidden somewhere in his house. Annette should really frame it for him as a joke.


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The UK's banks should boost their capital to help them withstand the expected global economic downturn, the Bank of England claims.
In the latest report by the Financial Stability Committee, the Bank reasserted its view that eurozone sovereign and banking risks are the "significant and immediate threat" to the health of the UK financial sector.
"UK banks have significant refinancing needs," the report says. "While their direct exposures to the sovereign debt of the most vulnerable economies are limited, they have larger exposures to the private sectors of some weaker euro-area economies.
"They also have significant exposures to major European banking systems, which in turn are highly exposed to weaker euro-area countries."
The committee says the UK banking sector has made "significant progress" in bolstering their capital and funding resilience since the height of the financial crisis. But it adds that progress has been set back by the strains in international bank funding markets.
Speaking at a press conference after the report's publication, Mervyn King, the governor of the Bank, said banks should consider issuing new shares to raise capital if their earnings prove insufficient.
King also suggested banks could limit their dividends, while calling on the Financial Services Authority to encourage banks to shore up their balance sheets and disclose their leverage ratios as part of their regular reporting.


text:
This foxy-red kitten is so cute isn't it? It is chasing a something in the grass, maybe a bug and has a very anxious look. There are stones and purple flowers on this high resolution photo. You can crop other popular image sizes by using our suggested cropping tool. You can find this "Cute foxy-red kitten" wallpaper in these categories: Cats (106). The size of this "Cute foxy-red kitten" image is 1920x1200 pixels.
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text:
This morning, and probably on-going for a while, we're going to cover the more important story. We're covering the one with the better press because we wouldn't want to kick a guy when he's down.
So, we're covering Bieber!
At The Curmudgeon we pride ourselves in only looking at serious subjects and we've decided that Barack Obama is no longer serious. Well heck, he hasn't been serious for decade, but everybody's been pretending he was and we played along. So anyway, it's Friday so we're catching up on the earth shattering news about Boober.
Here's the initial summary from yesterday…
Let's see if we can get the gist of what's going on here.
As we have come to understand all this, this Bleeper person is a 19 year old punk who is some kind of a "star" with the 10 year old girl set. He pretty much runs around with his shirt off showing off that silly tat thingy on what passes for his shoulder. We are given to understand that 10 year old girls swoon at the sight of him.
At this point in his life, although we're not sure it's fair to call anything any 19 year old guy who's not in the US military is doing a "life", we're guessing that a bunch of guys in the Dade County lockup were probably swooning over that tat too. Or "tat as well". Whatever, you get the idea. We wonder if Blabber did?
According to the "news" reports – this is kinda fun, we may start submitting stuff to E! – the kid was driving a rented Lamborghini. Side question, who in their right mind rents a 19 year old guy with a no-limit AmEx card a freaking Lamborghini? Well, to answer our own question, it IS a no-limit card…
Not only was he driving it, he didn't have a valid driver's license – we guess if you've got a no-limit card… He had been drinking and smoking lots of weed and taking prescription meds.
He was apparently driving it very fast. Well heck we'd have been driving it really fast, that's the whole point of having a Lamborghini, don't the Florida cops understand that? Come on!
He gets pulled over and proceeds to act like the cop should know him and cut him a break. It's not reported whether he asked the cop if he had a 10 year old daughter, but it was reported that Bumper dropped the "F bomb" on the cop at least three times. We understand, from second hand stories of course, that dropping the "F bomb" on a cop who's pulled you over for driving like a maniac is not a good idea.
At least Bugger didn't get taised. Honestly we kinda wish he had gotten taised, that would have been great video!
It seems as though, based on the Twitter feeds the announcer was reading off, that people are piling on to Blubber. We're betting a whole bunch of guys in the Dade County lockup would like to pile on him too; they just don't have access to Twitter from jail, jailers being media Nethanderthals and all.
The real question here, with punky being 19 and all, is where have his parents been? We saw a tweet yesterday (sorry, didn't capture it) to the effect that his parents have been pretty lenient with him because they were afraid of losing their allowance if they ever told him "no".
That, we can believe. We are definitely Bieliebers. Or something. (Content warning, follow that link, you'll want eye bleach.)
We'll wrap this up just a couple of notes.
We don't think this is the last time we're going to be hearing (endlessly) about Blobber in chrome bracelets. We're sort of looking forward to covering his marriage to Lindsay Lohan – they'll get married in jail and exchange bracelets instead of rings. We'll bet attending their 2-Step meetings will be a real hoot.
Finally, as idiotic as this saga may be, this little twit is doing less damage to us – not counting the 10 year old girls – than the alternate subject for a post today. Personally, we'd prefer Dweeber be POTUS than our alternative President.
Maybe Barack should invite him to the White House. They could smoke some choom on the porch, get giggly, and chow down on munchies – assuming Michelle is out for the afternoon.