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A Refinement of the Local Serrin--Type Regularity A refinement of the local Serrin--type regularity criterion for a suitable weak solution to the Navier--Stokes equations

We formulate a new criterion for regularity of a suitable weak solution v to the Navier-Stokes equations at the space-time point (x_0,t_0). The criterion imposes a Serrin-type integrability condition on v only in a backward neighbourhood of (x_0,t_0), intersected with the exterior of a certain space-time paraboloid with vertex at point (x_0,t_0). We make no special assumptions on the solution in the interior of the paraboloid.


Introduction
The Navier-Stokes system. Let Ω be a domain in R 3 and T > 0. Put Q T := Ω × (0, T ). We deal with the Navier-Stokes system such that v ∈ L ∞ (U ). Points in Q T that are not regular are called singular. A weak solution v of system (1.1), (1.2) is called a suitable weak solution if an associated pressure p belongs to L 5/4 (Q T ) and the pair (v, p) satisfies the so called generalized energy inequality for every non-negative function φ from C ∞ 0 (Q T ). It is also shown in [1] that the set of singular points of suitable weak solution v has the 1-dimensional parabolic measure (which dominates the 1-dimensional Hausdorff measure) equal to zero. This result follows (by a standard covering procedure) from the local regularity criterion (also proven in [1]), saying that there exists a universal constant ǫ > 0 such that if lim sup then (x 0 , t 0 ) is a regular point of v. Analogous results and some generalizations can also be found in papers [6], [7], [4], [19], and others. Some other local regularity criteria. The next criteria are often called ǫ-criteria because they state that there exists a universal constant ǫ > 0 (sufficiently small) such that if a certain quantity is less than or equal to ǫ then (x 0 , t 0 ) is a regular point of solution v. Constant ǫ is generally different in different criteria.
F. Lin [7] proved that the condition dx dt ≤ ǫ guarantees that v is Hölder continuous in the set B ρ (x 0 )×[t 0 −ρ 2 , t 0 ] (for some ρ > 0), which implies that (x 0 , t 0 ) is a regular point of solution v. Lin's criterion has been several times improved (see [8], [13] and [20]). Wolf's criterion (see [20]) says that if 3 ≤ r, s ≤ ∞ and holds for at least one δ > 0 then v is Hölder continuous in the set Particularly, if we choose r = s = 3 then we observe that if the inequality holds for at least one δ > 0 then (x 0 , t 0 ) is a regular point of solution v. A series of other local regularity criteria can be found in [14], [9], [2], etc. Let us finally recall that S. Takahashi [16] proved that if the norm of a weak solution v in L r w (t 0 − ρ 2 , t 0 ; L s (B ρ (x 0 )) (where L r w denotes the weak L r -space and 2/r + 3/s ≤ 1, 3 < s ≤ ∞) is less than or equal to ǫ then (x 0 , t 0 ) is a regular point of v.
Takahashi's criterion has been modified in paper [10], where v is supposed to be integrable with powers r ∈ [3, ∞) (in time) and s ∈ (3, ∞) (in space) not necessarily in the whole backward parabolic neighbourhood Q a,ρ := B √ aρ (x 0 ) × (t 0 − ρ 2 , t 0 ), but only in the intersection of this neighbourhood with the exterior of the space-time paraboloid (with vertex at (x 0 , t 0 )). Exponents r and s are required to satisfy the condition 2/r + 3/s < 1, and number a is supposed to satisfy the inequality 0 < a < 4ν in [10]. Moreover, it is also supposed in [10] that there exist real numbers R and h such that R > 1, 0 < h < R − 1 and for α ∈ [ r r−1 , ∞) and β ∈ ( 3 2 , ∞), satisfying the inequality 2/α + 3/β < 2. Note that the domain of the integral in (1.6) is the exterior of paraboloid P a , intersected with neighbourhood Q a,ρ .
On the result of this paper. In this paper, we improve the regularity criterion from [10] especially so that we remove the assumption on the pressure. Concretely, we show that condition (1.6) can be omitted. Moreover, in comparison to [10], we assume that 2/r + 3/s = 1 and we also use a weaker restriction on parameter a (see Theorem 1). This is enabled by finer estimates in Sections 3-5, and by a different treatment of the term containing the transformed pressure p ′ in Section 3, see Lemma 2.
Our Theorem 1 (formulated below) imposes the Serrin-type condition only on velocity v in an arbitrarily small region U a,ρ in Q T , which is defined as follows: (for a > 0 and 0 < ρ < √ t 0 ). In contrast to a series of other regularity criteria, we make no assumptions on v or p in the interior of paraboloid P a , concretely in set V a,ρ which is the interior of P a , intersected with neighbourhood Q a,ρ .
A generalization of Theorem 1, where parameter a does not appear, is presented in Section 6. We denote by λ S (B 1 ) be the least eigenvalue of the Dirichlet-Stokes operator in the unit ball B 1 in R 3 . Note that the question of how to calculate explicitly the eigenvalues and eigenfunctions of the Stokes operator in the ball was asked by O. A. Ladyzhenskaya in 2003. It can be deduced from [11] that λ S (B 1 ) ≤ µ 2 1 (J 3/2 ), where µ 1 (J 3/2 )= 4.4934 is the first positive root of the Bessel function J 3/2 . Using the variational representation of λ S (B 1 ) (i.e. that λ S (B 1 ) equals the infimum of ∇u 2 2; B 1 / u 2 2; B 1 over all non-zero divergence-free functions u ∈ W 1,2 0 (B 1 )) and the analogous representation of λ L (B 1 ) (the principal eigenvalue of the Dirichlet-Laplacian on the unit ball), we obtain λ S (B 1 ) ≥ λ L (B 1 ). The latter equals π 2 , see e.g. [11].
The main result of this paper says: Let v be a suitable weak solution of system (1.1), (1.2), (x 0 , t 0 ) ∈ Q T and 0 < a < 4νλ S (B 1 ). Let ρ > 0 be so small that Q a,ρ ⊂ Q T . Suppose that function v satisfies the integrability condition in set U a,ρ : for some r, s, satisfying the inequalities Theorem 1 shows that if a singularity eventually appears in a suitable weak solution of the Navier-Stokes system (1.1), (1.2) at the point (x 0 , t 0 ) then it cannot develop only around point x 0 itself (i.e. only in set V a,ρ ). On the other hand, "large" values of velocity must also be necessarily transferred to the point x 0 from the sides at times t < t 0 with the speed increasing to infinity as t → t 0 −.

Proof of Theorem 1 -part I
Notation and the used regularity criterion. We denote We split G(δ) to two parts: We will show that lim inf δ→0+ G(δ) = 0. An estimate of G I (δ). Assume that r > 3. Then G I (δ) can be estimated as follows: This implies, due to conditions (1.7) and (1.8), that We obtain the same information in the case r = 3, too.
Transformation to the new coordinates x ′ , t ′ . In order to estimate G II (δ), we transform the integral in (2.2) and the system (1.1), (1.2) to the new coordinates x ′ and t ′ , which are related to x and t through the formulas  The time interval (t 0 − ρ 2 , t 0 ) on the t-axis now corresponds to the interval 0, ∞ on the t ′axis. Equations (2.5) represent a one-to-one transformation of the parabolic region V a,ρ in the x, t-space onto the infinite stripe then functions v ′ , p ′ represent a suitable weak solution of the system of equations The symbols ∇ ′ and ∆ ′ denote the nabla operator and the Laplace operator with respect to the spatial variable x ′ .) In order to keep a consistent notation, we also denote by (2.7)). Suppose that ϕ is an infinitely differentiable function in R 3 such that The factor 2/(3 3 Lemma 1. Assume that 0 < α ≤ r, 0 < β ≤ s, R > 1, t ′ δ > 2a −1 ln R, and at least one of the two conditions (2.14) Proof. We use C as a generic constant independent of δ. In order to indicate that C may depend on other quantities, we often write e.g. C(R, a), C(R, β) or similar. We have Hence the right hand side of (2.15) is less than or equal to This tends to zero as δ → 0+ due to (1.7). If condition (b) holds then the right hand side of (2.15) is less than or equal to The last factor on the right hand side is This shows that the right hand side of (2.15) tends to zero for δ → 0+ in the case of condition (b) as well. The proof is completed.
The generalized energy inequality in the x ′ , t ′ -space. Since v ′ , p ′ is a suitable weak solution to the system (2.10), (2.11), it satisfies (by analogy with (1.3)) the generalized energy inequality for every non-negative function φ from C ∞ 0 (Q ′ a ). Due to technical reasons, we further assume that 0 < δ < ρ/4. This assumption implies that , where ϕ is defined in (2.12) and ϑ is a C ∞ -function in 2a −1 ln 4, ∞ with a compact support, we get , where χ is the characteristic function of the interval (t ′ δ , t ′ ) and R 1/m is a one-dimensional mollifier with the kernel supported in (−1/m, 1/m), and letting m → ∞, we obtain This inequality holds for a.a. t ′ δ > 2a −1 ln 4 and all t ′ ≥ t ′ δ . Note that it can also be formally obtained, multiplying (2.10) by 2v . The second term on the left hand side of inequality (2.18) comes from the integral of |ϕ( ] on the right hand side of (2.17). Using the formula we can further rewrite (2.18) as follows:

Estimates of the right hand side of inequality (2.19)
The right hand side of inequality (2.19) can be estimated from above by the sum of ϕv ′ ( . , t ′ δ ) 2 2; B ′ 4 and the two terms K I (δ), K II (δ), where Due to Lemma 1, K I (δ) → 0 for δ → 0+. K II (δ) can be estimated as follows: where In order to estimate P(δ), we use the next lemma.
Proof. Let η be an infinitely differentiable cut-off function in R 3 such that Function η can be further expressed in the form η 1 + η 2 , where both the functions η 1 and η 2 are from C ∞ 0 (R 3 ), with values in [0, 1], and such that η 1 = 1 on B ′ 1 and η 1 = 0 on R 3 B ′ 2 . Thus, function η 1 is supported in the closure of B ′ 2 and η 2 is supported in R 3 B ′ 1 .
The function ηp ′ satisfies the obvious identity Integrating by parts and using the formula which we obtain if we apply operator div ′ to equation (2.10)), we derive that Furthermore, applying the Calderon-Zygmund theorem, we obtain Using Lemma 2, we can now estimate P(δ) as follows: (3.7) Applying inequality (2.19), we deduce that the first term on the right hand side of (3.7) is The second term on the right hand hand side of (3.7) equals where C = C(a, ρ) and Thus, c 5 (δ) c 6 (δ) → 0 as δ → 0+. Hence the second term on the right hand hand side of (3.7) tends to zero as δ → 0+. Due to the well known result from [1], saying that the set of singular points of a suitable weak solution has the 1-dimensional Hausdorff measure equal to zero, we can assume (without loss of generality) that ρ and γ are such positive numbers that v and p are bounded on the set Then the third term on the right Using now (3.1) and estimating P(δ) by means of (3.7)-(3.10), we obtain Assuming that δ is sufficiently small, the term c 1 (δ) (6c 2 /a) r−1 r K II (δ) on the right hand side can be absorbed by the left hand side and we get the estimate In order to control the first term on the right hand side of (3.11), we shall use the next lemma. (0, ∞). Let ζ > 0. Then at least one of the two statements holds: for a.a. σ ∈ (σ 0 , ∞). Thus, statement (A) holds. The proof of Lemma 3 is completed.

Lemma 3. Let function f be integrable and nonnegative on the interval
If we apply Lemma 3 with , we obtain that either 16) or there exists a set E ′ ζ ⊂ (0, ∞) with the properties named in item (B) of Lemma 3 such that

Completion of the proof of Theorem 1 in the case of (3.17)
In this section, we assume that (3.17) holds.
A Friedrichs-type estimate of ϕ ξ Applying the so called Bogovskij operator in A ′ 1,1+ξ , one can construct a function w ξ ∈ W 1,2 is the first eigenvalue of the Dirichlet-Stokesian and π 2 is the principal eigenvalue of the Dirichlet-Laplacian in the unit ball -see Section 1.) Hence This implies that to each ξ > 0 and κ > 0 there exists c 12 (κ, ζ) > 0 such that Application of inequality (3.17). The integrand ϕv ′ 2 2; B ′ 4 on the right hand side of (3.17) can be expressed in the form for δ → 0+ for each fixed ξ due to Lemma 1. Substituting inequality (3.17), with the integrand on the right hand side expressed as in (5.3) to (3.11), we obtain where µ is an arbitrary positive number and as δ → 0+ (5.5) (due to (3.12)). Inequality (5.4) is satisfied for t ′ δ ∈ E ′ ζ , for corresponding δ = ρ e − 1 2 at ′ δ (we denote by E ζ the set of these δ) and for all t in the integral on the right hand side of (5.4) can be further estimated by means of inequality (5.2). The term on the right hand side of (5.2) is estimated as follows: . Now, inequalities (5.2) and (5.4) yield where Due to (2.14), c 15 (δ, κ, ξ) → 0 for all fixed κ, ξ and δ → 0+. We observe from (5.5) and from the inequality a < 4νλ S (B 1 ) (see the assumptions of Theorem 1) that there exist positive numbers δ 0 , ξ, ζ, µ, κ and ǫ such that for all 0 < δ ≤ δ 0 . Then inequality (5.7), with these fixed numbers ξ, ζ, µ κ and ǫ, and with δ ∈ E ζ , 0 < δ ≤ δ 0 , yields Thus, we deduce that an analogous expression to the left hand side of (4.2) tends to zero as δ → 0, δ ∈ E ζ . The proof of Theorem 1 can now be completed in the same way as in Section 4 after (4.2).

A generalization of Theorem 1
The assumption a < 4νλ S (B 1 ) in Theorem 1 represents a restriction on the shape of paraboloid P a : the paraboloid cannot be arbitrarily wide and set U a,ρ (where v is supposed to satisfy the Serrin-type condition, considered for fixed ρ) therefore cannot be arbitrarily small. The condition a < 4νλ S (B 1 ) is used only in Section 5, where it guarantees the validity of inequality (5.8). There arises a natural question whether Theorem 1 can be improved so that the Serrin-type integrability condition would be assumed on a smaller set than U a,ρ . We present such a possibility in this section.
In order to stress the dependence on parameter a, we further denote the function θ(t) ≡ a(t 0 − t) by θ a (t). Then θ 1 (t) = √ t 0 − t. Note that set U a,ρ (see Section 1) can also be defined as follows: Furthermore, since the least eigenvalue of the Dirichlet-Stokes operator in B √ a (any ball in R 3 with the radius √ a) is λ S (B √ a ) = λ S (B 1 )/a, the condition a < 4νλ S (B 1 ) is equivalent to These notes lead us to the generalization of Theorem 1: assume that D is a bounded open set in R 3 (not necessarily connected), with a Lipschitzian boundary and containing point 0. Let λ S (D) be the least eigenvalue of the Dirichlet-Stokes operator in D. We define set U ρ (analogous to the previous U a,ρ ) to be U ρ := (x, t) ∈ R 4 ; t 0 − ρ < t < t 0 , x ∈ B ρ (x 0 ) [x 0 + θ 1 (t) D] . Now we can formulate the theorem: Theorem 2. Let v be a suitable weak solution of system (1.1), (1.2), (x 0 , t 0 ) ∈ Q T and ρ > 0 be so small that Q 1,ρ ⊂ Q T . Assume that D is a bounded open set in R 3 with a Lipschitzian boundary, containing point 0, such that 1 < 4νλ S (D) and function v satisfies the integrability condition in set U ρ : |v(x, t)| s dx r s dt < ∞ (6.1) for some r, s, satisfying inequalities (1.8). Then (x 0 , t 0 ) is a regular point of solution v.
Theorem 2 can be proven in the same way as Theorem 1, up to smaller modifications. The most important ones are: we use function θ 1 instead of θ a , we obtain the sets V ′ = (x ′ , t ′ ) ∈ R 4 ; t ′ > 0 and x ′ ∈ D , instead of V a,ρ and U a,ρ , we deal with U ξ (D) (the ξ-neighbourhood of D) instead of B ′ 1+ξ and the cut-off function ϕ decreases from one to zero in A ′ R+1,R+2 instead of A ′ 3,4 (where R is so large that D ⊂ B R (0)).
Theorem 1 is a special case of Theorem 2, corresponding to the choice D = B √ a (0). However, D can generally have another shape than B √ a (0) (it can be e.g. "larger" than B √ a (0) in some directions) and it can still satisfy the condition 1 < 4νλ S (D). The question of dependence of λ S (D) on D is discussed in greater detail in [12] and [21].