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Local well-posedness of Euler-Poisson dominated molecular clouds in astrophysics

Motivated by the astrophysical problems of star formations from molecular clouds, we make the first step on the possible mathematical descriptions of molecular clouds. This article first obtain a model describing the physically reasonable molecular clouds by a proposal to remedy the defects of Makino's star model and removing its exterior regularities (i.e., remove its imposed nonphysically exterior free-falling equations). Since some technical and tricky problems occur, this question is not easy and has been asked by Makino's original paper [37] and Rendall's [48] for a long time. We, in this article, establish a reasonable model by Euler-Poisson system for describing the evolution of molecular clouds and obtain the local existence, uniqueness and continuation principle of the classical solution. In the companion article [31], we will develop this description to the long time region, and conclude under certain data (without any symmetry), the developments of the molecular clouds are either global with near-boundary mass accretions (leads to star formations), or blowup at finite time.


Introduction
Understanding the star formation and the evolution of molecular clouds are active and key quests in astrophysics recently (see [28,29,56] for introductions on stellar formation) and numerous studies in astrophysics (see, for example, [13,14,20,21,29,56] and references cited therein), both observational and theoretical, indicate that stars are formed or "born" in a truly vast cold interstellar gas clouds (or the giant molecular clouds 1 ). Intuitively, since the molecular clouds are enormous, usually containing up to a million solar masses of gas, the self-gravity is huge and may cause the interstellar gas to condense into new stars. Therefore, mathematically it is can be simplified as the blowup problem or the mass accretion problem of the density of perfect fluids (molecular clouds) with vacuum under Newtonian self-gravity.
Although the evolution of giant molecular clouds is a key ingredient for understanding of the star formation and there are numerous observational, numerical and theoretical works on this topic in astrophysical literature, the main formation mechanics and the origin of their physical conditions still remains uncertain and is an unsolved problem to date due to their extreme complexity (too flexible to capture the main properties) and the outstanding discrepancy between the observations and the theoretical models. In the meanwhile, to the best of our knowledge there is no literature in mathematics to rigorously model and analyze the fully nonlinear evolution of giant molecular clouds (it has been pointed out by Rendall [49, §7.2] that there are no results on Jeans instability available for the fully nonlinear case). We, in this article and the companion [31], attempt to make the first step on the mathematical model and nonlinear analysis of the large, expanding and irregularly-shaped self-gravitational molecular clouds and give possibilities of the star formation (mass accretions), fragmentation of the molecular clouds 1 Although interstellar gas clouds and the molecular clouds are not exactly the same, we do not distinguish these two concepts throughout this article since we only focus on the fluid properties and the differences can be ignored. See Appendix E for more detailed meanings. and other singularities. The complete proofs of the linearized and nonlinear Jeans instability are in progress.
Usually, the molecular cloud is extremely vast and some region of it is more hydrodynamicsdominated (like "fluid") and other region is more ballistics-dominated (full of free particles, see Appendix E.2 for details). In this article, we aim to (1) use a natural boundary condition, free-fall (or diffuse boundary describing the diffuse clouds in astrophysics), to separate hydrodynamics-dominated domain from the ballistic one, (2) use the Euler-Poisson system to model the hydrodynamics-dominated part of molecular clouds, (3) analyze the local wellposedness and continuation principle. From now on, we only refer to the hydrodynamicsdominated region of clouds when we mention the molecular clouds, and the ballistic region can be viewed as vacuum (density of the fluid vanishes) by neglecting the pressure and density.
1.1. The diffuse boundary problem of molecular clouds. Let us begin with stating the following model of molecular clouds.
(i) Variables: The molecular clouds are characterized by following variables: We use functions ρ : [0, T )×R 3 → R ≥0 , p : [0, T )×R 3 → R ≥0 and Φ : [0, T )×R 3 → R, for some constant T > 0, to describe the distribution of the mass density, pressure of the fluids and the Newtonian potential of gas clouds, respectively, and denote Ω(t) := supp ρ(t, ·) = {x ∈ R 3 | ρ(t, x) > 0} ⊂ R 3 is the changing volume occupied by the gas at time t. Then the vacuum is identified by ρ(t, x) = 0 in Ω c (t) and the fluids by ρ(t, x) > 0 in Ω(t). We focus on the isentropic ideal gas throughout this article, that is, the equation of state bridging the pressure p and the density ρ of the gas cloud is given by where γ > 1 and K ∈ R >0 are both given constants. The velocity of fluids is only defined on Ω(t) and denoted byẘ := (ẘ i ) : [0, T ) × Ω(t) → R 3 (note that there is no definition in Ω c (t)). We call Ω(t) the hydrodynamic region and Ω c (t) the ballistic region (viewed as vacuum in this model, see Appendix §E.2 for reasons). In ballistic region, rarefied gas dynamics prevails and rarefied gas means there is no mass accretion, which, therefore, does not interest us due to the motivations of star formations. However, in the hydrodynamic region Ω(t), the regime of hydrodynamics applies, and in order to clarify the evolution of the hydrodynamic region of clouds, we propose the following Euler-Poisson system with the diffuse boundary to frame out the hydrodynamic region and characterize this system. Later it turns out this system is well-posed and we emphasize again that we are interested in the potion of the hydrodynamic region of molecular clouds.
1.2.1. Vectors and components. We will use, unless otherwise stated, boldface, e.g., x, y, w for Latins and the Greeks, e.g., ξ, to denote vectors and normal font with indices, e.g., x i , y i , ξ i and w i , to denote the components of vectors (we also use these components to express the vector if it clear from the context).
1.2.2. Indices and summation convention. Throughout this article, unless stated otherwise, we adopt the notation system in general relativity (i.e. Einstein notation, see [55] for more). In specific, we use upper indices to represent components of vectors, lower indices to represent components of covectors, and use lower case Latin letters, e.g. i, j, k, for spatial indices that run from 1 to 3, and lower case Greek letters, e.g. α, β, γ, for spacetime indices that run from 0 to 3 (we also denote time coordinate t by x 0 := t). We use Einstein summation convention as well. That is, when an index variable (i.e. dummy index) appears twice, once in an upper superscript and once in a lower subscript, in a single term and is not otherwise defined, it implies summation of that term over all the values of the index. For example, for a summation 2 This model simplifies diffuse clouds of astrophysics by ignoring the chemical and radiant complexities (diffuse clouds include only atoms and molecules), see Appendix E.1 for details. 3 w i ∈ C 1 ([0, T ) × Ω(t), R 3 ) meansẘ i and ∂ jẘ i have continuous extensions to [0, T ) × Ω(t). 4 Comparing with the physical vacuum boundary condition of star models (see, for example, [11,12,22,34,35]), −∞ < ∇ n (c 2 s ) < 0 on ∂Ω(t), where n is the exterior unit normal vector to ∂Ω(t), diffuse boundaries do not have such singularities on them, that is why they describe molecular clouds. In fact, the conclusion of the companion paper [31] shows there is a possibility that physical vacuum boundary is formed on the boundary or even in the interior of the clouds. z = 3 i=1 x i y i , we simply use Einstein notation to denote z = x i y i . In addition, we raise and lower indices by the Euclidean metric δ ij and δ ij , that is, for example, W ij := W lk δ li δ kj , W ij := W lk δ li δ kj and W ij := W i k δ kj . 1.2.3. Lagrangian descriptions. Suppose a field f : [0, T ) × Ω(t) → V (or a property f in Eulerian description) and the flow χ : [0, T ) × Ω(0) → Ω(t) ⊂ R 3 generated by a vector field w := (w i ), such that χ(t, ξ) = x ∈ Ω(t) for every (t, ξ) ∈ [0, T ) × Ω(0) where T > 0 is a constant, Ω(t) ⊂ R 3 is a domain depending on t and V ⊂ R n for some n ∈ Z ≥1 , we denote describing the property f of the parcel labeled by the initial position ξ at time t (the property f along the flow, i.e., in Lagrangian description). Using this notation, we have where D t is the material derivative (i.e. D t := ∂ t + w i ∂ i , see, for instance, [10]). According to the definition of L ∞ , if χ(t, Ω(0)) = Ω(t), we conclude that

Sets.
We denote U is strictly contained in V by U ⊂⊂ V , which represents the closure of U is a compact subset of V . A set U is called precompact if its closure U is compact. We denote the following subset Ω of Ω for some constant > 0 by where dist(x, ∂Ω) := inf y∈∂Ω |x − y| and denoteΩ := Ω \ Ω . We also use B(x 0 , r) to denote a ball centered at x 0 with the radius r. Therefore, by (1.9), we denote The complement set of a set Ω is denoted by Ω c := R 3 \ Ω.
Let us lower the index of W i j by δ ki , i.e., W jk (t, x) := δ ki W i j (t, x) and decompose where Θ jk is the symmetric deformation component (Θ jk = Θ kj ) of W jk and Ω jk the antisymmetric rotation component (Ω jk = −Ω kj ) defined by Then we have the identities, W kj = Θ jk + Ω jk and W jk = Θ jk − Ω jk . (1.14) We also denote the divergence of the velocity  Theorem). Suppose the initial data (ρ 0 ,ẘ i 0 ) of the diffuse boundary problem is given by (1.6), then: (1) (Local existence and uniqueness) there is a constant T > 0, such that the diffuse boundary problem (i)-(iv) has a unique classical solution (ρ,ẘ i , Φ, Ω(t)) defined by Definition 1.1.
1.5. Related work. To our knowledge, the Euler-Poisson system with vacuum was started with Makino [37,40], which gave the local existence theorem by introducing the Makino's density and the unphysical concept of tame solutions. Later on, a series of works [33,38,39,41,47] continued to study the evolution of tame solutions to Euler equation with or without gravity. Note that the blowup results of Euler-Poisson system with vacuum [38, 39,47] of tame solutions are under the condition of spherical symmetry, but there is no symmetric assumptions for Euler equations in [41]. The advantage of this method is easy to prove blowup, but usually, it does not give more information about the detailed behavior of the finite time blowup solutions. We point out that [38] specified Makino's conjecture that any tame solution, including nonsymmetric tame solution, will become not tame after a finite time, and this current article can be viewed as a partial answer. Oliynyk [44], Brauer and Karp [6] and the references cited therein have improved the local existence and uniqueness of solutions to Euler-Poisson(-Makino) system with vacuum. Brauer [5], by generalizing the method in [4,9], gives a strong continuation principle for tame solutions, especially, a more detailed classification of the type of blowups. In our companion article [31], we will develop this description to the long time region, and conclude under certain data (without any symmetry), the developments of the molecular clouds are either global with near-boundary mass accretions (leads to star formations), or blowup at finite time .
1.6. Overview and main ideas. §2 aims to prove the main Theorem 1.3.(1), that is, the local existence and uniqueness theorem of the classical solution to the diffuse boundary problem. The idea of the proof is to connect the classical solution of diffuse boundary problem with the tame solution of the Makino's initial value problem (see §2.1). Then we introduce the local existence of the tame solution of Makino's problem in §2.2. The third step is to prove the local existence of the solution to the diffuse boundary problem by Calderón's extensions of some data of classical solutions and transforming the solution of the diffuse boundary problem to the tame solution of Makino's problem. The uniqueness of the classical solution is given in §2.5 based on Lemma B.1 (see Appendix B, a Lemma proven by using the relative entropy-entropy flux method given in [35, §2, Step 2]) and Stein Extension Theorem. One ingredient of proving the uniqueness theorem is the preservation of the regularity of the initial C 1 boundary and this will be presented in §2. 4.
In §3, we introduce the continuation principles of the diffuse boundary problem. These continuation principles allow us to continue the local solutions to a larger time interval provided the suitably bounded spatial derivatives. First we give the weak version (see §3.1) and it will help us extend the classical solutions to several types of singularities and the idea of the proof is to use Makino's formulation of Euler-Poisson system and energy estimates. However, the standard Gagliardo-Nirenberg-Moser estimates have been replaced by the revised Gagliardo-Nirenberg-Moser estimates on the Lipschitz bounded domain. The strong continuation principle (see §3.2) helps us reduce the types of singularities and remove certain singularities on the boundary. The key tool of the proof for the strong continuation principle is Ferrari-Shirota-Yanagisawa inequality (a bounded domain version of Beale-Kato-Majda estimate). This inequality allow us to control the velocity w i W 1,∞ "almost only" by the L ∞ -norms of expansion and rotation under certain assumptions. In order to use Ferrari-Shirota-Yanagisawa (FSY) inequality, we have to prove a C ∞ -approximation lemma C.6 of boundary in Appendix C.2 since FSY inequality requires domain is of class C 2,β , β ∈ (0, 1) and w i δ ij n j = 0 on ∂Ω(t) and n j is the unit outward normal at x ∈ ∂Ω(t). To overcome these two requirements and use FSY inequality, our idea is, near the boundary ∂Ω(t), we select a smooth surface (C ∞approximation lemma of boundary C.6 ensures this) and decompose the velocity field to two parts around this C ∞ surface, one of these parts is "tangent" to the C ∞ surface on it, then analyze these two parts respectively.

Local existence and uniqueness of solutions to diffuse boundary problem
This section contributes to the local existence and uniqueness theorem of the classical solution to the diffuse boundary problem. The idea of the proof is to connect the classical solution of diffuse boundary problem with the tame solution of the Makino's initial value problem (see §2.1). Then we introduce the local existence of the tame solution of Makino's problem in §2.2. The third step is to prove the local existence of the solution to the diffuse boundary problem by Calderón's extensions of some data of classical solutions and transforming the solution of the diffuse boundary problem to the tame solution of Makino's problem. The uniqueness of the classical solution is given in §2.5 based on Lemma B.1 (see Appendix B, a Lemma proven by using the relative entropy-entropy flux method given in [35, §2, Step 2]) and Stein Extension Theorem. One ingredient of proving the uniqueness theorem is the preservation of the regularity of the initial C 1 boundary and this will be presented in §2.4.
2.1. Regular solutions to the initial value problem of Euler-Poisson system. We have given the definition of diffuse boundary (see (1.7)) in the previous section §1. 1, and the following proposition claims there are two useful equivalent expressions of this boundary which help us translate the diffuse boundary problem to a initial value problem and the "Makino's problem". The proof of Proposition 2.1 is given in Appendix A for avoiding interrupting the continuity of the statements of this section and please note that we have no any regularity condition on [0, T ) × ∂Ω(t) in Proposition 2.1.
3), Φ is given by (1.5), (1.1) holds and γ > 1. Then, the following three conditions are equivalent: (1) Diffuse boundary: lim (3) Free-fall boundary: the velocity of the limiting Lagrangian observer on the boundary satisfies the free-fall equation, i.e., for (t, ξ) Remark 2.2. Note usually (2.2) is not equivalent to the following condition (in fact, this is stronger than (2.2)), Since for the diffuse boundary problem, the (extended) velocity . In addition, (2.2) can be expressed as D t w k (t, x) = −∂ k Φ(t, x).
Next we point out if we introduce a concept of regular solutions to the initial value problem of Euler-Poisson system (i)-(iii), then finding the classical solution of diffuse/free-fall boundary problem (i)-(iv) is equivalent to finding the regular solution of the initial value problem (i)-(iii). Let us first give the definition of the regular solution.
Definition 2.3. A set of functions (ρ,ẘ i , Φ, Ω(t)) is called a regular solution 5 to the initial value problem of Euler-Poisson system (i)-(iii) on t ∈ [0, T ) for T > 0 if it solves the system (i)-(iii) and satisfies the following conditions: (1) there is a C 1 -extension w i ofẘ i to the boundary, i.e., there exists Then, let us state the proposition trnsforming the diffuse boundary problem to a initial value problem.
To avoid deviating the main objective of this article (the long time behaviors of the classical solutions) too far, we leave the complete analysis of the local existence of the classical solution of diffuse/pressureless/free-fall boundary problem (i)-(iv) (or equivalently, the regular solution of the initial value problem (i)-(iii)) to another separated paper later, but here only focus on the local existence of a type of "stronger solutions" of the diffuse boundary problem by using Makino's tame solution (this means the initial data requires additional conditions). We call this "stronger solution" as Makino solution (roughly, it can be regarded as a restriction of Makino's tame solution on the support of density) and it is defined by the following Definition 2.5 which is a revised regular solution by replacing Definition 2.3.(3) to be the following stronger condition (3 ).
Definition 2.5. A set of functions (ρ,ẘ i , Φ, Ω(t)) is called a Makino solution to the initial value problem of Euler-Poisson system (i)-(iii) on t ∈ [0, T ) for T > 0 if it solves the system (i)-(iii) and satisfies the following conditions: (1) there is a C 1 -extension w i ofẘ i to the boundary, i.e., there exists Then immediately, we claim Proposition 2.6. Makino solutions to the initial value problem of Euler-Poisson system (i)-(iii) on t ∈ [0, T ) for T > 0 are regular solutions to the initial value problem of Euler-Poisson system (i)-(iii) on t ∈ [0, T ) and furthermore, they are classical solutions of diffuse/free-fall boundary problem (i)-(iv) as well.
2.2. Local existence of tame solution to the Makino problem. Before discussing the classical solutions of diffuse boundary problem, we first recall the relevant tame solutions and their local existences given by . The Cauchy problem of Euler-Poisson equations with variables defined in [0, T ) × R 3 for some constant T > 0 is given by where ρ and p are defined as the same as §1.1.(i), and the velocity w : The initial data are prescribed by and Ω(0) is precompact. (1) (ρ, w i ) ∈ C 1 ([0, T ) × R 3 , R 4 ) and Φ is given by (1.5); (2) ρ and w i satisfies an extra equation of the free falling, We refer to, throughout this article, the problem of finding the tame solutions of Euler-Poisson equation (2.3)-(2.6) as solving the Makino problem.
Remark 2.8. The equation of the free falling (2.7) can be interpreted as an equation of motion of a test particle. A test particle is an useful idealized model of an object whose mass is assumed to be negligible, that is, the mass is considered to be insufficient to alter the behavior of the rest of the system. (2.7) expounds that a test particle moves along the integral curves of its velocity field w i (t, x(t)) and the acceleration of this test particle is caused only by the Newtonian gravity.
A standard method to obtain the local existence of Euler-Poisson system is to transform this system to a symmetric hyperbolic system, then the theory of symmetric hyperbolic systems can be applied to derive the local existence. However, there is a severe difficulty when vacuum appears. The vanishing of density leads to the degenerated or unbounded coefficients of the symmetric system (detailed explanations of this difficulty and the ideas on how to overcome this difficulty can be found, for example, in [6,32,37]). Let us briefly recall Makino's ideas [37] in this section. In order to avoid above difficulty, we introduce the Makino's density Then, we rewrite (2.3)-(2.5), with the help of Makino density α, as a symmetric hyperbolic system with a nonlocal term ∂ k Φ, ∆Φ =ρ, (2. 10) with initial data Then using above formulation, by constructing contraction mappings and with the help of fixed point theorem, Makino [37] arrived at the following local existence theorem. We omit the detailed proofs which appeared in [37].
Theorem 2.9. (Local Existence of Tame Solution, see [37]) Assume the initial data (ρ 0 , w i 0 ) ∈ C 1 (R 3 , R 4 ), ρ 0 ≥ 0 and has compact support. If then there is a constant T > 0, such that the Cauchy problem (2.3)-(2.6) has a tame solution , and the solution (ρ where s = 3, 4 according (a) and (b) respectively. There are some improvements of the regularities of solutions with extra exterior velocity constraints in [6,33,44]. However, for simplicity, we currently only use above cited Makino's local existence theorem 2.9 directly for the followings and leave improvements in near future.
2.3. Local existence of classical solutions to the diffuse boundary problem. In this section, we give the local existence theorem of the classical solution to the diffuse boundary problem under certain initial data.
We first prove the following lemma which states the restrictions onto (t, x) ∈ [0, T ) × Ω(t) of the tame solution (ρ, w i ) of the Makino problem (2.3)-(2.6) is a regular solution to the diffuse boundary problem (1.2)-(1.1).
is a Makino solution and further a regular solution of the initial value problem (i)-(iii).
Proof. To conclude this Lemma, we only need to compare the Definition 2.5 of Makino solutions and Definition 2.7 of tame solutions. Since . Then, we complete the proof. Now let us give the local existence theorem of classical solutions to the diffuse boundary problem (i)-(iv) on t ∈ [0, T ) for T > 0.
Proof. We take three steps to prove this theorem: Step 1: Extendẘ i 0 from Ω(0) to R 3 . Since Ω(0) satisfies the Calderón's uniform cone condition, by applying the Calderón extension theorem (see Appendix, Theorem D.2), there exists a simple (s, 2)-extension operator E : Step 2: Makino's local existence implies an extended regular solution. Using the extended data (ρ 0 , w i 0 ) ∈ C 1 (R 3 , R 4 ) and noting that (ρ 0 , w i 0 ) satisfies the requirements (a) or (b) in Theorem 2.9, Theorem 2.9 implies there is a constant T > 0, such that the Cauchy problem (2.3)-(2.6) has a tame solution (ρ, by the Definition 2.7 of tame solutions. Step 3: Restrain tame solutions to be classical solutions. Now we have tame solutions (ρ, w i , Φ) to the Makino's problem (2.3)-(2.6). by defining Ω(t) := supp ρ(t, ·) and a functionẘ i : We are able to use Lemma 2.11, with the help of Proposition 2.6, to conclude (ρ,ẘ i , Φ, Ω(t)) is a regular solution of the initial value problem (i)-(iii) and a classical solutions of diffuse/free-fall boundary problem (i)-(iv) as well.
In the end, noting that for any due to the definition of the Calderón extension, we complete the proof.
Remark 2.13. There may be classical solutions of diffuse boundary problem (or regular solution to the initial value problem) which is not Makino solutions. However, if there is a Makino solution, by Uniqueness Theorem 2.21 later, it is the only solution and there is no other non-Makino classical solution. By weakening the current data, it is possiblities that there is a unique non-Makino solution.
2.4. Preservation of C 1 -boundary. Although it is possible to weaken the regularity of the boundary by more meticulous considerations, for simplicity and emphasizing the mechanisms of blowups, throughout this article, we assume the initial boundary ∂Ω(0) of the support of the fluids is a C 1 boundary, and we, in this section, prove C 1 regularity of this boundary preserves during the evolution, i.e., ∂Ω(t) ∈ C 1 for t in the existence interval of the classical solution.
(4) is due to the diffeomorphism invariance of the boundary, see Theorem C.11 (see Appendix C).
Proof. Firstly, let us derive a flow and Ω t . Step 1 and 2 from the proof of Theorem 2.12 imply there is a vector field where T 0 < 0 < T 1 (using these steps to future and past respectively). Then by letting Ω 0 := Ω(0) and Lemma 2.14, we obtain the flow ϕ described by preceding lemma. Denote the flow ϕ restricted on the submanifold [0, T )×{0}×D by χ : Then Ω t = χ t Ω(0) and ∂Ω t ∈ C 1 by Lemma 2.14. (5).
Secondly, we prove Ω(t) = Ω t = χ t Ω(0) . In other words, we have to verify that the image of the initial support of density is still the support of density, i.e., χ t supp ρ(0, ·) = supp ρ(t, ·).
Remark 2. 16. We point out from the proof of this theorem, the regularity of ∂Ω(t) depends on the regularity of initial ∂Ω(0) and the regularity of the vector field w. Since w ∈ C 1 ([0, T ) × Ω(t), R 3 ) leads to a C 1 flow, it seems we can obtain a Lipschitz boundary ∂Ω(t) provided ∂Ω(0) is Lipschitz. However, due to the implicit function theorem applied by Theorem C.4, the Lipschitz boundary can not be achieved directly.
We put the proof of this proposition in Appendix B.2 to avoid deviating from the main objective too far.

2.5.
Uniqueness of the solution to the diffuse boundary problem. In this section, we prove the uniqueness theorem of the classical solution of the diffuse boundary problem (the proof of this theorem is inspired by the relative entropy-entropy flux method given in [35, §2, Step 2]). This theorem helps us confirm that there is only one classical solution and this one has to be the restriction of the tame one, and rule out all other possibilities if the initial data are in the form of Theorem 2.12. (1) and (2). Although the exterior data of velocity w i of the corresponding tame solutions can be selected with certain freedoms, the classical solution is unique and independent of selections of the exterior data of velocity of the corresponding tame solutions once the interior data of velocity are fixed. This phenomenon is essentially due to the degenerated sound cones near the boundary (see [5] for details). Before the uniqueness theorem, we first present the following lemma which is a direct result of Lemma B.1 (see Appendix B) and states for the solutions on R 3 if the portion of data contained in the support of density are the same, then so are the portion of solutions contained in the support of density.
Lemma 2. 18. Under the assumptions of Lemma B.1, if initial data satisfy ∪ Ω 2 (t)) c and use the initial data = Ω 2 (0), then the above domain of integration can be shrunk to Ω 1 (t) ∪ Ω 2 (t) and above inequality (2.16), with the help of Then by the Gronwall's inequality, we arrive at ∪ Ω 2 (t)) and further Ω 1 (t) = Ω 2 (t) since they are supports of densities ρ 1 and ρ 2 , respectively. Then we complete this proof.
Note that Local Existence Theorem 2.12 gives the local classical solutions of diffuse problem by the Calderón extension of the data and the symmetric hyperbolic system theory. However, since the classical solution from Local Existence Theorem 2.12 is much stronger than the Def- (Ω(t)))), it does not mean every classical solutions of diffuse boundary problem comes from this procedure, that is, until now there may be some classical solutions of diffuse boundary problem which are not Makino solution (i.e., ρ ) and hence can not be derived by above extension and Makino's system. The next Uniqueness Theorem 2.20 indicates any solutions (not necessary to be the classical solutions of the diffuse boundary problem) with the same initial data in the hydrodynamic region Ω(t) are the same if these solutions are extendable. After giving Uniqueness Theorem 2.20, we point out, in Strong Uniqueness Theorem 2.21, every classical solution defined by Definition 1.1 is extendable and furthermore, with the help of Uniqueness Theorem 2.20, it follows the classical solution exists and unique under the data given in Local Existence Theorem 2.12.
, that is they are, respectively, two extendable solutions of the system, and the equation of state (1.1) and the diffuse boundary (1.7) hold. If initial data are the same, , then the solutions are the same as well, Proof. Firstly, direct calculations (note (2.17)×ẘ k +(2.18) yields the following second equation) lead to that for any (t, x) ∈ [0, T ) × Ω(t) the system (2.17)-(2.18) is equivalent to Since (ρ ,ẘ i , Φ , Ω (t)), for = 1, 2, are extendable (using Definition 2. 19), there are extensions = 0) and they both solve the Euler-Poisson system given in Lemma B.1. After verifying them, with the help of Lemma 2.18, we conclude this theorem.
Above Theorem 2.20 claims the extendable solution is unique, the next Theorem 2.21 states with the stronger conditions of the local existence Theorem 2.12, the classical solutions are all extendable, furthermore, under the conditions of the local existence Theorem 2.12, the classical solution is unique.
, Ω (t)), for = 1, 2, are two classical solutions to diffuse boundary problem (i)-(iv), defined by Definition 1.1 on [0, T ). If the initial domain Ω(0) is bounded and satisfies ∂Ω(0) is a C 1 boundary, and initial data are the same, , then the solutions are the same as well, Furthermore, under the conditions of the local existence Theorem 2.12, and ∂Ω(0) is a C 1 boundary of Ω(0), then there is a unique classical solution (ρ,ẘ i , Φ, Ω(t)) in t ∈ [0, T ] given by Theorem 2.12 for every 0 < T < T .
Proof. Let us first claim every classical solutions to diffuse boundary problem (i)-(iv) is extendable. In fact, since (ρ,ẘ i , Φ, Ω(t)) is a classical solutions to diffuse boundary problem (i)-(iv), by Definition 1. . Therefore, by Definition 2.19, the classical solution is extendable. Further, using Uniqueness Theorem 2.20 completes the proof of the first part.
If assuming conditions of the local existence Theorem 2.12 with C 1 boundary ∂Ω(0) of Ω(0) for t ∈ [0, T ), then there is at least one Makino solution given by Theorem 2.12 andẘ i ∈ C 0 ([0, T ), H s (Ω(t))) ∩ C 1 ([0, T ), H s−1 (Ω(t))). Using above result, we conclude any two classical solutions are the same, that is, there is a unique classical solution of the system and the unique solution has to be the Makino solution. We complete this proof.
Remark 2.22. We point out [5, Proposition 2] proved the uniqueness theorem of the Makino solutions (which allow us obtain a symmetric hyperbolic system) instead of all classical solutions, since [5] presumes two solutions solve Makino's formulations (make sure the hyperbolic system is not degenerated) of Euler-Poisson system and energy estimates lead to they are equal in the lenslike shape region if data are the same, this can be done only if ρ γ−1 2 (t) ∈ H s rather than more general condition ρ γ−1 (t) ∈ H s for all classical solutions. Therefore, the uniqueness of [5] implies the solution satisfying ρ

Continuation principles of diffuse boundary problem
In this section, we introduce the continuation principles of diffuse boundary problem. These continuation principles allow us to continue the local solutions to a larger time interval provided the suitably bounded spatial derivatives. First we give the weak version (see §3.1) and it will help us extend the classical solutions to several types of singularities and the basic idea of the proof is to use Makino's formulation of Euler-Poisson system and energy estimates. However, the standard Gagliardo-Nirenberg-Moser estimates have been replaced by the revised Gagliardo-Nirenberg-Moser estimates on the Lipschitz bounded domain. The strong continuation principle (see §3.2) helps us reduce the types of interior singularities and remove certain singularities on the boundary. The key tool of the proof for the strong continuation principle is Ferrari-Shirota-Yanagisawa inequality (a bounded domain version of Beale-Kato-Majda estimate). This inequality allow us to control the velocity w i W 1,∞ "almost only" by expansion and rotation. In order to use Ferrari-Shirota-Yanagisawa (FSY) inequality, we have to prove a C ∞ -approximation lemma C.6 of boundary in Appendix C.2 since FSY inequality requires domain is of class C 2,β , β ∈ (0, 1) and w i δ ij n j = 0 on ∂Ω(t) and n j is the unit outward normal at x ∈ ∂Ω(t). To overcome these two requirements and use FSY inequality, our idea is, near the boundary ∂Ω(t), we select a smooth surface (C ∞ -approximation lemma of boundary C.6 ensures this) and decompose the velocity field to two parts around this C ∞ surface, one of these parts is "tangent" to the C ∞ surface on it, then analyze these two parts respectively.
3.1. Weak continuation principle of diffuse boundary problem. Using energy estimates on bounded Ω(t), we obtain the following weak continuation principle. Note since we will use Makino's formulations, the derivations of this section work only for Makino solutions (recall Definition 2.5, a special class of regular solutions resulted from Theorem 2.12) instead of the general regular solutions.  then there is a constant T > T > 0, such that the classical solution (ρ,ẘ i , Φ, Ω(t)) exists on t ∈ [0, T ). Moreover, for t ∈ [0, T ), there is an estimate, where U := (α,ẘ k ) T .
Proof. Firstly, according to Makino's formulations (2.8)-(2.9), we reexpress the Euler-Poisson system (1.2)-(1.4) of the diffuse boundary problem in Ω(t) and rewriting them into the matrix form yields, for (t, x) ∈ [0, T ) × Ω(t), Secondly, let us derive a local energy estimate on Ω(t) by using the local calculus inequalities in Appendix D.3 (as the standard procedure, we, omit the details, first derive the estimate by assuming all the variables are smooth and then using the approximation theorems of Sobolev spaces in [16, §5.3] to conclude the derived inequality holds for Sobolev spaces as well). First acting on both sides of (3.3) by A 0 ∂ β (A 0 ) −1 (for |β| ≤ s and s = 3 or 4) yields Let us define the local energy, and calculate an important boundary term for later use. Note Then since w i is the extension ofẘ i to the boundary ∂Ω(t), let us calculate, by the divergence theorem and α| ∂Ω(t) ≡ 0, the boundary term vanishes, where ν i := δ ij ν j and ν j is the unit outward-pointing normal to ∂Ω(t).
At the end, we prove the local existence can be extended to t ∈ [0, T ). If, as [36, §2.2], [0, T ) (T < T ) is the maximal interval of the local existence, we can use (3.2) and the local existence Theorem 2.12 to start a solution at time T − for any small > 0 and continue this solution beyond T (or in specific, as the proof of theorem 2.12, one can extend the solution to R 3 , then the local existence of standard hyperbolic system with a proceeding time to exceed T , then restrain the solution to Ω(t)), then a contradiction follows, which means T = T . Then continue to begin a solution at time T − for any small > 0 and extend it beyond T to T > T . Then the proof is completed.

3.2.
Strong continuation principle of diffuse boundary problem. Inspired by the ideas of [5], we improve the above continuation principle to decrease the types of singularities. Firstly, Lemma 3.2 (resemble [17, Corollary 1]) is a useful tool for proving the strong version of continuation principle and can be derived directly from the Ferrari-Shirota-Yanagisawa inequality of Theorem D.10 (see Appendix D.4).
The next theorem 3.3 gives the continuation principle more meticulous portrayal which will provide a better blowup result when the boundary behavior is more "regular", see the companion [31].

Theorem 3.3 (Strong Continuation Principle of Diffuse Boundary Problem).
Suppose Ω(t) ⊂ R 3 is a bounded domain, the initial data satisfy requirements of Theorem 2.12, T > 0 is some given time. If a classical solution (ρ,ẘ i , Φ, Ω(t)) to the diffuse boundary problem (i)-(iv) exists on t ∈ [0, T ) for any T ∈ (0, T ], and if there is a constant > 0 such that an estimate holds 7 + Ω jk (s) L ∞ (Ω (s)) + ∇α(s) L ∞ (Ω(s)) ds < ∞. Proof. According to the estimate of Theorem 3.1, a direct idea of this proof is to verify that W jk (t) L ∞ (Ω(t)) is controlled by for some small > 0. However, in fact, besides (B.9), Lemma 3.2 implies the estimate of W jk (t) L ∞ also slightly depends on the H s norm of the unknowns U H s (Ω(t)) . An approach from [4] (it is widely used, for example, in [5,17], etc.) indicates Lemma 3.2 along with the energy estimate (3.12) and Gronwall inequality (see Theorem D.11) will help us derive an improved a priori estimate on U H s (Ω(t)) . Then a continuation argument as the last paragraph of the proof of Theorem 3.1 can be used to conclude this proof.
For later use, let us start with a useful estimate that Step 1: Re-express (2.9) in terms of Lagrangian descriptions (recall §1.2.3), Integrating it along the integral curves of w, with the help of [5, Lemma 1], (3.13), (3.15) and notation (1.8) in §1.2.3, yields Then, by (3.18), we arrive at Step 2: Now let us turn to control T 0 W jk (t) L ∞ (Ω(t)) dt. The idea of this proof is under the situation of Theorem 3.1, we try to use Lemma 3.2 to control w i W 1,∞ in terms of Θ L ∞ and Ω jk L ∞ . However, there are two difficulties obstructing this idea. In order to use Lemma 3.2, the first difficulty is the boundary in this theorem does not have to be C 2,β as Lemma 3.2 requires; the second one is the orthogonal condition of the boundary limiting velocity w i δ ij n j = 0 is not always true in this theorem. It is clear both of these difficulties come from the boundary. To overcome these, our idea is, near the boundary, we select a smooth surface (C ∞ -approximation lemma of boundary C.6 ensures this) and decompose the velocity field to two parts around this C ∞ surface, one of these part is "tangent" to the C ∞ surface on it, then analyze these two parts respectively.
First using the C ∞ -approximation lemma of boundary (see Lemma C.6 in Appendix §C.2), for every time t ∈ [0, T ) and for any small constant > 0, there is an open domain D(t) with C ∞ boundary, such thatΩ (t) ⊂ D(t) ⊂ Ω(t). Then by the extension theorems of the outward unit normal (see, for example, Proposition C.7 or Theorem C.9, see Appendix C. 3

), since D(t) is a C ∞ domain, then there exists O(t) ⊂ Ω (t), an open neighborhood of the boundary ∂D(t),
and a unit vector field n : O(t) → R 3 of class C ∞ in O(t) with the property that n| ∂D(t) is the outward unit normal to ∂D(t), then we further extend n from O(t) to Ω(t) by assuming N is a smaller neighborhood of ∂D(t) satisfying ∂D(t) ⊂ N and N ⊂ O and using Proposition C.10 (see Appendix C.3, we denote the extended vector field still by n and supp n ⊂ O).
Then, there is a cut-off function φ ∈ C ∞ 0 (see, for example, [30, Proposition 2.25]) such that Then w i can be decomposed as We can define the rotation and expansion of vector field w i − (w j n j )n i φ as (1.13) and (1.15), and denote them by Ω jk and Θ, respectively. Then in order to apply Lemma 3.2, for every t ∈ [0, T ) we verify, with the help of Corollary D.7, Since D(t) is a bounded, simply connected, C ∞ domain, and direct examinations imply (w i − (w j n j )n i φ)n i | ∂D(t) = 0, Lemma 3.2 yields, for every t ∈ [0, T ), ≤ C 1 + ln + w i − (w j n j )n i φ H s (D(t)) ( Θ L ∞ (D(t)) + Ω jk L ∞ (D(t)) ) + 1 .

Appendix A. Preliminary propositions of diffuse boundary
The purpose of this Appendix is to prove Proposition 2.1, but before that, let us fix notations and give a simple lemma. Note that we have no any regularity condition on the boundary [0, T ) × ∂Ω(t) in this Appendix. For anyx = (x µ ), a unit vector v = (v µ ) and x = (x µ ) = (x µ + hv µ ) where h ∈ R, we denote the one-sided derivatives and We first prove a simple but useful Lemma A.1.
After knowing above basic lemma, now let us prove Proposition 2.1.
, for = 1, 2, both solve the system, where the equation of state (1.1) holds. Define the relative entropy Then and there is an estimate

Proof. See
Step 2 in [35, for the detailed proof.
B.2. Proof of Proposition 2.17 for boundary gluing. Let us focus on D l ∪ D b is Lipschitz and D l ∪ D t can be proven similarly, and then conclude this proposition. The key idea of this proof is to construct a local Lipschitz function near D l ∩ D b and further the local boundary of D l ∪ D b near D l ∩ D b is Lipschitz. We also prove D l ∈ C 1 and with the help of D b ∈ C 1 , we obtain the complete boundary is Lipschitz. B.2.1.

B.2.2.
Step 2: Construction of local graph. In order to prove D l ∪ D b is Lipschitz, we focus on the points on D l ∩ D b and see if in neighborhoods of these points, D l ∪ D b can be represented by a local graph of a Lipschitz function. In order to construct a local Lipschitz function (and to avoid the multi-valued function appears), we introduce a rotation in R 4 , and will rotate the original coordinate {x µ } to a new one {x µ }, such that D l ∪ D b is locally graph-represented by a suitable Lipschitz function in {x µ }.
From this, we also know the inverse of the rotation is R −1 ϑ = R −ϑ . Then we obtain a coordinate transform x µ = R ϑx µ for fixed ϑ defined by (B.6). By reexpressing Ψ(x µ ) = 0, for x µ ∈D l , in terms ofx µ , we obtain Ψ • R ϑ (x µ ) = 0. We still, with loss of generality, focus on the case x µ = (0, ξ i ) = (0, 0, 0, 0) ∈ D l ∩ D b . For other general cases if x µ is not the origin, we use a rigid map to move it to the origin as before.
In the end, let us verify that F is a Lipschitz function. We only focus on the case for two pointsx i 1 andx i 2 ∈ B(0, δ), such that F(x i 1 ) =f (x i 1 ) and F(x i 2 ) =x 3 2 tan ϑ (i.e.,f (x 1 1 ,x 2 1 ,x 3 1 ) ≥ x 3 1 tan ϑ andf (x 1 2 ,x 2 2 ,x 3 2 ) ≤x 3 2 tan ϑ). For other cases, they are either similar to above case or, by usingf ∈ C 1 andx 3 tan ϑ ∈ C 1 , we can obtain the Lipschitz inequality directly. Sincê Then we obtain F is a Lipschitz function. This means D l ∪ D b is a Lipschitz boundary. Furthermore, the similar proofs arrive at (D l ∪ D b ) ∪ D t is of Lipschitz as well, then we complete the proof.
Appendix C. Tools of geometry C.1. Regularity of boundary and the geometric description. Let us first recall the definition of regularities of the boundary. See, for example, [3].
Definition C.1. For a proper, non-empty, open subset Ω ⊂ R n , we say the boundary ∂Ω is of class C k (or Lipschitz) locally at a pointx ∈ ∂Ω, if it can be expressed as a graph of a C k (or Lipschitz) function locally at a neighborhood ofx, that is, at the pointx ∈ ∂Ω, there is a composition of a rotation and a translation (i.e., rigid map) T : R n → R n with T (x) = 0, and a neighborhood C := B d (0) × (−h, h) ⊂ R n where constants d, h > 0, along with a C k (or Lipschitz) function φ : R n−1 ⊃ B d (0) → (−h, h), such that φ(0) = 0, ∇φ(0) = 0, and Furthermore, we say the boundary ∂Ω is of class C k (or Lipschitz) or Ω is a C k (or Lipschitz) domain, if ∂Ω is of class C k (or Lipschitz) locally at every pointx ∈ ∂Ω, and we simply denote this by ∂Ω ∈ C k (or ∂Ω ∈ Lip).
Another important domain we will mention is the one with the Calderón's uniform cone condition, this domain has a good extension properties (see §D.1) allowing us to obtain the expected extension. (1) every point of ∂Ω is the center of a sphere of radius entirely contained in one of these sets U j ; (2) every point of U j ∩ Ω is the vertex of a translate of C j entirely contained in Ω.
From these definitions, we obtain Lemma C.3. If Ω is a bounded domain and ∂Ω is a C 1 boundary, then it is a Lipschitz boundary and also satisfies the Calderón's uniform cone condition.
Proof. For a bounded domain Ω, there is a finite open cover U j of ∂Ω which can be rigidly moved to C = B d (0) × (−h, h) for some constant d, h > 0. If ∂Ω is C 1 , then there is a C 1 function φ satisfies the requirements of the definition and it is direct that this function φ is a Lipschitz function on B d (0), which in turn implies this domain is a Lipschitz domain. Since there is a finite open cover in the form of C = B d (0) × (−h, h), we can take small enough to ensure Definition C.2.(1) holds. For every point of 10 U j ∩ Ω, by taking the height of the cone to be smaller than h/2 and the angle of the vertex to be min{arctan 1 |φ | , arctan h d } > 0 where φ ∈ C 0 (B d (0)), Definition C.2.(2) holds. Then it satisfies the Calderón's uniform cone condition.
Due to the implicit function theorem, the following theorem gives C k boundary another description in terms of C k submanifold. We highlight this equivalence only hold for k ≥ 1 because of the implicit function theorem. Therefore, this excludes the equivalence between Lipschitz boundary and submanifold (see [18,19]).
C.2. Sard's Theorem and C ∞ -approximation lemma of boundary. For readers' convenience, we cite the famous Sard's theorem here, and more information can be found in, for instance, [30]. We will use it to prove a useful lemma of C ∞ -approximation of the boundary. We have to prove the boundary ∂D is of class C ∞ next. This is because f (x) = δ for every regular point, under some local coordinate, x = (x 1 , · · · , x n−1 , x n ) ∈ ∂D and by letting C ∞ F (x) := f (x) − δ, the implicit function theorem implies there is a small neighborhood around every x, such that there is a function φ ∈ C ∞ , x n = φ(x 1 , · · · , x n−1 ). Then by Definition C.1, with suitable rotations and translations, we conclude the boundary is smooth and further complete this proof.
C.3. Extension of the outward unit normal. The following theorems come from [3, Theorem 5.3.1 and Lemma 4.6.18] which are useful tools used in the proof of strong continuation principle (i.e., Theorem 3.3). We only cite the statements without proofs.
Proposition C.7 (see Lemma 4.6.18 in [3]). Suppose that Ω is a C k domain in R m , m ≥ 2, for some k ∈ N ∪ {∞}, k ≥ 2. Then the outward unit normal n is a function of class C k−1 . That is, there exists an open set U ⊂ R m , which contains ∂Ω, along with a function N : U → R m of class C k−1 in U with the property that N| ∂Ω = n.
Definition C.8. Given an arbitrary set Ω ⊂ R m , the signed distance (to its boundary ∂Ω) is the function d : R m → R, defined by is a vector-valued function of class C k−1 in U which has the following properties: (1) N(x) = 1 for every x ∈ U; (2) N| ∂Ω = n, the outward unit normal to Ω; (3) ∂ j N k = ∂ k N j in U, for all j, k ∈ {1, · · · , m}; (4) for every j ∈ {1, · · · , m}, the directional derivative D N N j vanished in U.
The next proposition from [30,Lemma 8.6] extends a smooth vector field from a closed subset to the whole manifold with a compact support. Theorem C.11 (Diffeomorphism Invariance of Boundary). Suppose M and N are C k (k ≥ 1) manifolds with boundary and F : M → N is a C k diffeomorphism. Then F (∂M ) = ∂N , and F restricts to a C k diffeomorphism from IntM to IntN .
We denote V := (V 1 , · · · , V n ) a continuous vector field and use a dot to denote an ordinary derivative with respect to t in this Appendix. Let us see the ordinary differential system, y i (t) =V i t, y 1 (t), · · · , y n (t) , (C.1) for i = 1, · · · , n, where (t 0 , c) := (t 0 , c 1 , · · · , c n ) is an arbitrary point. We first introduce an important Theorem C.12 without proofs, see [ (1) (Existence) For any s 0 ∈ J and x 0 ∈ U , there exists an open interval J 0 ⊂ J containing s 0 and an open subset U 0 ⊂ U containing x 0 , such that for each t 0 ∈ J 0 and c = (c 1 , · · · , c n ) ∈ U 0 , there is a C k map y : J 0 → U that solves (C.1)-(C.2).
Proof. See [2, Theorem 5.28] or [8] for the construction of this extension and the detailed proof.
In the region where the gas cloud is not quite cold enough for hydrogen molecules to survive, there is a cloud of neutral hydrogen atoms (known as H I region in astrophysics). In this article, we will focus on such cold interstellar medium including only molecules, atoms and dusts as the interstellar gas clouds or the molecular clouds (in astrophysics, they are also known as the stellar nursery, etc.).
In addition, to complete the backgrounds, we only point out that there is another interstellar gas of the H II regions (a cloud of ionized hydrogen is called an H II region) which located near hot stars. The ultraviolet radiation from the stars also ionizes the hydrogen. The very hot stars required to produce H II regions are rare, and only a small fraction of interstellar matter is close enough to such hot stars to be ionized by them. Most of the volume of the interstellar medium is filled with neutral (non-ionized) hydrogen. Ultra-hot interstellar gas comes from supernova.

E.2.
Remarks on the mathematical model with diffuse boundary. The molecular clouds, or more generally, the interstellar mediums, are inherently complex in their structures. We prefer a family of clouds which is in a decent "onion-like" structure described in [51, §2]. In this model, the dense cloud is in the core, surrounded by translucent gas, then in turn, surrounded by more diffuse gas which is the most tenuous cloud, fully exposed to starlight. Due to the photo-dissociation, nearly all molecules in the outermost layer of the diffuse gas (the inner part of the diffuse clouds can also be molecular) are quickly destroyed and become the neutral atomic form. There, of course, is no distinct concept of "boundary" in the astrophysical model. We attempt to abstract the crucial properties of the above outermost diffuse part to be the mathematical "boundary". E.2.1. Astrophysicists' ideas. In astronomer Elmegreen's article [15, §5.2] on the classifications of spiral, he expounded that the pressure boundary for a molecular cloud is usually ignored in discussions of molecular cloud structure and dynamics because the internal or average cloud pressure is much larger than the ambient interstellar pressure when gravity is strong, as in a star-forming cloud, and he pointed out that the flow pattern of the diffuse clouds tends to be more ballistic and is away from the regime of hydrodynamics. In fact, this is because the Knudsen number K n (see, for example [25], for the theory of classical ballistic transport of rarefied gases), which is defined by the ratio of the molecular mean free path length to a characteristic physical length scale, of the outermost layer of the diffuse clouds is close to or greater then 1, that implies that the mean free path of a molecule is comparable to the diameter of the clouds, and the continuum assumption of fluid mechanics is not a good approximation any more. Consequently, the ballistic transport is observed near the "boundary" of the molecular clouds. That means the pressure exerted around a small parcel can be ignored due to the very large mean free path of the particles on the boundary. The diffuse boundary condition, thus, is a proper abstraction of this situation. In other words, the fictitious "flow" (or test particles) on the boundary are free particles only attracted by the Newtonian gravity without any thermal pressure resisting gravity. We also point out this boundary surely can not be a surface in practice, since the diffuse clouds may extended very far. However, due to the very low pressure as stated by Elmegreen [15, §5.2], we ignore all this part and view it as vacuum in the model. E.2.2. Mathematical abstractions. In terms of mathematics, as we mentioned in §1.1, we call Ω(t) the hydrodynamic region and Ω c (t) the ballistic region (viewed as vacuum in this model), since in Ω(t), the dynamics of the matter is the ideally hydrodynamics-dominated gas and in Ω c (t), rarefied ballistics-dominated gas. In ballistic region Ω c (t), rarefied gas dynamics prevails and the rarefied gas follows Boltzmann equations (or Vlasov equation, i.e., collisionless Boltzmann equation) usually due to the large enough Knudsen number. However, rarefied gas implies there is no mass accretion (since the density of rarefied gas is very small and can be ignored to be viewed as vacuum), which does not interest us due to the motivations of star formations. By contrast, in the hydrodynamic region Ω(t), the regime of hydrodynamics applies, and usually, when the Knudsen is very small, the Euler equation governs this system, while the Knudsen is bigger but still smaller than 0.1, the Navier-Stokes equation prevails. However, the temperature is extremely low for the interested molecular clouds (dark cold clouds), therefore, for gas model, the viscosity is very small. By ignoring the extremely small viscosity, the Navier-Stokes equation becomes Euler equation. On the other hand, the decent gravitational model is the Einstein equation of the general relativity. However, due to its extreme complexities, we focus on its Newtonian limit gravity, i.e., the Newtonian gravity. Therefore, in the hydrodynamic region Ω(t), the Euler-Poisson system applies.
By viewing the ballistic region Ω c (t) as vacuum for the hydrodynamic system we are interested in, there is a property characterized by the diffuse boundary near the moving interface between the cloud and the exterior vacuum. The wellposedness shown in this paper implies we have assumed the portion of the hydrodynamic region can be isolated from the clouds and the interaction from the exterior diffuse clouds or the ballistic region can be neglected (due to the interaction is negligible in practice). E.2.3. About physical vacuum condition. At the end, let us briefly contrast the physical vacuum condition with the diffuse one. The physical vacuum condition implies that the flows near the boundary are still in the regime of the hydrodynamics and the Knudsen number is small up to the boundary and jumps to very large in the exterior region. The models of stars usually do not contain the diffuse area due to the existence of the phase interface.