Local Smooth Solution and Non-Relativistic Limit of Radiation Hydrodynamics Equations

Boundary Value Problems, Aug 2010

We investigate a multidimensional nonisentropic radiation hydrodynamics model. We study the local existence and the convergence of the nonisentropic radiation hydrodynamics equations via the non-relativistic limit. The local existence of smooth solutions to both systems is obtained. For well-prepared initial data, the convergence of the limit is rigorously justified by an analysis of asymptotic expansion, an energy method, and an iterative scheme. We also establish uniform a priori estimates with respect to .

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Local Smooth Solution and Non-Relativistic Limit of Radiation Hydrodynamics Equations

Boundary Value Problems Hindawi Publishing Corporation Local Smooth Solution and Non-Relativistic Limit of Radiation Hydrodynamics Equations Jianwei Yang 1 2 Shu Wang 0 1 Yong Li 0 1 Donal O'Regan 0 College of Applied Sciences, Beijing University of Technology , PingLeYuan100, Chaoyang District, Beijing 100022 , China 1 and energy , as see 2, 3, 5 2 College of Mathematics and Information Science, North China University of Water Resources and Electric Power , Zhengzhou 450011 , China We investigate a multidimensional nonisentropic radiation hydrodynamics model. We study the local existence and the convergence of the nonisentropic radiation hydrodynamics equations via the non-relativistic limit. The local existence of smooth solutions to both systems is obtained. For well-prepared initial data, the convergence of the limit is rigorously justified by an analysis of asymptotic expansion, an energy method, and an iterative scheme. We also establish uniform a priori estimates with respect to . - 1. Introduction div ρu 0, ∂t ρu div ρu ⊗ u ∇ p 1 3 θ4 0, E p θ4 u 1 1.1 for x, t ∈ R3 × 0, T , T > 0, where ρ, u u1, . . . , ud T , p, and θ denote the density, velocity, thermal pressure, and absolute temperature, respectively, 8π5k4/15h3c3 > 0 is a radiation constant, and c is the light speed, and is the total energy, e e ρ, θ is the internal energy, and u2 id 1 ui2 is the square of the macroscopic velocity. From 1.1 and 1.2 , we see that the system includes both gas and radiative contributions to flow dynamics. The quantities 1/3 θ4 and θ4 represent the radiative pressure and radiative energy density, respectively. To complete system 1.1 , one needs the equation of state for the pressure p p ρ, θ . In this paper, for the purpose of our test problems, we will limit our study to the polytropic ideal gases, namely: p Rρe γ − 1 ρe with γ > 1 being the specific heat ratio and e cV θ with cV being the specific heat; we assume cV 1 without loss of generality. We point out that if one assumes → 0 in 1.1 , then system 1.1 reduces to the usual inviscid Euler equations: ∂tρ0 div ρ0u0 0, which are nonisentropic and compressible Euler equations. The aim of this paper is to justify rigorously the local existence of smooth solutions of system 1.1 and the convergence of system 1.1 to this formal limit equations 1.3 . Concerning the non-relativistic limit c → ∞, that is, → 0, there are only partial results. Indeed, we know that the phenomenon of non-relativistic is important in many physical situations involving various nonequilibrium processes. For example, important examples occur in inviscid radiation hydrodynamics 6 , in quantum mechanics 7 , in KleinGordon-Maxwell system 8 , in Vlasov-Poisson system 9 , in Euler equations 10 , in Euler Maxwell equations 11, 12 , and so on. In this paper, we are interested in the nonrelativistic limit → 0 in the problem 1.1 for the radiation hydrodynamics equations. We prove the existence of smooth solutions to the problem 1.1 and their convergence to the solutions of the compressible and nonisentropic Euler equations in a time interval independent of . For this propose, we use the method of iteration scheme and classical energy method. The convergence of the radiation hydrodynamics equations to the compressible and nonisentropic Euler equations is achieved through the energy estimates for error equations derived from 1.1 and it’s formal limit equations 1.3 . The remainder of this paper is arranged as follows: In the next section, we give the local smooth solutions to both system 1.1 and 1.3 . Section 3 is devoted to justify the convergence of 1.1 to 1.3 . By formal analysis, we show that the leading profiles of the density, velocity, and temperature with respect to satisfy a compressible nonisentropic Euler equations, and their next order profiles satisfy the corresponding linearized equations. The Cauchy problem for this nonisentropic Euler equations is solved in this section. The final part is devoted to rigorously justifying the asymptotic expansion developed in Section 3 and obtaining the convergence of solutions to the multidimensional compressible nonisentropic Euler system in a time interval independent of . Notations and Preliminary Results 1 Throughout this paper, ∇ ∇x is the gradient, α α1, . . . , αd and β are multiindeices, and Hs Rd denotes the standard Sobolev’s space in Rd, which is defined by Fourier transform, namely, f ∈ Hs Rd if and only if f s2 2π d 1 |k| 2 s Ff k 2 < ∞, k∈Zd where Ff k Rd f x e−ikxdx is the Fourier transform of f ∈ Hs Rd . 2 Also, we need the following basic Moser-type calculus inequalities see, Klainerman and Majda 13, 14 : for f, g, v ∈ Hs and any nonnegative multi-index α, |α| ≤ s, s s Dxg L2 g L∞ s Dxf L2 , s ≥ 0, Dxα f g − f Dxαg L2 ≤ Cs Dxf L∞ Dxs−1g g L∞ s Dxf L2 , s ≥ 1, L2 DxsA v L2 ≤ Cs DxsA v L∞ 1 ∇v L∞ s−1 Dxsv L2 , s ≥ 1. 3 Sobolev’s inequality . For s > d/2, 4 If s > d/2, then for f, g ∈ Hs and |α| ≤ s, f L∞ ≤ Cs f s. Dxα f g L2 ≤ Cs f s g s. 1.4 1.5 1.6 1.7 1.8 1.9 2. The Local Existence In this section, we give our main result about local existence. For this purpose, we first rewrite the system 1.1 as a symmetric hyperbolic system of first order. Then, we prove the local existence and uniqueness of smooth solutions to the Cauchy problem for 1.1 . For smooth solutions, the system 1.1 can be rewritten as follows: ∂tρ In fact, 2.1 is a non-relativistic, non-isotropic, and compressible Euler equations. For convenience, we introduce the following two functions: Then, 2.1 can be rewriten as follows: Denote the vector and matrix V ρ, u, θ T , Aj V uj I d 2 × d 2 f1 ρ, θ f2 ρ, θ 4θ3 3ρ , 4/3 − 4R θ4 ρ 4 θ4 . ∂tρ ⎜⎜⎜⎝⎜⎜⎜⎜ 0 0 Rθ ρ ej ρejT 0 Rθ f2 ejT R ⎞ 0 ⎟⎟ f1 ej ⎟⎟⎠⎟⎟⎟ , 0 2.1 2.2 2.3 2.4 2.5 where e1, . . . , ed is the canonical basis of Rd and yi denotes the ith component of y ∈ Rd. Thus, we can rewrite the system 2.3 as follows: We will study the Cauchy problem for 2.5 together with the initial data d It is not difficult to see that the equations of V in 2.5 are symmetrizable and hyperbolic. If we introduce the d 2 × d 2 matrix A0 V ⎜⎜⎜⎜⎜⎜⎝⎜ which is positive definite for 1, then Aj V A0 V Aj V are symmetric for all 1 ≤ j ≤ d. Note that for smooth solutions, 2.3 is equivalent to that of 2.5 . Noticing the above facts and using the standard iteration techniques of local existence theory for symmetrizable hyperbolic system see 15 , we have the following. Theorem 2.1. Assume that V0 ∈ Hs, s > d/2 1, V0 x ∈ G1, G1 ⊂⊂ G {V : ρ, θ ≥ C1 > 0}, and C1 is a positive constant. Then there exists a time interval 0, T with T > 0, such that 2.5 and 2.6 have a unique solution V x, t ∈ C1 Rd × 0, T , with V x, t ∈ G2, G2 ⊂⊂ G for x, t ∈ Rd × 0, T . Furthermore, V ∈ C 0, T , Hs ∩ C1 0, T , Hs−1 , and T depends on , V0 s and G1. 3. Asymptotic Analysis 3.1. Formal Asymptotic Expansions Let ρ , u , θ be the smooth solution to the system 2.3 . In this section, we are going to study the formal expansions of ρ , u , θ as → 0. To this end, we assume that initial data ρ0, u0, θ0 have the asymptotic expansion with respect to : Then, we take the following ansatz: j uj x, t , in terms of for the solutions to the system 2.3 . Substituting the expansion 3.2 into the system 2.3 , we have the following. 1 The leading terms p0, u0, θ0 satisfy the following problem: ∂tρ0 0 0 div ρ u 0, ∂tu0 u0 · ∇ u0 Rθ0 ρ0 ∇ρ 0 R∇θ0 0, ∂tθ0 Rθ0 div u0 u0 · ∇ θ0 0, ρ0, u0, θ0 . 3.3 2 For any j ≥ 1, the profiles ρj , uj , θj satisfy the following problem for linearized equations: j div ρkuj−k 0, j k 0 ∂tuj ∇θj uk · ∇ θj−k θk div uj−k gj−1 2 0, R j ρj , uj , θj ρj , uj , θj , where gi0 0 i 1, 2 for j ≥ 1. In fact, gij−1 i can be obtained from the following relation: 1, 2 depends only on {ρk, uk, θk}k≤j−1 and − R θj ∇ ln ρ0 θ0∇ ln’ρ0ρj gj−1 1 gj−1 2 R dj ⎡ ⎛ ⎛ j! d j ⎣ ⎝ ⎝ θ 0 j1! ddjj ⎡⎣ f2⎛⎝ ρ0 j≥1 j≥1 ⎞ ⎛ j θj⎠ ∇ ln⎝ ρ0 j≥1 ⎞ ⎞ j ρj⎠ ⎠ j ρj , θ0 j θj ⎠ ⎞ ⎛ div⎝ u0 , j≥1 ⎛ f1⎝ ρ0 j≥1 j uj ⎠ ⎞ j≥1 ⎤ ⎦ j ρj , θ0 0 . j≥1 ⎞ j θj⎠ ⎤ ⎦ 3.2. Determination of Formal Expansions 3.2.1. Preliminary From 3.4 , we know that once ρ0, u0, θ0 are solved from the problem 3.3 , ρ1, u1, θ1 are solutions to the following problem for a linearized equations: ∂tρ1 0 1 div ρ u 1 0 ρ u 0, ∂tu1 u0 · ∇ u1 u1 · ∇ u0 R θ1∇ ln ρ0 θ0∇ln ρ0ρ1 ∇θ1 where Inductively, suppose that pk, uk, θk k≤j−1 are solved already for some j ≥ 2, from 3.4 , we know that pj , uj , θj satisfy the following linear problem: j k 0 ∂tuj ∇θj −g1j−1, R j θk div uj−k j uk · ∇ θj−k −g2j−1, Thus, in order to determine the profiles ρ , u , θ , we require to solve the nonlinear problem 3.3 for ρ0, u0, θ0 and the linear system 3.8 . 3.2.2. Existence and Uniqueness of Solution ρ0, u0, θ0 Obviously, 3.3 are nonisentropic and compressible Euler equations. Thus, we recall the following the classical result on the existence of sufficiently regular solutions of the compressible Euler equations, see 15 . 3.6 3.7 3.8 Proposition 3.1. Assume that ρ0, u0, θ0 ∈ Hs 1 ∩ L∞ Rd with ρ0, θ0 ≥ C1 > 0 and s > d/2 1. Then, there is a finite time T ∈ 0, ∞ , depending on the Hs and L∞ norms of the initial data, such tCh1at t0h,eTC;aHucshy. problem 3.3 has a unique bounded smooth solution ρ, u, θ ∈ C 0, T ; Hs 1 ∩ 3.2.3. Existence and Uniqueness of Solution ρj , uj , θj for j ≥ 1 Now, let us briefly describe the solvability of ρj , uj , θj for any j ≥ 1 from the problem 3.3 and 3.8 provided that we have known ρk, uk, θk k≤j−1 already. Thus, ρj , uj , θj satisfy the following linear system: where ∂tρj 0 j div ρ u ρj u0 div ρkuj−k , − j−1 θ0∇ ln ρ0ρj ∇θj Gj1−1, ∂tθj R θ0 div uj θj div u0 u0 · ∇ θj uj · ∇ θ0 Gj2−1, Gj−1 2 −g2j−1 − R θk div uj−k − uk · ∇ θj−k. j−1 It is not difficult to see that the system 3.9 can be rewritten as a symmetrizable hyperbolic system. Thus, by the standard existence theory of local smooth solutions of symmetrizable hyperbolic equations see 15 , we have Proposition 3.2. Let T0 ∈ 0, T , and assume that ρj , uj , θj ∈ Hs ∩ L∞, s > d/2 1. Then, there e∩xi1is0tCs ia 0ti,mTe0 i,nHtesr−vialRd0, T.0 , such that 3.9 or 3.8 has a unique smooth solution ρj , uj , θj ∈ Remark 3.3. In particular, if the initial data is C∞, the solution of 3.9 or 3.8 belongs to C∞ 0, T0 × Rd . 4. Convergence to Compressible Euler Equations In this section, we are devoted to prove the convergence of system 2.3 to compressible Euler equations. ρa,m x, t ua,m x, t θa,m x, t 4.1. Derivation of Error Equations For any fixed integers m ≥ 1 and s0 > d/2 2, set 4.1 4.2 4.3 4.4 with ρj , uj , θj being given by Proposition 3.2. From the asymptotic analysis of Section 3.1, we know that ρa,m, ua,m, θa,m satisfy the following problem: where and the remainders Rρ, Ru, and Rθ satisfy f1a,m, f2a,m f1, f2 ρa,m, θa,m , sup 0≤t≤T0 Rρ, Ru, Rθ Hs0 < M m, for some constant M > 0 independent of . Now, we let ρ , u , θ be the smooth solution to the system 2.3 and denote N , U , Θ ρ − ρa,m, u − ua,m, θ − θa,m . Obviously, N , U , Θ satisfy the following problem: ∂tN div N U u a,m ρa,mU −Rρ, R Θ θa,m N ρa,m ∂tU U ua,m · ∇ U ∇N R f1 ∇Θ U · ∇ ua,m Rρa,mΘ − RN θa,m ρa,m N ρa,m ∇ρa,m f1 − f1a,m ∇θa,m −Ru, ∂tΘ R Θ θa,m f div U 2 U ua,m · ∇Θ RΘ f2 − f2a,m div u a,m U · ∇ θa,m −Rθ, N , U , Θ |t 0 N0, U0, Θ0 , where Thus, the problem 4.6 for the unknown V can be rewritten as 3 R , It is easy to see that the equations of V in 4.6 are symmetrizable and hyperbolic if we introduce A0 V ⎛ Θ θa,m θa,m f2 ⎞ ⎟⎟⎟ ⎟⎟ , ⎟⎠⎟⎟⎟ which is positively definite. When N ρa,m, Θ A0 V Aj V c are symmetric for all 1 ≤ j ≤ d. 4.2. Proof of Convergence Obviously, the existence and uniqueness of smooth solutions of 2.3 are equivalent to that of 4.6 or 4.9 . Then, in order to rigorously justify the convergence of 2.3 to 1.3 , it suffices to obtain their uniform estimates with respect to the light speed c. This will be done by using iteration techniques for the symmetrizable hyperbolic problem. More amply, we solve the nonlinear problem 4.9 by the following iteration for linear problems cf. 15 : d j 1 ∂tV ,k 1 Aj V ,k ∂xj V ,k 1 H1 V ,k −H2 V ,k c−3 R , θa,m ≥ C > 0 for 1, then Aj V with V ,k 1|t 0 V0 , V ,0 x, t V0 x . To study the problem 4.9 and 4.11 , we introduce the Sobolev’s norms: ⎛ V t l ⎝ |α|≤l ∂xV t 2L2 Rd ⎠ α ⎞ 1/2 , V t l,T sup V t l, l ∈ N∗. 0≤t≤T The key point for proving the convergence as → 0 is the following a priori estimate. 4.9 4.10 Lemma 4.1. Let s and l be two integers such that d/2 1 < l ≤ s. Assume that for some constant M1 > 0 independent of . Then, there exist constants M2 > 0, M3 > 0, 0 > 0, and T1 ∈ 0, T0 , such that for all ≤ 0 the solutions V ,k of 4.11 satisfy V0 l ≤ M1 m, V ,k l,T1 ≤ M2 m, ∀k ∈ N, ∂tV ,k l−1,T1 ≤ M3 m, ∀k ∈ N. Proof. Let α ∈ Nd with |α| ≤ l. We define Vα ,k by Vα ,k that α ∈ Nd satisfies the following problem: ∂αxV ,k; thus, it is not difficult to know where Rα,k is defined by A0 V ,k ∂tVα ,k 1 A0 V ,k Aj V ,k ∂xj Vα ,k 1 Rα,k, d α ∂xV0 , Rα ,k A0 V ,k ∂αx −H1 V ,k − H2 V ,k Rc d A0 V ,k Aj V ,k Vα ,k 1 − ∂αx Aj V ,k V ,k 1 . which implies, together with 4.11 1, that V ,k 1 In what follows we let Mi i ≥ 4 be various positive constants independent of , k ∈ N, M2, and M3. Equation 4.15 implies that the matrix A0 V ,k is positively definite, uniformly with respect to , k ∈ N, M2, and for all ≤ 0 with some 0 > 0. Because l > d/2 1, we obtain satisfying V ,k l,T1 ≤ 1, ∂tV ,k l−1,T1 ≤ 1, div Aj V ,k ∂tA0 V ,k ∂xj Aj V ,k , d div Aj V ,k L∞ Rd× 0,T1 ≤ M4. Employing the classical energy estimate of symmetric hyperbolic equations to the problem 4.17 1, we can obtain sup Vα ,k 1 L2 Rd ≤ M5eM5T1 0≤t≤T1 α ∂xV0 L2 Rd T1 0 Rα,k τ L2 Rd dτ . By the definition of Rα,k in 4.18 , the classical Moser-type inequality 1.5 , 1.7 , and Sobolev’s embedding lemma with l > d/2 1, we deduce that T1 0 Rα,k τ L2 Rd dτ ≤ C M2 T1 m V ,k 1 l,T1 . Here the constant C M2 > 0 may depend on M2. Now, substituting 4.25 into 4.24 and using 4.14 , one gets Then Vα ,k 1 L2 Rd ≤ M5eM5T1 M1 C M2 T1 m M5eM5T1 C M2 T1 Vα ,k 1 L2 Rd . Now, we choose T1 > 0 such that eM5T1 ≤ 2, C M2 T1 ≤ 1, 1 M5eM5T1 C M2 T1 ≤ 2 V c,k 1 1 . The proof of Lemma 4.1 is complete. Returning to the problem 2.3 and 4.6 , we conclude the following. 4.21 4.22 Theorem 4.2. For any fixed integers s > d/2 1, suppose that Then the solution of 2.3 satisfies m j ρj , uj , θj s s,T1 ≤ M m, Acknowledgments The authors cordially acknowledge partial support from the National Science Foundation of China Grant no. 10771099,10901011 , the Beijing Science Foundation Grant no. 1082001 , Beijing Municipal Commission of Education, Funding Project for Academic Human Resources Development in Institutions of High Learning Under the Jurisdiction of Beijing Municipality PHR200906103 , and Research Initiation Project for High-level Talents no. 201025 of North China University of Water Resources and Electric Power. 1 D. Mihalas and B. W. Mihalas , Foundations of Radiation Hydrodynamics, Oxford University Press, New York, NY, USA, 1984 . 2 G. C. Pomraning , The Equations of Radiation Hydrodynamics, Pergamon Press, Oxford, UK, 1973 . 3 J. I. Castor , Radiation Hydrodynamics, Cambridge University Press, Cambridge, Mass, USA, 2004 . 4 A. M. Anile , A. M. Blokhin , and Yu . L. 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Jianwei Yang, Shu Wang, Yong Li. Local Smooth Solution and Non-Relativistic Limit of Radiation Hydrodynamics Equations, Boundary Value Problems, 2010, 716451,