In , after the studies of Lowrie, Morel and Hittinger , and Buet and Després , we considered a singular limit for a compressible inviscid radiative flow, where the motion of the fluid is given by the Euler system for the evolution of the density , the velocity field and the absolute temperature , and where radiation is described in the limit by an extra temperature . All of these quantities are functions of the time and the Eulerian spatial coordinate .
In  we proved that the associated Cauchy problem admits a unique global smooth solution, provided that the data are small enough perturbations of a constant state.
In  we coupled the previous model to the electromagnetic field through the so-called magnetohydrodynamic (MHD) approximation, in presence of thermal and radiative dissipation. Hereafter, we consider the perfect non-isentropic Euler–Maxwell system and we also consider a radiative coupling through a pure convective transport equation for the radiation (without dissipation). Then we deal with a pure hyperbolic system with partial relaxation (damping on velocity).
More specifically the system of equations to be studied for the unknowns reads
where is the density, the velocity, the temperature of matter, is the total mechanical energy, is the radiative energy related to the temperature of radiation by , and is the radiative pressure given by , with . Finally, is the electric field and is the magnetic induction.
We assume that the pressure and the internal energy are positive smooth functions of their arguments, with
and we also suppose for simplicity that (where is a momentum-relaxation time) and and are positive constants.
A simplification appears if one observes that, provided that equations (1.7) and (1.8) are satisfied at , they are satisfied for any time , and consequently they can be discarded from the analysis below.
Notice that the reduced system (1.1)–(1.4) is the non-equilibrium regime of radiation hydrodynamics, introduced by Lowrie, Morel and Hittinger  and, more recently, by Buet and Després , and studied mathematically by Blanc, Ducomet and Nečasová . Extending this last analysis, our goal in this work is to prove global existence of solutions for system (1.1)–(1.8) when the data are sufficiently close to an equilibrium state, and study their large time behavior.
For the sake of completeness, we mention that related non-isentropic Euler–Maxwell systems have been the subject of a number of studies in the recent past. Let us quote the recent works [9, 10, 12, 14, 18, 21].
In the following, we show that the ideas used by Ueda, Wang and Kawashima in [20, 19] in the isentropic case can be extended to the (radiative) non-isentropic system (1.1)–(1.6). To this end, we follow the following plan. In Section 2 we present the main results and then, in Section 3, we prove the well-posedness of system (1.1)–(1.6). Finally, in Section 4, we prove the large time asymptotics of the solution.
2 Main results
Let be a constant state, with , and , and compatibility condition , and suppose that . There exists such that for any initial state satisfying
In addition, this solution satisfies the following energy inequality:
for some constant which does not depend on .
The large time behavior of the solution is described as follows.
Moreover, if , then
Note that, due to lack of dissipation by viscous, thermal and radiative fluxes, the Kawashima–Shizuta stability criterion (see  and ) is not satisfied for the system under study, and the techniques of  relying on the existence of a compensating matrix do not apply. However, we will check that radiative sources play the role of relaxation terms for the temperature and radiative energy and this will lead to global existence for the system.
3 Global existence
3.1 A priori estimates
Introducing the entropy of the fluid by the Gibbs law and denoting by the radiative entropy, equation (1.4) is rewritten as
The internal energy equation is
and by dividing it by , we get the entropy equation for matter
By introducing the Helmholtz functions
we check that the quantities and are non-negative and strictly coercive functions reaching zero minima at the equilibrium state .
Let and be given positive constants. Let and be the sets defined by
Then there exist positive constants and such that
for all , and
for all .
The first assertion is proved in , and we only sketch the proof for convenience. According to the decomposition
one checks that is strictly convex and reaches a zero minimum at , while is strictly decreasing for and strictly increasing for , according to the standard thermodynamic stability properties, see . Computing the derivatives of leads directly to estimate (3.5).
The second assertion follows from the properties of
Using the previous entropy properties, we have the following energy estimate.
Let the assumptions of Theorem 2.1 be satisfied with
By defining, for any , the auxiliary quantities
we can bound the spatial derivatives as follows.
Assume that the hypotheses of Theorem 2.1 are satisfied. Then, for , we have
and applying to this system, we get
Then, by taking the scalar product of each of the previous equations, respectively, by
and adding the resulting equations, we get
By integrating (3.8) on space, one gets
Integrating now with respect to and summing on , with , yields
By observing that
and using the commutator estimates (see the Moser-type calculus inequalities in )
we see that
Then integrating with respect to time gives
for any . Similarly, we estimate
Then we get
Then integrating with respect to time yields
for any . ∎
The above results, together with (3.6), allow us to derive the following energy bound.
Assume that the assumptions of Proposition 3.2 are satisfied. Then
Under the assumptions of Theorem 2.1, and supposing that , we have the following estimate for any :
By rearranging the left-hand side of (3.17), we get
Integrating (3.18) over space and using Young’s inequality yields
Integrating (3.19) over space and time yields
By observing that
and summing (3.20) on for , we get
where we used Corollary 3.4.
Let us estimate the last integral in (3.20). We have
for . Then
Plugging bounds (3.21) into the last inequality gives
which completes the proof of Lemma 3.5. ∎
Finally, we check from [20, Lemma 4.4] that the following result for the Maxwell’s system holds true for our system with a similar proof.
Under the assumptions of Theorem 2.1, and supposing that , for any , the following estimate (here, we set ) holds:
Integrating in space gives
By integrating on time and summing for , we have
where we used the bound
3.2 Proof of Theorem 2.1
Putting this last estimate into (3.22) yields
Now, by observing that, provided , one has , and, provided , one has for some positive constant , we see that
In order to prove global existence, we argue by contradiction, and assume that is the maximum time existence. Then we necessarily have
where is defined by
Thus, we are left to prove that is bounded. For this purpose, we use the argument used in . After the previous calculation, we have
Hence, setting , we have
By studying the variation of , we see that , and that is increasing on the interval and decreasing on the interval . Hence,
Hence, we can choose small enough to have for all , where , and we see that , which contradicts (3.25).
4 Large time behavior
We have the following analogue of Proposition 3.2 for time derivatives.
By using system (3.7), we see that
S. Benzoni-Gavage and D. Serre, Multidimensional Hyperbolic Partial Differential Equations. First-Order Systems and Applications, Oxford Math. Monogr., Oxford University Press, Oxford, 2007. Google Scholar
X. Blanc, B. Ducomet and Š. Nečasová, Global existence of a diffusion limit with damping for the compressible radiative Euler system coupled to an electromagnetic field, Topol. Methods Nonlinear Anal., to appear. Google Scholar
A. R. Choudhuri, The Physics of Fluids and Plasmas: An Introduction for Astrophysicists, Cambridge University Press, Cambridge, 1998. Google Scholar
E. Feireisl and A. Novotný, Singular Limits in Thermodynamics of Viscous Fluids, Adv. Math. Fluid Mech., Birkhäuser, Basel, 2009. Google Scholar
Y.-H. Feng, S. Wang and S. Kawashima, Global existence and asymptotic decay of solutions to the non-isentropic Euler–Maxwell system, Math. Models Methods Appl. Sci. 24 (2014), no. 14, 2851–2884. Web of ScienceCrossrefGoogle Scholar
Y.-H. Feng, S. Wang and X. Li, Stability of non-constant steady-state solutions for non-isentropic Euler–Maxwell system with a temperature damping term, Math. Methods Appl. Sci. 39 (2016), no. 10, 2514–2528. Web of ScienceCrossrefGoogle Scholar
B. Hanouzet and R. Natalini, Global existence of smooth solutions for partially dissipative hyperbolic systems with a convex entropy, Arch. Ration. Mech. Anal. 169 (2003), no. 2, 89–117. CrossrefGoogle Scholar
J. W. Jerome, The Cauchy problem for compressible hydrodynamic-Maxwell systems: A local theory for smooth solutions, Differential Integral Equations 16 (2003), no. 11, 1345–1368. Google Scholar
S. Kawashima, Systems of a hyperbolic-parabolic composite type, with applications to the equations of magnetohydrodynamics, Doctoral Thesis, Kyoto University, 1984. Google Scholar
A. Majda, Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables, Appl. Math. Sci. 53, Springer, New York, 1984. Google Scholar
Y. Shizuta and S. Kawashima, Systems of equations of hyperbolic-parabolic type with applications to the discrete Boltzmann equation, Hokkaido Math. J. 14 (1985), no. 2, 249–275. CrossrefGoogle Scholar
Z. Tan and Y. Wang, Large-time behavior of solutions to the compressible non-isentropic Euler–Maxwell system in , Nonlinear Anal. Real World Appl. 15 (2014), 187–204. Google Scholar
Y. Ueda and S. Kawashima, Decay property of regularity-loss type for the Euler–Maxwell system, Methods Appl. Anal. 18 (2011), no. 3, 245–267. Google Scholar
Y. Ueda, S. Wang and S. Kawashima, Dissipative structure of the regularity-loss type and time asymptotic decay of solutions for the Euler–Maxwell system, SIAM J. Math. Anal. 44 (2012), no. 3, 2002–2017. Web of ScienceCrossrefGoogle Scholar
About the article
Published Online: 2018-03-21
Funding Source: Agence Nationale de la Recherche
Award identifier / Grant number: ANR-15-CE40-0011
Funding Source: Grantová Agentura České Republiky
Award identifier / Grant number: 201-16-03230S
Šárka Nečasová acknowledges the support of the GAČR (Czech Science Foundation) project 16-03230S in the framework of RVO: 67985840. Bernard Ducomet is partially supported by the ANR project INFAMIE (ANR-15-CE40-0011).