Abstract
We study the Cauchy problem for a system of equations corresponding to a singular limit of radiative hydrodynamics, namely, the 3D radiative compressible Euler system coupled to an electromagnetic field. Assuming smallness hypotheses for the data, we prove that the problem admits a unique global smooth solution and study its asymptotics.
1 Introduction
In [3], after the studies of Lowrie, Morel and Hittinger [15], and Buet and Després [5], 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
In [3] 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 [4] 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
where
We assume that the pressure
and we also suppose for simplicity that
A simplification appears if one observes that, provided that equations (1.7) and (1.8) are satisfied at
Notice that the reduced system (1.1)–(1.4) is the non-equilibrium regime of radiation hydrodynamics, introduced by Lowrie, Morel and Hittinger [15] and, more recently, by Buet and Després [5], and studied mathematically by Blanc, Ducomet and Nečasová [3]. 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
We are going to prove that system (1.1)–(1.8) has a global smooth solution close to any equilibrium state. Namely, we have the following theorem.
Theorem 2.1.
Let
and
there exists a unique global solution
In addition, this solution satisfies the following energy inequality:
for some constant
The large time behavior of the solution is described as follows.
Theorem 2.2.
Let
Moreover, if
Remark 2.3.
Note that, due to lack of dissipation by viscous, thermal and radiative fluxes, the Kawashima–Shizuta stability criterion (see [17] and [1]) is not satisfied for the system under study, and the techniques of [13] 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
Multiplying (1.2) by
Introducing the entropy
The internal energy equation is
and by dividing it by
So adding (3.3) and (3.2), we obtain
By subtracting (3.4) from (3.1) and using the conservation of mass, we get
By introducing the Helmholtz functions
we check that the quantities
Lemma 3.1.
Let
Then there exist positive constants
for all
for all
Proof.
The first assertion is proved in [8], and we only sketch the proof for convenience. According to the decomposition
where
one checks that
The second assertion follows from the properties of
Using the previous entropy properties, we have the following energy estimate.
Proposition 3.2.
Let the assumptions of Theorem 2.1 be satisfied with
Consider a solution
Proof.
We define
multiply (3.4) by
By defining, for any
and
we can bound the spatial derivatives as follows.
Proposition 3.3.
Assume that the hypotheses of Theorem 2.1 are satisfied.
Then, for
Proof.
By rewriting system (1.1)–(1.6) in the form
and applying
where
Then, by taking the scalar product of each of the previous equations, respectively, by
and adding the resulting equations, we get
where
By integrating (3.8) on space, one gets
Integrating now with respect to
By observing that
and using the commutator estimates (see the Moser-type calculus inequalities in [16])
we see that
Then integrating with respect to time gives
for any
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.
Corollary 3.4.
Assume that the assumptions of Proposition 3.2 are satisfied. Then
Our goal is now to derive bounds for the integrals in the right- and left-hand sides of equation (3.9). For this purpose we adapt the results of Ueda, Wang and Kawashima [20].
Lemma 3.5.
Under the assumptions of Theorem 2.1, and supposing that
Proof.
We linearize the principal part of system (1.1)–(1.3) as follows:
with coefficients
and sources
and
By multiplying (3.11) by
where
By rearranging the left-hand side of (3.17), we get
where
Integrating (3.18) over space and using Young’s inequality yields
In fact, in the same way one obtains estimates for the derivatives of
where
Integrating (3.19) over space and time yields
By observing that
and summing (3.20) on
where we used Corollary 3.4.
Let us estimate the last integral in (3.20). We have
for
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.
Lemma 3.6.
Under the assumptions of Theorem 2.1, and supposing that
Proof.
By applying
where
and
Integrating in space gives
By integrating on time and summing for
where we used the bound
obtained in the same way as in the proof of Lemma 3.5. The proof of Lemma 3.6 is completed. ∎
We are now in position to conclude with the proofs of Theorems 2.1 and 2.2.
3.2 Proof of Theorem 2.1
We first point out that local existence for the hyperbolic system (1.1)–(1.6) may be proved using standard fixed-point methods. We refer to [16] for the proof.
Now, by plugging (3.22) into (3.10) with
Putting this last estimate into (3.22) yields
Then, from (3.10), (3.23) and (3.24), we get
or, equivalently,
Now, by observing that, provided
In order to prove global existence, we argue by contradiction, and assume that
where
Thus, we are left to prove that
Hence, setting
By studying the variation of
Hence, we can choose
4 Large time behavior
We have the following analogue of Proposition 3.2 for time derivatives.
Corollary 4.1.
Let the assumptions of Theorem 2.1 be satisfied, and consider the solution
Proof.
By using system (3.7), we see that
and
Then, for
4.1 Proof of Theorem 2.2
By using Corollary 4.1, we get
This implies that
and
and then
Now, by applying the Gagliardo–Nirenberg inequality and (2.1), we get
So
Similarly,
and then
Finally,
Then, arguing as before,
So
which completes the proof.
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
Funding statement: 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).
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