LOCAL WELL-POSEDNESS OF THE FULL COMPRESSIBLE NAVIER-STOKES-MAXWELL SYSTEM WITH VACUUM

. In this paper, we prove the local well-posedness of strong solutions for a compressible Navier-Stokes-Maxwell system, provided the initial data satisfy a natural compatibility condition. We do not assume the positivity of initial density, it may vanish in an open subset (vacuum) of Ω.


1.
Introduction. The Navier-Stokes-Maxwell system is a coupling of the compressible Navier-Stokes equations with the Maxwell equations through the Lorentz force in three dimensional space, which is a plasma physical model describing the motion of charged particles (ions and electrons) in electromagnetic field [5], [19], [25]. The derivation of the Navier-Stokes-Maxwell system from the Vlasov-Maxwell-Boltzmann system is given in the Appendix in [5].
We also mention some works where the model seems a little bit different though it is also called compressible Navier-Stokes-Maxwell system. For the one-fluid nonisentropic compressible Navier-Stokes-Maxwell system in R 3 , the global existence of solutions near constant steady states is established and the time-decay rates of perturbed solutions are obtained in [23] with different right terms from those in this paper. The global existence of regular solutions to the 2D Navier-Stokes-Maxwell system is proved by using energy estimates and Brezis-Gallouet inequality, and a blow-up criterion for solutions to 3D Navier-Stokes-Maxwell system is obtained in [16]. For the isentropic compressible Navier-Stokes-Maxwell system in one space dimension, existence and uniqueness of global strong solutions with large initial data and vacuum are established [12] for the initial boundary value problem when there is initial vacuum. The global existence and large time behavior of this model have been studied by Duan [5]. The global existence of spherical symmetric classical solution to the Navier-Stokes-Maxwell system is obtained with large initial data and vacuum [10]. For the bipolar compressible Navier-Stokes-Maxwell system in R 3 , under the assumption that the initial values are close to a equilibrium solutions, asymptotic behavior of global smooth solutions to the Cauchy problem is proved in [9] without decay rate. Moreover, the essential difference between the one-fluid Navier-Stokes-Maxwell system and the bipolar compressible Navier-Stokes-Maxwell system is shown via the phenomenon on the charge transport. The decay rate of the global smooth solutions is obtained [26] based on a detailed analysis to the Green's function of the linearized system and some elaborate energy estimates. Hou and Zhu [11] show a regularity criterion for a compressible Navier-Stokes-Maxwell system.
Enlightened by the known results on well-posedness, we investigate the initial boundary value problem to the compressible non-isentropic Navier-Stokes-Maxwell system with vacuum in three space dimensions, where the right terms are different from those in [23]. The main difficulty in this paper is how to prove the estimate (8), which will be proved in section 2. Under the assumption that the natural compatibility conditions hold, we prove the local well-posedness of the non-isentropic compressible Navier-Stokes-Maxwell system in three space dimensions with vacuum by the compactness principle.
The natural compatibility conditions are given below: there exists (g 1 , g 2 ) ∈ L 2 such that Before stating our main results, we first give a proposition below: Then there exists a positive time T δ > 0 such that the problem (1)-(2) has a unique solution (ρ, u, θ, E, b), satisfying that ρ ≥ δ, θ ≥ 0 in Ω × (0, T δ ), and for k = 0, 1, 2, Remark 1. The local existence can be proved in a similar way as in [28] if ρ ≥ 1 C . Therefore, we omit the details of the proof. (4) can be replaced by Now we state our main results.
for some T > 0.
Define that M (t) as below: Theorem 1.2. Let T δ be the maximal time of existence for the problem (1)-(2) in the sense of Proposition 1. Then for any t ∈ [0, T δ ), it holds that for some given nondecreasing continuous functions C 0 (·) and C(·).
Then T δ > T 1 for 0 < δ ≤ 1. Otherwise, by using the above uniform estimates and applying Proposition 1 repeatedly, one can extend the time interval of existence to [0, T 1 ], which contradicts to the maximality of T δ . Therefore, M (t) ≤ D for any t ∈ [0, T 1 ] where T 1 is independent of 0 < δ ≤ 1. Clearly, the conclusion is also true for T δ = ∞ by applying the same argument.
Therefore, by taking δ → 0 and the standard compactness principle, the proof of existence is complete.
The proof of uniqueness can be finished by the very similar calculations as that in [6], [3], we omit the details here.
The proof of Theorem 1.1 is completed. Now, we turn to prove Theorem 1.2. In other words, we prove the inequality (8).
Proof of Theorem 1.2. For simplicity, we drop the superscript "δ" in ρ δ , u δ , θ δ , E δ and b δ . The physical constants C V and R do not cause any essential difficulties in our arguments. Therefore, we take C V = R = 1. First, testing the first equation in (1) by ρ q−1 , we see that and thus On the other hand, using the Gagliardo-Nirenberg inequality Taking ∇ to the first equation in (1), testing by |∇ρ| 4 ∇ρ, we find that which gives It is easy to see that Similarly, we find that Applying ∂ t to the second equation in (1), testing by u t and using the first equation in (1), we discover that We bound 1 , · · · , 5 as follows.
Inserting the above estimates into (13) and integrating it over (0, t), we have By the H 2 -theory of elliptic system, using the first equation and second one in (1), (9), (10), (11) and (12), (14), we derive Here we used the Gagliardo-Nirenberg inequality It follows from the first equation in (1), (9), (10) and (15) that Using the fourth equation and the seventh one in (1), we observe that Using the fifth equation and the seventh one in (1), and (16), we infer that Taking rot 2 to the fourth equation and the fifth equation in (1), testing the results by rot 2 E and rot 2 b, respectively, summing up the results and using (17), we get Integrating the above estimate over (0, t), we have Taking ∇div to the fourth equation in (1) and testing by ∇div , we have Taking ∂ t rot to the fourth equation and the fifth equation in (1), testing by ∂ t rot E and ∂ t rot b, respectively, summing up the results and using (16), we obtain Applying ∂ t div to the fourth equation in (1) and testing by ∂ t div E, we have Now we recall the following Poincaré inequality [22]: for any w ∈ H 1 (Ω) with w · n = 0 or w × n = 0 on ∂Ω.
by taking small enough.