UNIFORM REGULARITY OF FULLY COMPRESSIBLE HALL-MHD SYSTEMS

. In this article we study a fully compressible Hall-MHD system. These equations include shear viscosity, bulk viscosity of the ﬂow, and heat conductivity and resistivity coeﬃcients. We prove uniform regularity estimates.

Applications of the Hall-MHD system cover a very wide range of physical objects, for example, magnetic reconnection in space plasmas, star formation, neutron stars, and geo-dynamo. For well-posedness, regularity and decay properties, and related incompressible models, we refer to works [5,6,7,10,11,20,21,22,23] and references therein.
Before stating our main results, we recall the existence of local smooth solutions to (1.1)-(1.6). Since the system (1.1)-(1.6) is parabolic-hyperbolic, we have the following result.
The aim of this article is to prove uniform regularity estimates in (λ, µ, κ, η), as stated in the following theorem.
holds for some positive constants C and T 0 (≤ T ) independent of λ, µ, η and k. Remark 1.3. By the uniform estimates, one can easily take the limits of λ, µ, η and k to zero, hence we omit the details here. Our estimates are uniform in a with a := (λ, µ, η, k) while the ones in existence results of Hall-MHD depend on a.
We define (1.8) Here we note that v := u − w.
From (1.9) it follows that [1,3,16] M (t) ≤ C. (1.10) In the following proofs, we will use the bilinear commutator and product estimates due to Kato-Ponce [15], with s > 0 and 1 . We only need to show Theorem 1.4, which is given in the next section. Our proof consists of two steps. In step 1, we give the lower order estimates and in step 2, we show the higher order estimates.