Stability of Non-constant Equilibrium Solutions for Bipolar Full Compressible Navier–Stokes–Maxwell Systems

We study the stability of smooth solutions near non-constant equilibrium states for a bipolar full compressible Navier–Stokes–Maxwell system in a three-dimensional torus $$\mathbb {T}= (\mathbb {R}/\mathbb {Z})^3$$T=(R/Z)3. This system is quasilinear hyperbolic-parabolic. In the first part, by using the maximum principle, we find a non-constant steady state solution with small amplitude for this system. In the second part, with the help of suitable choices of symmetrizers and classic energy estimates, we prove that global smooth solutions exist and converge to the non-constant steady states as the time goes to infinity. As a byproduct, we obtain the global stability for the bipolar full compressible Navier–Stokes–Poisson system.

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