Global weak solutions with higher regularity to the two-dimensional isentropic compressible Navier-Stokes and magnetohydrodynamic equations with far-field vacuum and unbounded density

We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes and magnetohydrodynamic (MHD) equations on the whole plane under a suitably small initial energy. The solutions constructed here admit far-field vacuum and unbounded densities. Moreover, they possess an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This particularly extends our previous half-plane case with Dirichlet boundary conditions (arXiv:2601.11852) to the whole-plane MHD coupling, and we also generalize the works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Suen and Hoff (Arch. Ration. Mech. Anal. 205 (2012), pp. 27-58) by allowing vacuum states and unbounded density. Our analysis lies in a new perspective that exploits the spatial integrability of the density and the resulting integrability of the pressure, together with the specific structure of the MHD system.

💡 Research Summary

The paper addresses the global existence of a class of weak solutions for the two‑dimensional isentropic compressible Navier–Stokes and magnetohydrodynamic (MHD) equations posed on the whole plane ℝ². The authors work under a small‑energy hypothesis and allow both far‑field vacuum (ρ→0 as |x|→∞) and unbounded density (ρ may become arbitrarily large in finite regions). Their solutions occupy an intermediate regularity regime that bridges the finite‑energy weak solutions of Lions–Feireisl (which tolerate vacuum but have very low regularity) and the “Hoff‑type” intermediate weak solutions (which enjoy higher regularity but usually require a uniform L∞ bound on the density).

Main assumptions.
The initial density ρ₀ is required to belong to L^θ(ℝ²) with
θ = 12γ(2α+1)/(α−1) > 60γ,
for some α∈(1,2). Moreover, a weighted integrability condition
∫_{ℝ²} (e+|x|²)^{α/2} ρ₀(x) dx < ∞
is imposed, which controls the mass distribution at infinity. The initial velocity u₀ and magnetic field B₀ belong to the homogeneous Sobolev space D^{1,2}(ℝ²) and satisfy div B₀=0. The total initial energy
C₀ = ∫