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This is a Quantitative Study study.

October 13, 2025Open Access

Global well-posedness of 3-D density-dependent incompressible MHD equations with variable resistivity

Key Points

Global existence of weak solutions proved for 3-D incompressible MHD equations with variable viscosity.
Initial density bounded between positive constants ensures solutions exist in critical space for small initial conditions.
Assuming kinematic viscosity equals 1, uniqueness of solutions follows with additional conditions on initial velocity.
Results extend to scenarios where both variables adhere to specific critical norm constraints.

Abstract

In this paper, we investigate the global existence of weak solutions to 3-D inhomogeneous incompressible MHD equations with variable viscosity and resistivity, which is sufficiently close to 1 in L^ (R³), provided that the initial density is bounded from above and below by positive constants, and both the initial velocity and magnetic field are small enough in the critical space Ḣ^1{2} (R³). Furthermore, if we assume in addition that the kinematic viscosity equals 1, and both the initial velocity and magnetic field belong to Ḃ^1{2}₂, ₁ (R³), we can also prove the uniqueness of such solution.

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