The Cauchy problem for the magneto-hydrodynamic system

Abstract

We study the uniqueness and regularity of Leray-Hopf's weak solutions for the MHD equations with dissipation and resistance in different frameworks. Using different kinds of spacetime estimates in conjunction with the Littlewood-Paley-Bony decomposition, we present some general criteria of uniqueness and regularity of weak solutions to the MHD system, and prove the uniqueness and regularity criterion in the framework of mixed space-time Besov spaces by applying Tao's trichotomy method.

Regularity criteria for the 3D magneto-micropolar fluid equations in the Morrey–Campanato space

“…If, further, the vortex viscosity χ = 0, the velocity u does not depend on the micro-rotation field ω, and the first equation reduces to the classical Navier-Stokes equation which has been greatly analyzed, see, for example, the classical books by Ladyzhenskaya [13], Lions [15] or Lemarié-Rieusset [14]. If we ignore the micro-rotation of particles, it reduces to the viscous incompressible magnetohydrodynamic equations, which has also been studied extensively [1][2][3]9,20]. It is worthy to note that He and Xin [9], Zhou [25,26] proved the regularity criteria of weak solutions to the magneto-hydrodynamic equations, which only need the velocity u or its gradient ∇u or the vorticity ∇ × u or p and b or ∇p and b to satisfy some conditions.…”

Existence and regularizing rate estimates of solutions to a generalized magneto-hydrodynamic system in pseudomeasure spaces

“…When γ = 1, the system (1.1)-(1.4) becomes the usual MHD system, which has drawn much attention during the past twenty more years (cf. [2,3,13,16] and the references cited therein). In particular, when γ = 1 and b ≡ 0, the system (1.1)-(1.4) becomes the well-known Navier-Stokes equations (see [1,5,8,10,11]).…”

“…Thus, by taking a = n + 1 − 2γ , we are able to get existence of some self-similar solutions if we have proved certain global existence for initial data in P M a . Self-similar solutions are an important class of solutions possessing a special structure that have significant applications in physics, so that they have drawn great interest of many authors (cf., [1,2,7] and the references cited therein).…”

Well-posedness of the upper convected Maxwell fluid in the limit of infinite Weissenberg number

“…The kinematic viscosity and the magnetic diffusivity are taken into account. In this case, for the Cauchy problem (17), Cannone et al study the uniqueness and regularity of Leray-Hopf's weak solutions in [5], whereas He and Xin [6] give two classes of sufficient conditions which guarantee that the weak solutions are regular.…”

“…Here 4 (M, T ), 5 (M, T ), 6 (M, T ) are controllable and 4 , 6 → 0 as T → 0. In Steps 6 and 7, we know l = (v n+1 , q n+1 ) =l+(∇ n+1 , 0).…”