Abstract
This paper is concerned with an improved pressure regularity criterion of the three-dimensional (3D) magnetohydrodynamic (MHD) equations in the largest critical Besov spaces. Based on the Littlewood-Paley decomposition technique, the weak solutions are proved to be smooth if the pressure lies in the largest critical Besov spaces, $\pi(x,t) \in L^{p}(0,T;\dot{B}^{0}_{q,r}(\mathbf{R}^{3}))$ for $\frac{2}{p}+\frac{3}{q}=2$ , $1\leq r\leq\frac{2q}{3}$ , $\frac{3}{2}< q\leq\infty$ .
Highlights
Introduction and main resultsIn this paper, we consider the regularity criterion of the Cauchy problem to the threedimensional ( D) magnetohydrodynamic (MHD) equations in R × (, T),⎧ ⎪⎨∂tv + v · ∇v – b · ∇b + ∇π = γ v, ⎪⎩∂∇t b · +v v= ·∇, b b· ∇∇v = η · b =, b, ( . )associated with the initial condition v(x, ) = v, b(x, ) = b .Here v(x, t), b(x, t), π(x, t) are the unknown velocity field, magnetic field and pressure scalar field, respectively. γ ≥ is the kinematic viscosity, η ≥ is the magnetic diffusivity
Due to its importance in mathematics, there is a large literature on the well-posedness of the MHD equations [ ]
Many efforts have been made on the regularity criteria of weak solutions
Summary
We consider the regularity criterion of the Cauchy problem to the threedimensional ( D) magnetohydrodynamic (MHD) equations in R × ( , T),. Fan et al [ ] (see Zhang et al [ ], Chen and Zhang [ ], and Dong and Chen [ ]) proved the pressure regularity criterion of the Navier-Stokes equations in the homogeneous Besov space (see the definition ). Motivated by the previous regularity criteria results on the MHD and Navier-Stokes equations, the main purpose of this paper is to investigate the pressure regularity criteria of the D MHD equations in the largest critical Besov spaces without any additional assumption on the magnetic field b. In order to complete our proofs, we need the following results on the local smooth solutions and blow up criterion of the D MHD equations.
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