This paper studies the Cauchy problem for the nonlinear fractional power dissipativebeing a homogeneous pseudo-differential operator of order d ∈ [0, 2α) and f (u) is a function of u which behaves like |u| b u with b > 0. AMS Subject Classification 2000: 35K05, 35K15.
This paper examines the global (in time) regularity of classical solutions to the 2D incompressible magnetohydrodynamics (MHD) equations with only magnetic diffusion. Here the magnetic diffusion is given by the fractional Laplacian operator (−∆) β . We establish the global regularity for the case when β > 1. This result significantly improves previous work which requires β > 3 2 and brings us closer to the resolution of the well-known global regularity problem on the 2D MHD equations with standard Laplacian magnetic diffusion, namely the case when β = 1.Theorem 1.1. Consider (1.1) with β > 1. Assume that (u 0 , b 0 ) ∈ H s (R 2 ) with s > 2, ∇ · u 0 = 0, ∇ · b 0 = 0 and j 0 = ∇ × b 0 satisfying ∇j 0 L ∞ < ∞.
Abstract. In this paper we study the ideal incompressible magneto-hydrodynamics system, and prove the local existence and uniqueness of solutions in critical Besov spaces B 1+n/p p,1 for 1 ≤ p ≤ ∞.
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