Abstract
This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in {mathbb{R}^d}, where d = 2, 3, with initial data {B_0in H^s(mathbb{R}^d)} and {u_0in H^{s-1+epsilon}(mathbb{R}^d)} for {s > d/2} and any {0 < epsilon < 1}. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking {epsilon=0} is explained by the failure of solutions of the heat equation with initial data {u_0in H^{s-1}} to satisfy {uin L^1(0,T; H^{s+1})}; we provide an explicit example of this phenomenon.
Highlights
In this paper we consider the equations of MHD with zero magnetic resistivity, ut − u + (u · ∇)u + ∇ p = (B · ∇)B, ∇ · u = 0, Bt + (u · ∇)B = (B · ∇)u, ∇ · B = 0,(1.1a) (1.1b) along with specified initial data u(0) = u0 and B(0) = B0
Jiu and Niu [7] established the local existence of solutions in 2D for initial data in H s for integer s 3, and more recently Ren et al [12] and Lin et al [10] have established the existence of global-in-time solutions for initial data sufficiently close to certain equilibrium solutions
In a previous paper [6] we proved a local existence result for these equations taking arbitrary initial data in u0, B0 ∈ H s(Rd ) with s > d/2 for d = 2, 3
Summary
In a previous paper [6] we proved a local existence result for these equations taking arbitrary initial data in u0, B0 ∈ H s(Rd ) with s > d/2 for d = 2, 3. To expect that such a local existence result should be possible with less regularity for u0 than for B0 To this end, it was shown in [5] that one can prove local existence when the initial data B0 ∈ B2d,/12(Rd ) and u0 ∈ B2d,/12−1(Rd ). By making use of maximal regularity results for the heat equation (which we recall ) we are able to prove the local existence of a solution that remains bounded in these spaces (see Theorem 3.1 for a precise statement). Su 2 + u 2, s > 0, which is equivalent to the standard H s norm when s is a positive integer
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