Local existence for the non-resistive MHD equations in Besov spaces

By Jean-Yves Chemin, D. McCormick, James C. Robinson, José L. Rodrigo. 2015

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In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of $\mathbb{R}^{n}$, $n=2,3$, for divergence-free initial data in certain Besov spaces, namely $\boldsymbol{u}_{0} \in B^{n/2-1}_{2,1}$ and $\boldsymbol{B}_{0} \in B^{n/2}_{2,1}$. The a priori estimates include the term $\int_{0}^{t} \| \boldsymbol{u}(s) \|_{H^{n/2}}^{2} \, \mathrm{d} s$ on the right-hand side, which thus requires an auxiliary bound in $H^{n/2-1}$. In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate in $H^{1/2}$ is required, which we prove using the splitting method of Calder\'on (Trans. Amer. Math. Soc. 318(1), 179--200, 1990). By contrast, we prove that such solutions are unique in 3D, but the proof of uniqueness in 2D is more difficult and remains open.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/c22cfb9a131c2884e0646c3ab31acf27af3b645d

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Citations in the corpus

Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces (2016, unknown): “To this end, it was shown in Chemin et al. (2016) that one can prove local existence when B0 ∈ Bd/22,1 (Rd) and u0 ∈ B d/2−1 2,1 (R d).”
Local existence and uniqueness for the non-resistive MHD equations in homogeneous Besov spaces (2016, background): “in [2] proved the local existence of solutions to (1.”