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Register a new userHigher order commutator estimates and local existence for the non-resistive MHD equations and related models
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This paper establishes the local-in-time existence and uniqueness of strong solutions in $H^{s}$ for $s > n/2$ to the viscous, non-resistive magnetohydrodynamics (MHD) equations in $mathbb{R}^{n}$, $n=2, 3$, as well as for a related model where the advection terms are removed from the velocity equation. The uniform bounds required for proving existence are established by means of a new estimate, which is a partial generalisation of the commutator estimate of Kato & Ponce (Comm. Pure Appl. Math. 41(7), 891-907, 1988).
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In this paper, we address the space-time decay properties for strong solutions to the incompressible viscous resistive Hall-MHD equations. We obtained the same space-time decay rates as those of the heat equation. Based on the temporal decay results in cite{cs}, we find that one can obtain weighted estimates of the magnetic field $B$ by direct weighted energy estimate, and then by regarding the magnetic convection term as a forcing term in the velocity equations, we can obtain the weighted estimates for the vorticity, which yields the corresponding estimates for the velocity field. The higher order derivative estimates will be obtained by using a parabolic interpolation inequality proved in cite{k01}. It should be emphasized that the the magnetic field has stronger decay properties than the velocity field in the sense that there is no restriction on the exponent of the weight. The same arguments also yield the sharp space-time decay rates for strong solutions to the usual MHD equations.
In this paper, we mainly investigate the Cauchy problem of the non-resistive MHD equation. We first establish the local existence in the homogeneous Besov space $dot{B}^{frac{d}{p}-1}_{p,1}times dot{B}^{frac{d}{p}}_{p,1}$ with $p<infty$, and give a lifespan $T$ of the solution which depends on the norm of the Littlewood-Paley decomposition of the initial data. Then, we prove that if the initial data $(u^n_0,b^n_0)rightarrow (u_0,b_0)$ in $dot{B}^{frac{d}{p}-1}_{p,1}times dot{B}^{frac{d}{p}}_{p,1}$, then the corresponding existence times $T_nrightarrow T$, which implies that they have a common lower bound of the lifespan. Finally, we prove that the data-to-solutions map depends continuously on the initial data when $pleq 2d$. Therefore the non-resistive MHD equation is local well-posedness in the homogeneous Besov space in the Hadamard sense. Our obtained result improves considerably the recent results in cite{Li1,chemin1,Feffer2}.
In this paper, we prove the a priori estimates in Sobolev spaces for the free-boundary compressible inviscid magnetohydrodynamics equations with magnetic diffusion under the Rayleigh-Taylor physical sign condition. Our energy estimates are uniform in the sound speed. As a result, we can prove the convergence of solutions of the free-boundary compressible resistive MHD equations to the solution of the free-boundary incompressible resistive MHD equations, i.e., the incompressible limit. The key observation is that the magnetic diffusion together with elliptic estimates directly controls the Lorentz force, magnetic field and pressure wave simultaneously.
In this paper, we study the energy equality for weak solutions to the non-resistive MHD equations with physical boundaries. Although the equations of magnetic field $b$ are of hyperbolic type, and the boundary effects are considered, we still prove the global energy equality provided that $u in L^{q}_{loc}left(0, T ; L^{p}(Omega)right) text { for any } frac{1}{q}+frac{1}{p} leq frac{1}{2}, text { with } p geq 4,text{ and } b in L^{r}_{loc}left(0, T ; L^{s}(Omega)right) text { for any } frac{1}{r}+frac{1}{s} leq frac{1}{2}, text { with } s geq 4 $. In particular, compared with the existed results, we do not require any boundary layer assumptions and additional conditions on the pressure $P$. Our result requires the regularity of boundary $partialOmega$ is only Lipschitz which is the minimum requirement to make the boundary condition $bcdot n$ sense. To approach our result, we first separate the mollification of weak solutions from the boundary effect by considering a non-standard local energy equality and transform the boundary effects into the estimates of the gradient of cut-off functions. Then, by establishing a sharp $L^2L^2$ estimate for pressure $P$, we use zero boundary conditions of $u$ to inhibit the boundary effect and obtain global energy equality by choosing suitable cut-off functions.
We call pattern any non-constant stable solution of a semilinear elliptic equation with Neumann boundary conditions. A classical theorem of Casten, Holland [19] and Matano [49] states that stable patterns do not exist in convex domains. In this article, we show that the assumptions of convexity of the domain and stability of the pattern in this theorem can be relaxed in several directions. In particular, we propose a general criterion for the non-existence of patterns, dealing with possibly non-convex domains and unstable patterns. Our results unfold the interplay between the geometry of the domain, the magnitude of the nonlinearity, and the stability of patterns. We propose several applications, for example, we prove that (under a geometric assumption) there exist no patterns if the domain is shrunk or if the nonlinearity has a small magnitude. We also refine the result of Casten Holland and Matano and show that it is robust under smooth perturbations of the domain and the nonlinearity. In addition, we establish several gradient estimates for the patterns of (1). We prove a general nonlinear Cacciopoli inequality (or an inverse Poincar{e} inequality), stating that the L2-norm of the gradient of a solution is controlled by the L2-norm of f(u), with a constant that only depends on the domain. This inequality holds for non-homogeneous equations. We also give several flatness estimates. Our approach relies on the introduction of what we call the Robin-curvature Laplacian. This operator is intrinsic to the domain and contains much information on how the geometry of the domain affects the shape of the solutions. Finally, we extend our results to unbounded domains. It allows us to improve the results of our previous paper [53] and to extend some results on De Giorgis conjecture to a larger class of domains.
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