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Register a new userThe estimate of lifespan and local well-posedness for the non-resistive MHD equations in homogeneous Besov spaces
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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}.
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In this paper, we first establish the local well-posedness (existence, uniqueness and continuous dependence) for the Fornberg-Whitham equation in both supercritical Besov spaces $B^s_{p,r}, s>1+frac{1}{p}, 1leq p,rleq+infty$ and critical Besov spaces $B^{1+frac{1}{p}}_{p,1}, 1leq p<+infty$, which improves the previous work cite{y2,ho,ht}. Then, we prove the solution is not uniformly continuous dependence on the initial data in supercritical Besov spaces $B^s_{p,r}, s>1+frac{1}{p}, 1leq pleq+infty, 1leq r<+infty$ and critical Besov spaces $B^{1+frac{1}{p}}_{p,1}, 1leq p<+infty$. At last, we show that the solution is ill-posed in $B^{sigma}_{p,infty}$ with $sigma>3+frac{1}{p}, 1leq pleq+infty$.
In this paper, we consider the three-dimensional full compressible viscous non-resistive MHD system. Global well-posedness is proved for an initial-boundary value problem around a strong background magnetic field. It is also shown that the unique solution converges to the steady state at an almost exponential rate as time tends to infinity. The proof is based on the celebrated two-tier energy method, due to Guo and Tice [emph{Arch. Ration. Mech. Anal.}, 207 (2013), pp. 459--531; emph{Anal. PDE.}, 6 (2013), pp. 287--369], reformulated in Lagrangian coordinates. The obtained result may be viewed as an extension of Tan and Wang [emph{SIAM J. Math. Anal.}, 50 (2018), pp. 1432--1470] to the context of heat-conductive fluids. This in particular verifies the stabilization effects of vertical magnetic field in the full compressible non-resistive fluids.
In the paper, by constructing a initial data $u_{0}in B^{sigma}_{p,infty}$ with $sigma-2>max{1+frac 1 p, frac 3 2}$, we prove that the corresponding solution to the higher dimensional Camassa-Holm equations starting from $u_{0}$ is discontinuous at $t=0$ in the norm of $B^{sigma}_{p,infty}$, which implies that the ill-posedness for the higher dimensional Camassa-Holm equations in $B^{sigma}_{p,infty}$.
We prove that the Cauchy problem for the dispersion generalized Benjamin-Ono equation [partial_t u+|partial_x|^{1+alpha}partial_x u+uu_x=0, u(x,0)=u_0(x),] is locally well-posed in the Sobolev spaces $H^s$ for $s>1-alpha$ if $0leq alpha leq 1$. The new ingredient is that we develop the methods of Ionescu, Kenig and Tataru cite{IKT} to approach the problem in a less perturbative way, in spite of the ill-posedness results of Molinet, Saut and Tzvetkovin cite{MST}. Moreover, as a bi-product we prove that if $0<alpha leq 1$ the corresponding modified equation (with the nonlinearity $pm uuu_x$) is locally well-posed in $H^s$ for $sgeq 1/2-alpha/4$.
In this paper, we are concerned with the motion of electrically conducting fluid governed by the two-dimensional non-isentropic viscous compressible MHD system on the half plane, with no-slip condition for velocity field, perfect conducting condition for magnetic field and Dirichlet boundary condition for temperature on the boundary. When the viscosity, heat conductivity and magnetic diffusivity coefficients tend to zero in the same rate, there is a boundary layer that is described by a Prandtl-type system. By applying a coordinate transformation in terms of stream function as motivated by the recent work cite{liu2016mhdboundarylayer} on the incompressible MHD system, under the non-degeneracy condition on the tangential magnetic field, we obtain the local-in-time well-posedness of the boundary layer system in weighted Sobolev spaces.
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