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 $T_n\rightarrow T$, 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 $p\leq 2d$. 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 the paper, we consider the Cauchy problem of the non-resistive MHD equations in homogeneous Besov spaces. We prove the local existence and uniqueness of the solution to the non-resistive MHD equations by using the iterative scheme and compactness arguments. Our obtained result improves considerably the recent results in \cite{C,W}.
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 i…
This work is the continuation of the recent paper \cite{D2} devoted to the density-dependent incompressible Euler equations. Here we concentrate on the well-posedness issue in Besov spaces of type $B^s_{\infty,r}$ embedded in the set of Lipschitz continuous functions, a functional framework which contains the particular case of Hölder spaces and of the endpoint Besov space $B^1_{\infty,1}.$ For such data and under the nonvacuum assumption, we establish the local well-posedness and a continuation criterion in the spirit of that of Beale, Kato and Majda in \cite{BKM}. In the last part of the pap…
In the paper, we consider the initial value problem to the Camassa-Holm equation in the real-line case. Based on the local well-posedness result and the lifespan, we proved that the data-to-solution map of this problem is not uniformly continuous in nonhomogeneous Besov spaces in the sense of Hadamard. Our obtained result improves considerably the result in \cite{H-K}.
This paper solves the global well-posedness and stability problem on a special $2\frac12$-D compressible viscous non-resistive MHD system near a steady-state solution. The steady-state here consists of a positive constant density and a background magnetic field. The global solution is constructed in $L^p$-based homogeneous Besov spaces, which allow general and highly oscillating initial velocity. The well-posedness problem studied here is extremely challenging due to the lack of the magnetic diffusion, and remains open for the corresponding 3D MHD equations. Our approach exploits the enhanced …
In this paper, we mainly investigate the Cauchy problem for the incompressible Navier-Stokes equations in homogeneous Besov spaces $\dot{B}^{\frac{d}{p}-1}_{p,r}$ with $1\leq p<\infty,\ 1\leq r\leq \infty, \ d\geq 2$. Firstly, we prove the local existence of the solution and give a lower bound of the lifespan $T$ of the solution. The lifespan depends on the Littlewood-Paley decomposition of the initial data, that is $\dotΔ_j u_0$. Secondly, if the initial data $u^n_0\rightarrow u_0$ in $\dot{B}^{\frac{d}{p}-1}_{p,r}$, then the corresponding lifespan $T_n\rightarrow T$. Thirdly, we prove that t…
The present paper is devoted to the study of a zero-Mach number system with heat conduction but no viscosity. We work in the framework of general non-homogeneous Besov spaces $B^s_{p,r}(\mathbb{R}^d)$, with $p\in[2,4]$ and for any $d\geq 2$, which can be embedded into the class of globally Lipschitz functions. We prove a local in time well-posedness result in these classes for general initial densities and velocity fields. Moreover, we are able to show a continuation criterion and a lower bound for the lifespan of the solutions. The proof of the results relies on Littlewood-Paley decomposition…
In the present paper, we show an improved lower bound for the lifespan of the solutions to the ideal MHD equations in the case of space dimension $d=2$. In particular, for small initial magnetic fields $b_0$ of size (say) $\varepsilon>0$, the lifespan $T_\varepsilon>0$ of the corresponding solution goes to $+\infty$ in the limit $\varepsilon\rightarrow0^+$. Such a result does not follow from standard quasi-linear hyperbolic theory. For proving it, three are the crucial ingredients: first of all, to work in endpoint Besov spaces $B^s_{\infty,r}$, under the condition $s>1$ and $r\in[1,+\infty]$ …
This paper examines the uniqueness of weak solutions to the d-dimensional magnetohydrodynamic (MHD) equations with the fractional dissipation $(-Δ)^αu$ and without the magnetic diffusion. Important progress has been made on the standard Laplacian dissipation case $α=1$. This paper discovers that there are new phenomena with the case $α<1$. The approach for $α=1$ can not be directly extended to $α<1$. We establish that, for $α<1$, any initial data $(u_0, b_0)$ in the inhomogeneous Besov space $B^σ_{2,\infty}(\mathbb R^d)$ with $σ> 1+\frac{d}{2}-α$ leads to a unique local solution. For the case …
The present paper is the continuation of work [14], devoted to the study of an inviscid zero-Mach number system in the framework of \emph{endpoint} Besov spaces of type $B^s_{\infty,r}(\mathbb{R}^d)$, $r\in [1,\infty]$, $d\geq 2$, which can be embedded in the Lipschitz class $C^{0,1}$. In particular, the largest case $B^1_{\infty,1}$ and the case of Hölder spaces $C^{1,α}$ are permitted. The local in time well-posedness result is proved, under an additional $L^2$ hypothesis on the initial inhomogeneity and velocity field. A new a priori estimate for parabolic equations in endpoint spaces $B^s_…
In this paper, we prove the global well-posedness for the three-dimensional magnetohydrodynamics (MHD) equations with zero viscosity and axisymmetric initial data. First, we analyze the problem corresponding to the Sobolev regularities $ H^s\times H^{s-2}$, with $ s>5/2$. Second, we address the same problem but for the Besov critical regularities $ B_{p,1}^{3/p+1}\times B_{1,p}^{3/p-1}$, $2\leq p\leq \infty$. This case turns out to be more subtle as the Beale-Kato-Majda criterion is not known to be valid for rough regularities.
In this paper, we mainly investigate the Cauchy problem of the non-viscous MHD equations with magnetic diffusion. We first establish the local well-posedness (existence,~uniqueness and continuous dependence) with initial data $(u_0,b_0)$ in critical Besov spaces $ {B}^{\frac{d}{p}+1}_{p,1}\times{B}^{\frac{d}{p}}_{p,1}$ with $1\leq p\leq\infty$, 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 the global existence in critical Besov spaces. In particular, the results of global existence also hold in Sobole…
In this paper, we establish an optimal blow-up criterion for classical solutions to the incompressible resistive Hall-magnetohydrodynamic equations. We also prove two global-in-time existence results of the classical solutions for small initial data, the smallness conditions of which are given by the suitable Sobolev and the Besov norms respectively. Although the Sobolev space version is already an improvement of the corrresponding result in \cite{Chae-Degond-Liu}, the optimality in terms of the scaling property is achieved via the Besov space estimate. The special property of the energy estim…
We study in this paper the low Mach number limit for the 2d isentropic Euler system with ill-prepared initial data belonging to the critical Besov space $B_{2,1}^2$. By combining Strichartz estimates with the special structure of the vorticity we prove that the lifespan goes to infinity as the Mach number goes to zero. We also prove the strong convergence in the space of the initial data $B_{2,1}^2$ of the incompressible parts to the solution of the incompressible Euler system. There are at least two main difficulties: the first one concerns the Beale-Kato-Majda criterion which is not known to…
The present paper is devoted to the well-posedness issue for a low-Mach number limit system with heat conduction but no viscosity. We will work in the framework of general Besov spaces $B^s_{p,r}(\R^d)$, $d\geq 2$, which can be embedded into the class of Lipschitz functions. Firstly, we consider the case of $p\in[2,4]$, with no further restrictions on the initial data. Then we tackle the case of any $p\in\,]1,\infty]$, but requiring also a finite energy assumption. The extreme value $p=\infty$ can be treated due to a new a priori estimate for parabolic equations. At last we also briefly consid…
This paper is devoted to the study of the low Mach number limit for the isentropic Euler system with axisymmetric initial data without swirl. In the first part of the paper we analyze the problem corresponding to the subcritical regularities, that is $H^s$ {with $s>\frac52$.} Taking advantage of the Strichartz estimates and using the special structure of the vorticity we show that the {lifespan $T_ε$} of the solutions is bounded below by $\log\log\log\frac1ε$, where $ε$ denotes the Mach number. Moreover, we prove that the incompressible parts converge to the solution of the incompressible Eule…
