Related papers: Low regularity Sobolev well-posedness for Vlasov--…

We prove a local in time well-posedness result for quasi-linear Hamiltonian Schr\"odinger equations on $\mathbb{T}^d$ for any $d\geq 1$. For any initial condition in the Sobolev space $H^s$, with $s$ large, we prove the existence and…

We study the mKdV equation with periodic boundary conditions. We establish low regularity well -posedness in $H^{\frac{1}{4}+}(T)$. The proof involves a non-linear, solution dependent gauge transformation, similar to the one considered in…

The Cauchy problem for the Yang-Mills system in three space dimensions with data in Fourier-Lebesgue spaces $\hat{H}^{s,r}$ , $1 < r \le 2$ , is shown to be locally well-posed, where we have to assume only almost optimal minimal regularity…

We consider the Cauchy problem for the fourth order cubic nonlinear Schr\"odinger equation (4NLS). The main goal of this paper is to prove low regularity well-posedness and mild ill-posedness for (4NLS). We prove three results. First, we…

The 1D Cauchy problem for the Zakharov system is shown to be locally well-posed for low regularity Schr\"odinger data u_0 \in \hat{H^{k,p}} and wave data (n_0,n_1) \in \hat{H^{l,p}} \times \hat{H^{l-1,p}} under certain assumptions on the…

The local well-posedness problem for the Maxwell-Klein-Gordon system in Coulomb gauge as well as Lorenz gauge is treated in two space dimensions for data with minimal regularity assumptions. In the classical case of data in $L^2$-based…

We consider the initial value problem associated to a system consisting modified Korteweg-de Vries type equations \begin{equation*} \begin{cases} \partial_tv + \partial_x^3v + \partial_x(vw^2) =0,&v(x,0)=\phi(x),\\ \partial_tw +…

We prove the local well-posedness for the two-dimensional Zakharov-Kuznetsov equation in $H^s(\mathbb{R}^2)$, for $s\in [1,2]$, on the background of an $L^\infty(\mathbb{R}^3)$-function $\Psi(t,x,y)$, with $\Psi(t,x,y)$ satisfying some…

In this paper we continue our study [DSS20] of the nonlinear Schr\"odinger equation (NLS) with bounded initial data which do not vanish at infinity. Local well-posedness on $\mathbb{R}$ was proved for real analytic data. Here we prove…

In this paper, we study local well-posedness for the Navier-Stokes \linebreak equations with arbitrary initial data in homogeneous Sobolev spaces $\dot{H}^s_p(\mathbb{R}^d)$ for $d \geq 2, p > \frac{d}{2},\ {\rm and}\ \frac{d}{p} - 1 \leq s…

The Cauchy problem for the derivative nonlinear Schr\"odinger equation with periodic boundary condition is considered. Local well-posedness for periodic initial data u_0 in the space ^H^s_r, defined by the norms ||u_0||_{^H^s_r}=||<xi>^s…

In this paper we consider the hyperbolic-elliptic Ishimori initial-value problem. We prove that such system is locally well-posed for small data in $H^{s}$ level space, for $s> 3/2$. The new ingredient is that we develop the methods of…

In this paper, we study local well-posedness for the Navier-Stokes equations (NSE) with the arbitrary initial value in homogeneous Sobolev-Lorentz spaces $\dot{H}^s_{L^{q, r}}(\mathbb{R}^d):= (-\Delta)^{-s/2}L^{q,r}$ for $d \geq 2, q > 1, s…

We establish local well-posedness of the Hall-magneto-hydrodynamics (Hall-MHD) system in the Sobolev space $\left(H^s(\mathbb{R}^n)\right)^2$ with $s>\frac n2$. The previously known local well-posedness space was…

We study the local well-posedness theory for the Schr\"odinger Maps equation. We work in $n+1$ dimensions, for $n \geq 2$, and prove a local well-posedness for small initial data in $H^{\frac{n}{2}+\e}$.

We study the Cauchy problem to the KP-I equation posed on $\R^2$. We prove that it is $C^0$ locally well-posed in $H^{s,0}(\R\times \R)$ for $s>1/2$, which improves the previous results in \cite{GPW,GMo}.

In this paper, we first prove the local well-posedness of strong solutions to the incompressible Hall-MHD system for initial data $(u_0,B_0)\in H^{\frac{1}{2}+\sigma}(\mathbb{R}^3)\times H^{\frac{3}{2}}(\mathbb{R}^3)$ with $\sigma\in…

We study the theory of relativistic viscous hydrodynamics introduced in arXiv:1109.0985 and arXiv:1907.12695, which provided a causal and stable first-order theory of relativistic fluids with viscosity in the case of barotropic fluids. The…

We consider the NLS hierarchy with the nonzero boundary condition $q(t, x) \rightarrow q_\pm \in \mathbb{S}^1$ as $x \rightarrow \pm \infty$ and prove that it is global well-posedness for initial data of high regularity. Specifically, we…

We prove well-posedness for higher-order equations in the so-called NLS hierarchy (also known as part of the AKNS hierarchy) in almost critical Fourier-Lebesgue spaces and in modulation spaces. We show the $j$th equation in the hierarchy is…