Related papers: Sharp ill-posedness for the non-resistive MHD equa…
In this paper, we study the global well-posedness and decay rates of strong solutions to an incompressible Vlasov-MHD model arising in magnetized plasmas. This model is consist of the Vlasov equation and the incompressible…
In this paper, we prove a sharp nonuniqueness result for the incompressible Navier-Stokes equations in the periodic setting. In any dimension $d \geq 2$ and given any $ p<2$, we show the nonuniqueness of weak solutions in the class $L^{p}_t…
Several fluid systems are characterised by time reversal and parity breaking. Examples of such phenomena arise both in quantum and classical hydrodynamics. In these situations, the viscosity tensor, often dubbed ``odd viscosity'', becomes…
We show the local in time well-posedness of the Cauchy problem for the Kadomtsev-Petviashvili II equation for initial data in the non-isotropic Sobolev space H^{s_1,s_2}(R^2) with s_1 > -1/2 and s_2 \geq 0. On the H^{s_1,0}(R^2) scale this…
We establish the local Hadamard well-posedness of a certain third-order nonlinear Schr\"odinger equation with a multi-term linear part and a general power nonlinearity known as the higher-order nonlinear Schr\"odinger equation, formulated…
We consider the 3D Boltzmann equation with the constant collision kernel. We investigate the well/ill-posedness problem using the methods from nonlinear dispersive PDEs. We construct a family of special solutions, which are neither near…
We consider the incompressible Navier-stokes equations (NS) in $\mathbb{R}^{n}$ for $n\geq2$. Global well-posedness is proved in critical Besov-weak-Herz spaces (BWH-spaces) that consist in Besov spaces based on weak-Herz spaces. These…
In the present paper, we show the ill-posedness of the free boundary problem of the incompressible ideal magnetohydrodynamics (MHD) equations in two spatial dimensions for any positive vacuum permeability $\mu_0$, in Sobolev spaces. The…
We prove local well-posedness in the Sobolev spaces $\dot H^s(\mathbb{T})$, with $s>7/2$, for an initial value problem for a nonlocal, cubically nonlinear, dispersive equation that provides an approximate description of the evolution of…
In this paper, we prove the local well-posedness for the Ideal MHD equations in the Triebel-Lizorkin spaces and obtain blow-up criterion of smooth solutions. Specially, we fill a gap in a step of the proof of the local well-posedness part…
In this paper we consider the cubic Schrodinger equation in two space dimensions on irrational tori. Our main result is an improvement of the Strichartz estimates on irrational tori. Using this estimate we obtain a local well-posedness…
In this paper we consider the supercritical generalized Korteweg-de Vries equation $\partial_t\psi + \partial_{xxx}\psi + \partial_x(|\psi|^{p-1}\psi) = 0$, where $5\leq p\in\R$. We prove a local well-posedness result in the homogeneous…
In this paper, the well-posedness is studied for the initial boundary value problem of the two-dimensional compressible ideal magnetohydrodynamic (MHD) equations in bounded perfectly conducting domains with corners. The presence of corners…
The study of low regularity Cauchy data for nonlinear dispersive PDEs has successfully been achieved using modulation spaces $M^{p,q}$ in recent years. In this paper, we study the inhomogeneous nonlinear Schr\"odinger equation (INLS) $$iu_t…
In this paper, we prove a sharp and strong non-uniqueness for a class of weak solutions to the three-dimensional magneto-hydrodynamic (MHD) system. More precisely, we show that any weak solution $(v,b)\in L^p_tL^{\infty}_x$ is non-unique in…
We consider the Cauchy problem associated to the recently derived higher order hamiltonian model for unidirectional water waves and prove global existence for given data in the Sobolev space $H^s$, $s\geq 1$. We also prove an ill-posedness…
We provide a complete local well-posedness theory in $H^s$ based Sobolev spaces for the free boundary incompressible Euler equations with zero surface tension on a connected fluid domain. Our well-posedness theory includes: (i) Local…
We study the $2\frac{1}{2}$D electron magnetohydrodynamics (MHD): the electron MHD system that has $3$D magnetic field but is independent of $z$-variable. We establish a "half" strong ill-posedness result in $2\frac{1}{2}$D electron MHD…
We establish that the quadratic non-linear Schr\"odinger equation $$ iu_t + u_{xx} = u^2$$ where $u: \R \times \R \to \C$, is locally well-posed in $H^s(\R)$ when $s \geq -1$ and ill-posed when $s < -1$. Previous work of Kenig, Ponce and…
In this paper, we study the wellposedeness of the Hall-magnetohydrodynamic system augmented by the effect of electron inertia. Our main result consists of generalising the wellposedness one in \cite{Zhao} from the Sobolev context to the…