Related papers: Well-posedness and singularity formation for Vlaso…

We consider the Cauchy problem for the Chern-Simons-Dirac system on $\mathbb{R}^{1+1}$ with initial data in $H^s$. Almost optimal local well-posedness is obtained. Moreover, we show that the solution is global in time, provided that initial…

Consider the Vlasov-Poisson-Landau system with Coulomb potential in the weakly collisional regime on a $3$-torus, i.e. $$\begin{aligned} \partial_t F(t,x,v) + v_i \partial_{x_i} F(t,x,v) + E_i(t,x) \partial_{v_i} F(t,x,v) = \nu…

The relativistic membrane equation can be rewritten as a first order hyperbolic system. Making use of the characteristic decomposition method, a new blow-up theorem is established. As an application, it demonstrates the formation of…

We study the Cauchy problem for a generalized derivative nonlinear Schr\"odinger equation with the Dirichlet boundary condition. We establish the local well-posedness results in the Sobolev spaces $H^1$ and $H^2$. Solutions are constructed…

We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global…

Consideration here is a generalized $\mu$-type integrable equation, which can be regarded as a generalization to both the $\mu$-Camassa-Holm and modified $\mu$-Camassa-Holm equations. It is shown that the proposed equation is formally…

In this paper, we study the Cauchy problem to the density-dependent liquid crystal system in $\mathbb R^3$. We establish the local existence and uniqueness of strong solutions to this system. In order to overcome the difficulties caused by…

We study existence and uniqueness of the solution to the Vlasov-Poisson system describing a plasma constituted by different species evolving in $\mathbb{R}^3$, whose particles interact via the Coulomb potential. The species can have both…

The Cauchy problem for Zakharov-Kuznetsov equation on $\mathbb{R}^2$ is shown to be global well-posed for the initial date in $H^{s}$ provided $s>-\frac{1}{13}$. As conservation laws are invalid in Sobolev spaces below $L^2$, we construct…

In this paper we prove full local well-posedness for the Cauchy problem for the compressible 3D Euler equation, i.e. local existence, uniqueness, and continuous dependence on initial data, with initial velocity, density and vorticity…

This paper considers the two-dimensional Cauchy problem of the full compressible Navier-Stokes equations with far-field vacuum in $\mathbb{R}^2$, where the viscosity and heat-conductivity coefficients depend on the absolute temperature…

We consider relativistic plasma particles subjected to an external gravitation force in a $3$D half space whose boundary is a perfect conductor. When the mean free path is much bigger than the variation of electromagnetic fields, the…

In this paper, we study a new fifth-order Camassa-Holm type equation derived by Li \cite{Li.Z}. We firstly establish the local well-posedness in the sense of Hadamard for the Cauchy problem of the new fifth-order Camassa-Holm type equation…

The initial value problem for the Vlasov-Poisson system is by now well understood in the case of an isolated system where, by definition, the distribution function of the particles as well as the gravitational potential vanish at spatial…

We study the long-time behavior of radially symmetric solutions to the Vlasov-Poisson equation consisting of an attractive point mass and a small, suitably localized and absolutely continuous distribution of particles: if the latter is…

This paper is dedicated to the local existence theory of the Cauchy problem for a general class of symmetrizable hyperbolic partially diffusive systems (also called hyperbolic-parabolic systems) in the whole space $\mathbb{R}^d$ with $d\ge…

We consider the Cauchy problem for the equations of pressureless gases in two space dimensions. For a generic set of smooth initial data (density and velocity), it is known that the solution loses regularity at a finite time $t_0$, where…

This paper studies the local existence of strong solutions to the Cauchy problem of the 2D fluid-particle interaction model with vacuum as far field density. Notice that the technique used by Ding et al.\cite{SBH} for the corresponding 3D…

This paper studies the Cauchy problem for three-dimensional viscous, compressible, and heat conducting magnetohydrodynamic equations with vacuum as far field density. We prove the global existence and uniqueness of strong solutions provided…

For the Landau-Poisson system with Coulomb interaction in $\R^3_x$, we prove the global existence, uniqueness, and large time convergence rates to the Maxwellian equilibrium for solutions which start out sufficiently close.