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In this paper, we are concerned with the well-posedness and large time behavior of Cauchy problem for 3D incompressible Navier-Stokes-Cahn-Hilliard equations. First, using Banach fixed point theorem, we establish the local well-posedness of…
In their seminal work, Bourgain and Li establish strong ill-posedness of the 2D incompressible Euler equations with vorticity in the critical Sobolev space $W^{s,p}(\mathbb{R}^2)$ for $sp=2$ and $p\in(1,\infty)$. In this note, we establish…
We investigate a one-dimensional nonlinear wave system which arises from a variational principle modeling a type of cholesteric liquid crystals. The problem treated here is the Cauchy problem for the same wave speed case with initial data…
For the $d$-dimensional incompressible Euler equation, the standard energy method gives local wellposedness for initial velocity in Sobolev space $H^s(\mathbb R^d)$, $s>s_c:=d/2+1$. The borderline case $s=s_c$ was a folklore open problem.…
In this paper we consider the motion of a rigid body in a viscous incompressible fluid when some Navier slip conditions are prescribed on the body's boundary. The whole system "viscous incompressible fluid + rigid body" is assumed to occupy…
In this paper, we study the Cauchy problem of 2d Dirac equation with Hartree type nonlinearity $c(|\cdot|^{-\gamma} * \langle \psi, \beta \psi\rangle)\beta\psi$ with $c\in \mathbb R\setminus\{0\} $, $0 < \gamma < 2$. Our aim is to show the…
We prove that for sufficiently small $H^3$-perturbations of an affine solution, the Cauchy problem for the compressible nonlinear elastodynamics in $\mathbb{R}^d$, for $d=2,3$, admits a unique global strong solution. Moreover, we establish…
In Part II of the paper, we prove linear instability of a certain class of radially symmetric flows of an ideal incompressible fluid in dimension two used in Part I
The motion of two contiguous incompressible and viscous fluids is described within the diffuse interface theory by the so-called Model H. The system consists of the Navier-Stokes equations, which are coupled with the Cahn-Hilliard equation…
Local structures, beyond the well-known `frozen-in' to the barotropic flows of the generalized vorticities, of the two-fluid model of plasma flows are presented. More general non-barotropic situations are also considered. A modified Euler…
In this article, the Cauchy problem of three-dimensional (3-D) incompressible magnetohydrodynamic system was investigated. If the initial $\mathcal{M}^{1,1}$ norms of the vorticity $\omega$ and the current density $j$ are both sufficiently…
In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schr\"{o}dinger (INLS) equation \[iu_{t} +\Delta u=|x|^{-b} f(u), ~u(0)=u_{0} \in H^{s} (\mathbb R^{n} ),\] where $n\ge3$, $1\le s<\frac{n}{2} $, $0<b<2$…
The nonlinear Schr\"odinger equations with nonlinearities $|u|^{2k}u$ on the $d$-dimensional torus are considered for arbitrary positive integers $k$ and $d$. The solution of the Cauchy problem is shown to be unique in the class $C_tH^s_x$…
In this paper, the Cauchy problem for the three-dimensional (3-D) full compressible Navier-Stokes equations (CNS) with zero thermal conductivity is considered. First, when shear and bulk viscosity coefficients both depend on the absolute…
We study the Cauchy problem for the compressible Euler equations in two spatial dimensions under any physical barotropic equation of state except that of a Chaplygin gas. We prove that the well-known phenomenon of shock formation in simple…
We investigate the Cauchy problem for a two-component generalization of the Novikov equation with cubic nonlinearity -- an integrable system whose solutions may develop strong nonlinear phenomena such as gradient blow-up and interactions…
We are concerned with the Cauchy problem of the two-dimensional (2D) nonhomogeneous incompressible Navier-Stokes equations with vacuum as far-field density. It is proved that if the initial density decays not too slow at infinity, the 2D…
The linear stability with variable coefficients of the vortex sheets for the two-dimensional compressible elastic flows is studied. As in our earlier work on the linear stability with constant coefficients, the problem has a free boundary…
We investigate global well-posedness to the Cauchy problem of three-dimensional compressible viscous and heat-conducting micropolar fluid equations with zero density at infinity. By delicate energy estimates, we establish global existence…
In this paper, we study the Cauchy problem for a two-component higher order Camassa-Holm systems with fractional inertia operator $A=(1-\partial_x^2)^r,r\geq1$, which was proposed by Escher and Lyons. By the transport equation theory and…