The author propose what is the principal part of linear systems of partial differential equations in the Cauchy problem through the normal form of systems in the meromorphic formal symbol class and the theory of weighted determinant. As applications, he choose the necessary and sufficient conditions for the analytic well-posedness ( Cauchy-Kowalevskaya theorem ) and $C^{\infty }$ well-posedness (Levi condition).

We consider the Cauchy problem for a generalized Klein-Gordon-Schrödinger system with Yukawa coupling. We prove the existence of global weak solutions by the compactness method and, through a special choice of the admissible pairs to match two types of equations, we prove the uniqueness of those solutions by an approach similar to the method presented by J. Ginibre and G. Velo for the pure Klein-Gordon equation or pure Schrödinger equation. Though it is very simple in form, the method has an unnatural...

We consider the homogeneous time-dependent Oseen system in the whole space ${ℝ}^3$. The initial data is assumed to behave as $O\left(\right|x{|}^{-1-ϵ})$, and its gradient as $O\left(\right|x{|}^{-3/2-ϵ})$, when $\left|x\right|$ tends to infinity, where $ϵ$ is a fixed positive number. Then we show that the velocity $u$ decays according to the equation $\left|u(x,t)\right|=O\left(\right|x{|}^{-1})$, and its spatial gradient ${\nabla }_{x}u$ decreases with the rate ${\left|x\right|}^{-3/2}$, for $\left|x\right|$ tending to infinity, uniformly with respect to the time variable $t$. Since these decay rates are optimal even in the stationary case, they should also be the best possible...

The local well-posedness for the Cauchy problem of the liquid crystals system in the critical Besov space ${\dot{B}}_{p,1}^{n/p-1}\left({ℝ}^n\right)\times {\dot{B}}_{p,1}^{n/p}\left({ℝ}^n\right)$ with $n<p<2n$ is established by using the heat semigroup theory and the Littlewood-Paley theory. The global well-posedness for the system is obtained with small initial datum by using the fixed point theorem. The blow-up results for strong solutions to the system are also analysed.

We study the uniqueness and regularity of Leray-Hopf's weak solutions for the MHD equations with dissipation and resistance in different frameworks. Using different kinds of space-time estimates in conjunction with the Littlewood-Paley-Bony decomposition, we present some general criteria of uniqueness and regularity of weak solutions to the MHD system, and prove the uniqueness and regularity criterion in the framework of mixed space-time Besov spaces by applying Tao's trichotomy method.

The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. In the 3D case Guo [7] first constructed a global smooth irrotational solution by using the dispersive Klein-Gordon effect. It has been conjectured that same results should hold in the two-dimensional case. In our recent work [13], we proved the existence of a family of smooth solutions by constructing the wave operators for the 2D system....

In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index s &gt; 2 to obtain local solutions for any initial data, and global solutions for small initial data.

In this paper we study the Cauchy problem for second order strictly hyperbolic operators of the form $Lu:={\sum }_{j,k=0}^n{\partial }_{y_j}\left(a_{j,k}{\partial }_{y_k}u\right)+{\sum }_{j=0}^n\{b_j{\partial }_{y_j}u+{\partial }_{y_j}\left(c_{j}u\right)\}+du=f,$ when the coefficients of the principal part are not Lipschitz continuous, but only “Log-Lipschitz” with respect to all the variables. This class of equation is invariant under changes of variables and therefore suitable for a local analysis. In particular, we show local existence, local uniqueness and finite speed of propagation for the noncharacteristic Cauchy problem. This provides an invariant...

The change in the electric potential due to lightning is evaluated. The potential along the lightning channel is a constant which is the projection of the pre-flash potential along a piecewise harmonic eigenfunction which is constant along the lightning channel. The change in the potential outside the lightning channel is a harmonic function whose boundary conditions are expressed in terms of the pre-flash potential and the post-flash potential along the lightning channel. The expression for the...

The Boltzmann–Poisson system modeling the electron flow in semiconductors is used to discuss the validity of the Child–Langmuir asymptotics. The scattering kernel is approximated by a simple relaxation time operator. The Child–Langmuir limit gives an approximation of the current-voltage characteristic curves by means of a scaling procedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child–Langmuir regime by performing detailed numerical comparisons...

The Boltzmann–Poisson system modeling the electron flow in semiconductors is used to discuss the validity of the Child–Langmuir asymptotics. The scattering kernel is approximated by a simple relaxation time operator. The Child–Langmuir limit gives an approximation of the current-voltage characteristic curves by means of a scaling procedure in which the ballistic velocity is much larger that the thermal one. We discuss the validity of the Child–Langmuir regime by performing detailed numerical...