Virial type blow-up solutions for the Zakharov system with magnetic field in a cold plasma

In this paper, we study blow-up solutions of virial type to the Zakharov system with magnetic field in a cold plasma in RN (N=2,3). After obtaining some a priori estimates on those terms generated by the magnetic field, we obtain a virial type blow-up result to the system under consideration. The result suggests that the magnetic field in a cold plasma doesn?t affect the virial type blow-up character of the Zakharov system.     

On the limit behavior of the magnetic Zakharov system

In this paper,we prove that the solutions of magnetic Zakharov system converge to those of generalized Zakharov system in Sobolev space H s,s > 3/2,when parameter β→∞.Further,when parameter (α,β) →∞ together,we prove that the solutions of magnetic Zakharov system converge to those of Schro¨dinger equation with magnetic effect in Sobolev space H s,s > 3/2.Moreover,the convergence rate is also obtained.     

Blow Up in Finite Time for the Zakharov System

In [1],Merle guessed that the solution of Zakharov system always blows up in finite time.In accordance with the guess,in this paper we study the finite time blow-up results for the solution to the Cauchy problem of the generalized Zakharov system with combined power-type nonlinearities in R3:     

On the Cauchy problem for the magnetic Zakharov system

In this paper, we study the well-posedness results for the magnetic type Zakharov system. Such system describes the pondermotive force and magnetic field generation effects resulting from the nonlinear interaction between plasma-wave and particles. By using energy methods together with commutator estimate, we first derive a priori estimates for a regularized system. Then by approximation arguments, we obtain local existence results as well as global existence for small initial data.     

On the global existence and small dissipation limit for generalized dissipative Zakharov system

This paper deals with the existence and uniqueness of the global solutions to the initial boundary value problem for a generalized Zakharov system with direct self‐interaction of the dispersive waves and weak dissipation in the nondispersive subsystem. We prove the global existence of the generalized solution to the problem by a priori estimates and Galerkin method. We also establish the regularity of the global generalized solution and the existence and uniqueness of the global classical solution. Moreover, we obtain the convergence of the solutions of the generalized Zakharov system with dissipation as the dissipative coefficient approaches zero.     

广义Zakharov方程解的最佳收敛速度

本文考虑一类含自生磁场效应的广义Zakhaorv方程,并对它的非线性Schrdinger极限行为开展研究,主要考虑解的收敛速度问题.通过对非线性项的细致估计,文中给出收敛速度与第一初始层及第二初始层之间的具体依赖关系.该工作改进了Zhang和Guo在2011年得到的收敛性结果.     

Regularity properties of the Zakharov system on the half line

In this paper, we study the local and global regularity properties of the Zakharov system on the half line with rough initial data. These properties include local and global wellposedness results, local and global smoothing results, and the behavior of higher order Sobolev norms of the solutions. Smoothing means that the nonlinear part of the solution on the half line is smoother than the initial data. The gain in regularity coincides with the gain that was observed for the periodic Zakharov and the Zakharov on the real line. Uniqueness is proved in the class of smooth solutions. When the boundary value of the Schrödinger part of the solution is zero, uniqueness can be extended to the full range of local solutions. Under the same assumptions on the initial data, we also prove global-in-time existence and uniqueness of energy solutions. For more regular data, we prove that all higher Sobolev norms grow at most polynomially-in-time.     

Scattering theory for the Schrödinger equation in some external time-dependent magnetic fields

We study the theory of scattering for a Schrödinger equation in an external time-dependent magnetic field in the Coulomb gauge, in space dimension 3. The magnetic vector potential is assumed to satisfy decay properties in time that are typical of solutions of the free wave equation, and even in some cases to be actually a solution of that equation. That problem appears as an intermediate step in the theory of scattering for the Maxwell-Schrödinger (MS) system. We prove in particular the existence of wave operators and their asymptotic completeness in spaces of relatively low regularity. We also prove their existence or at least asymptotic results going in that direction in spaces of higher regularity. The latter results are relevant for the MS system. As a preliminary step, we study the Cauchy problem for the original equation by energy methods, using as far as possible time derivatives instead of space derivatives.     

Global well-posedness and the classical limit of the solution for the quantum Zakharov system

In this paper, we study the quantum Zakharov system, which describes the nonlinear interaction between the quantum Langmuir and quantum ion-acoustic waves. The global well-posedness result of this system in the energy and above energy spaces is obtained in the case d = 1, 2, 3. Moreover, the classical limit behavior of the quantum Zakharov system is also investigated as the quantum parameter tends to zero.     

Regularity of weak solution to Maxwell's equations and applications to microwave heating

In this paper we first study the regularity of weak solution for time-harmonic Maxwell's equations in a bounded anisotropic medium Ω. It is shown that the weak solution to the linear degenerate system, , is Hölder continuous under the minimum regularity assumptions on the complex coefficients γ(x) and ξ(x). We then study a coupled system modeling a microwave heating process. The dynamic interaction between electric and temperature fields is governed by Maxwell's equations coupled with an equation of heat conduction. The electric permittivity, electric conductivity and magnetic permeability are assumed to be dependent of temperature. It is shown that under certain conditions the coupled system has a weak solution. Moreover, regularity of weak solution is studied. Finally, existence of a global classical solution is established for a special case where the electric wave is assumed to be propagating in one direction.     

广义Захаров方程组的初边值问题

本文用积分估计的方法对于一类更广泛的Захаров方程组的初边值问题（其中一维问题为第三初边值问题，二维为第一初边值问题），证明了广义解的存在性。     

A probabilistic model associated with the pressureless gas dynamics

Using a method of stochastic perturbation of a Langevin system associated with the non-viscous Burgers equation we introduce a system of PDE that can be considered as a regularization of the pressureless gas dynamics describing sticky particles. By means of this regularization we describe how starting from smooth data a δ-singularity arises in the component of density. Namely, we find the asymptotics of solution at the point of the singularity formation as the parameter of stochastic perturbation tends to zero. Then we introduce a generalized solution in the sense of free particles (FP-solution) as a special limit of the solution to the regularized system. This solution corresponds to a medium consisting of non-interacting particles. The FP-solution is a bridging step to constructing solutions to the Riemann problem for the pressureless gas dynamics describing sticky particles. We analyze the difference in the behavior of discontinuous solutions for these two models and the relations between them. In our framework we obtain a unique entropy solution to the Riemann problem in 1D case.     

Control of error in the homotopy analysis of solutions to the Zakharov system with dissipation

We apply the method of homotopy analysis to the Zakharov system with dissipation in order to obtain analytical solutions, treating the auxiliary linear operator as a time evolution operator. Evolving the approximate solutions in time, we construct approximate solutions which depend on the convergence control parameters. In the situation where solutions are strongly coupled, there will be multiple convergence control parameters. In such cases, we will pick the convergence control parameters to minimize a sum of squared residual errors. We explain the error minimization process in detail, and then demonstrate the method explicitly on several examples of the Zakharov system held subject to specific initial data. With this, we are able to efficiently obtain approximate analytical solutions to the Zakharov system of minimal residual error using approximations with relatively few terms.     

Unique solvability of equations of motion for ferrofluids

We study the differential system governing the flow of an incompressible ferrofluid under the action of a magnetic field. The system is a combination of the Navier-Stokes equations, the angular momentum equation, the magnetization equation and the magnetostatic equations. No regularizing term is added to the magnetization equation. We prove the local-in-time existence of the unique strong solution to the system posed in a bounded domain of R³ and equipped with initial and boundary conditions.     

Stability and instability of the standing waves for the Klein‐Gordon‐Zakharov system in one space dimension

The orbital instability of standing waves for the Klein‐Gordon‐Zakharov system has been established in two and three space dimensions under radially symmetric condition by Ohta‐Todorova in 2007. In the one space dimensional case, for the nondegenerate situation, we first check that the Klein‐Gordon‐Zakharov system satisfies Grillakis‐Shatah‐Strauss' assumptions on the stability and instability theorems for abstract Hamiltonian systems; see Grillakis‐Shatah‐Strauss (J. Funct. Anal. 1987). As to the degenerate case that the frequency , we follow the recent splendid work of Wu (2017) to prove the instability of the standing waves for the Klein‐Gordon‐Zakharov system, by using the modulation argument combining with the virial identity. For this purpose, we establish a modified virial identity to overcome several troublesome terms left in the traditional virial identity.     

Klein‐Gordon‐Zakharov system in energy space: Blow‐up profile and subsonic limit

In this paper, we prove finite‐time blowup in energy space for the three‐dimensional Klein‐Gordon‐Zakharov (KGZ) system by modified concavity method. We obtain the blow‐up rates of solutions in local and global space, respectively. In addition, by using the energy convergence, we study the subsonic limit of the Cauchy problem for KGZ system and prove that any finite energy solution converges to the corresponding solution of Klein‐Gordon equation in energy space.     

Yudovich type solution for the 2D inviscid Boussinesq system with critical and supercritical dissipation

In this paper we consider the Yudovich type solution of the 2D inviscid Boussinesq system with critical and supercritical dissipation. For the critical case, we show that the system admits a global and unique Yudovich type solution; for the supercritical case, we prove the local and unique existence of Yudovich type solution, and the global result under a smallness condition of _θ0${\theta }_0$. We also give a refined blowup criterion in the supercritical case.     

Orbital stability of periodic traveling wave solutions to the generalized zakharov equations

This paper investigates the orbital stability of periodic traveling wave solutions to the generalized Zakharov equations

$\{\begin{array}{c}i{u}_t+u_{xx}=uv+|u{|}^{2}u,\\ v_{tt}-v_{xx}=\left(|u{|}^2\right)xx\end{array}.$

First, we prove the existence of a smooth curve of positive traveling wave solutions of dnoidal type with a fixed fundamental period L for the generalized Zakharov equations. Then, by using the classical method proposed by Benjamin, Bona et al., we show that this solution is orbitally stable by perturbations with period L. The results on the orbital stability of periodic traveling wave solutions for the generalized Zakharov equations in this paper can be regarded as a perfect extension of the results of [15, 16, 19].     

On the regularity of weak solutions to the magnetohydrodynamic equations

In this paper, we study the regularity of weak solution to the incompressible magnetohydrodynamic equations. We obtain some sufficient conditions for regularity of weak solutions to the magnetohydrodynamic equations, which is similar to that of incompressible Navier-Stokes equations. Moreover, our results demonstrate that the velocity field of the fluid plays a more dominant role than the magnetic field does on the regularity of solution to the magneto-hydrodynamic equations.     

具量子效应Zakharov方程弱解的存在性

Zakharov方程具有丰富的物理背景．通过Arzela－Ascoli定理、Faedo－Galerkin方法和紧性原理,得到等离子体模型中具量子效应Zakharov方程弱整体解的存在性．