arXiv subjects
Publications and source records attributed to Xixia Ma.
We study both the topological structure stability and the relations of the steady Magnetohydrodynamic equations when $ν,η$ are given different values in muti-connected bounded domain. We also show the solutions's existence for fixed $ν, η.$ The theoretical is the Morse-Sard theorem on Sobolev spaces.
By using the Wigner transform, it is shown that the nonlinear Schr$\ddot{\textmd{o}}$dinger equation can be described, in phase space, by a kinetic theory similar to the Vlasov equation which is used for describing a classical collisionless plasma. In this paper we mainly show Landau damping in the quantum sense, namely,quantum Landau damping exists for the Wigner-Poisson system. At the same time, we also prove the existence and the stability of the nonlinear Schr$\ddot{\textmd{o}}$dinger equation under the quantum stability assumption.
We prove cyclotron damping for the collisionless Vlasov-Maxwell equations on $\mathbb{T}_{x}^{3}\times\mathbb{R}_{v}^{3}$ under the assumptions that the electric induction is zero and $(\mathcal{\mathbf{PSC}})$ holds. It is a crucial step to solve the stability problem of the Vlasov-Maxwell equations. Our proof is based on a new dynamical system of the plasma particles, originating from Faraday Law of Electromagnetic induction and Lenz's Law. On the basis of it, we use the improved Newton iteration scheme to show the damping mechanism.
In this paper, we consider the nonlinear Vlasov-Poisson equations in a weakly collisional regime and study the linear Boltzmann collision operator. We prove that Landau damping still occurs in this case.
In this paper, we discuss Bernoulli's principle to the 3-dimensional incompressible Euler equation in a bounded local Lipschitz domain $Ω\subset\mathbb{R}^{3}$ with a Lipschitz boundary. Using topological properties of the level set and Morse-Sard theorem, we will prove Bernoulli's principle on Sobolev space.
In this paper, we study the singular set of 3-dimensional Navier-Stokes equations. Under the condition$\frac{1}{R^{\frac{3s}{q}+2-s}}\int^{R^{2}}_{0}(\int_{B_{R}}|u|^{q}dx)^{\frac{s}{q}}ds <C,$ for $(q,s)\in\{(2,5),(5,2)\},$ we use the backward uniqueness of parabolic equations to show that the Hausdorff dimension of the singular set is less than 1.
In this paper, we have showed $L^{4,\infty}$-solutions to the cauchy problem for the four-dimensional Navier-Stokes equations are smooth through backward uniqueness and analytic functions properties .