In this paper, under some assumptions on the flow with a low Mach number, we study the nonexistence of a global nontrivial subsonic solution in an unbounded domain Ω which is one part of a 3D ramp. The flow is assumed to be steady, isentropic and irrotational, namely, the movement of the flow is described by the potential equation. By establishing a fundamental a priori estimate on the solution of a second order linear elliptic equation in Ω with Neumann boundary conditions on Ω and Dirichlet boundary value at some point of Ω, we show that there is no global nontrivial subsonic flow with a low Mach number in such a domain Ω.
We study the partial regularity of weak solutions to the 2-dimensional Landau- Lifshitz equations coupled with time dependent Maxwell equations by Ginzburg-Landau type approximation. Outside an energy concentration set of locally finite 2-dimensional parabolic Hausdorff measure, we prove the uniform local C∞ bounds for the approaching solutions and then extract a subsequence converging to a global weak solution of the Landau-Lifshitz-Maxwell equations which are smooth away from finitely many points.
We study the time-decay properties of weighted norms of solutions to the Stokes equations and the Navier-Stokes equations in the half-space Rn+ (n 2). Three kinds of the weighted Lp-Lr estimates are established for the Stokes semigroup generated by the Stokes operator in the half-space R+n (n 2). As an application of the weighted estimates of the Stokes semigroup, a class of local and global strong solutions in weighted Lp (R+n) are constructed, following the approach given by Kato.
HE Cheng1 & WANG LiZhen21Division of Mathematics, Department of Mathematical and Physical Sciences, National Natural Science Foundation of China, Beijing 100085, China
In this paper, we introduce a new notion named as SchrSdinger soliton. The so-called SchrSdinger solitons are a class of solitary wave solutions to the SchrSdinger flow equation from a Riemannian manifold or a Lorentzian manifold M into a K//hler manifold N. If the target manifold N admits a Killing potential, then the SchrSdinger soliton reduces to a harmonic map with potential from M into N. Especially, when the domain manifold/~r is a Lorentzian manifold, the Schr6dinger soliton is a wave map with potential into N. Then we app][y the geometric energy method to this wave map system, and obtain the local well-posedness of the corresponding Cauchy problem as well as global existence in 1 + 1 dimension. As an application, we obtain the existence of SchrSdinger soliton solution to the hyperbolic Ishinmri system.