Under the assumptions that the initial density ρ0 is close enough to 1 and ρ0-1∈Hs+1(R2);u0∈Hs(R2)∩H_∈(R2) for s>2 and 0<ε<1;the authors prove the global existence and uniqueness of smooth solutions to the 2-D inhomogeneous Navier-Stokes equations with the viscous coefficient depending on the density of the fluid.Furthermore,the L2 decay rate of the velocity field is obtained.
In this article, we first present an equivalent formulation of the free boundary problem to 3-D incompressible Euler equations, then we announce our local wellposedness result concerning the free boundary problem in Sobolev space provided that there is no self-intersection point on the initial surface and under the stability assumption that $\frac{{\partial p}}{{\partial n}}(\xi )\left| {_{t = 0} } \right. \leqslant - 2c_0 < 0$ being restricted to the initial surface.
Ping ZHANG~1 Zhi-fei ZHANG~2 1 Academy of Mathematics and Systems Science,Chinese Academy of Sciences,Beijing 100080,China
Given a domain Ω R^n, let λ 〉 0 be an eigenvalue of the elliptic operator L ：= ∑i,j^n= 1δ/δxi on Ω for Dirichlet condition. For a function f ∈ L2（Ω）, it is known that the linear resonance equation Lu ＋ λu = f in Ω with Dirichlet boundary condition is not always solvable. We give a new boundary condition Pλ（u｜δΩ） = g, called to be projective Dirichlet condition, such that the linear resonance equation always admits a unique solution u being orthogonal to all of the eigenfunctions corresponding to λ which satisfies ｜｜u｜｜2,2 ≤ C（｜｜f｜｜2 ＋ ｜｜g｜｜2,2） under suitable regularity assumptions on δΩ and L, where C is a constant depends only on n, Ω, and L. More a priori estimates, such as W^2~＇P-estimates and the C^2,α-estimates etc., are given also. This boundary condition can be viewed as a generalization of the Dirichlet condition to resonance equations and shows its advantage when applying to nonlinear resonance equations. In particular, this enables us to find the new indicatrices with vanishing mean （Cartan） torsion in Minkowski geometry. It is known that the geometry of indicatries is the foundation of Finsler geometry.
The authors mainly study the generalized symplectic mean curvature flow in an almost Einstein surface,and prove that this flow has no type-I singularity.In the graph case,the global existence and convergence of the flow at infinity to a minimal surface with metric of the ambient space conformal to the original one are also proved.
Motivated by the results of J. Y. Chemin in ＂J. Anal. Math., 77, 1999, 27- 50＂ and G. Furioli et al in ＂Revista Mat. Iberoamer., 16, 2002, 605-667＂, the author considers further regularities of the mild solutions to Navier-Stokes equation with initial data uo ∈ L^d（R^d）. In particular, it is proved that if u C ∈（[0, T^＊）; L^d（R^d）） is a mild solution of （NSv）, then u（t,x）- e^vt△uo ∈ L^∞（（0, T）;B2/4^1,∞）~∩L^1 （（0, T）; B2/4^3 ,∞） for any T 〈 T^＊.
Let U be a multiply-connected fixed attracting Fatou domain of a rational map f.We prove that there exist a rational map g and a completely invariant Fatou domain V of g such that(f,U) and(g,V) are holomorphically conjugate,and each non-trivial Julia component of g is a quasi-circle which bounds an eventually superattracting Fatou domain of g containing at most one postcritical point of g.Moreover,g is unique up to a holomorphic conjugation.
