Scattering theory for nonlinear Schrödinger equations with inverse-square potential

We study the long-time behavior of solutions to nonlinear Schrödinger equations with some critical rough potential of a|x|⁻²$a{|x|}^{-2}$ type. The new ingredients are the interaction Morawetz-type inequalities and Sobolev norm property associated with _Pa=−Δ+a|x|⁻²$P_a=-\mathrm{\Delta }+a{|x|}^{-2}$. We use such properties to obtain the scattering theory for the defocusing energy-subcritical nonlinear Schrödinger equation with inverse square potential in energy space H¹(Rⁿ)$H^1({\mathbb{R}}^n)$.     

Stability of solitary waves for a system of nonlinear Schrödinger equations with three wave interaction

In this paper we consider a three components system of nonlinear Schrödinger equations related to the Raman amplification in a plasma. We study the orbital stability of scalar solutions of the form (e2iωtφ,0,0)$(e^{2i\omega t}\phi ,0,0)$, (0,e2iωtφ,0)$(0,e^{2i\omega t}\phi ,0)$, (0,0,e2iωtφ)$(0,0,e^{2i\omega t}\phi )$, where φ is a ground state of the scalar nonlinear Schrödinger equation.     

Profile decompositions of fractional Schrödinger equations with angularly regular data

We study the fractional Schrödinger equations in R^1+d${\mathbb{R}}^{1+d}$, d?3$d?3$, of order d/(d−1)<α<2$d/(d-1)<\alpha <2$. Under the angular regularity assumption we prove linear and nonlinear profile decompositions which extend the previous results [9] to data without radial assumption. As applications we show blowup phenomena of solutions to mass-critical fractional Hartree equations.     

An efficient algorithm for the Schrödinger–Poisson eigenvalue problem

We present a new implementation of the two-grid method for computing extremum eigenpairs of self-adjoint partial differential operators with periodic boundary conditions. A novel two-grid centered difference method is proposed for the numerical solutions of the nonlinear Schrödinger–Poisson (SP) eigenvalue problem.We solve the Poisson equation to obtain the nonlinear potential for the nonlinear Schrödinger eigenvalue problem, and use the block Lanczos method to compute the first k   eigenpairs of the Schrödinger eigenvalue problem until they converge on the coarse grid. Then we perform a few conjugate gradient iterations to solve each symmetric positive definite linear system for the approximate eigenvector on the fine grid. The Rayleigh quotient iteration is exploited to improve the accuracy of the eigenpairs on the fine grid. Our numerical results show how the first few eigenpairs of the Schrödinger eigenvalue problem are affected by the dopant in the Schrödinger–Poisson (SP) system. Moreover, the convergence rate of eigenvalue computations on the fine grid is O(h³)$\mathrm{O}(h^3)$.     

A unified proof on the partial regularity for suitable weak solutions of non-stationary and stationary Navier–Stokes equations

The partial regularity of the suitable weak solutions to the Navier–Stokes equations in Rⁿ${\mathbb{R}}^n$ with n=2,3,4$n=2,3,4$ and the stationary Navier–Stokes equations in Rⁿ${\mathbb{R}}^n$ for n=2,3,4,5,6$n=2,3,4,5,6$ are investigated in this paper. Using some elementary observation of these equations together with De Giorgi iteration method, we present a unified proof on the results of Caffarelli, Kohn and Nirenberg [1], Struwe [17], Dong and Du [5], and Dong and Strain [7]. Particularly, we obtain the partial regularity of the suitable weak solutions to the 4d non-stationary Navier–Stokes equations, which improves the previous result of [5], where Dong and Du studied the partial regularity of smooth solutions of the 4d Navier–Stokes equations at the first blow-up time.     

Low regularity for the nonlinear Klein–Gordon systems

In this paper, we study the global well-posedness below the energy norm of the Cauchy problem for the Klein–Gordon system in R³${\mathbb{R}}^3$. We prove the H^s$H^s$-global well-posedness with s<1$s<1$ of the Cauchy problem for the Klein–Gordon system. The method invoked is different from the well-known Bourgain’s method [Jean Bourgain, Refinements of Strichartz’s inequality and applications to 2D-NLS with critical nonlinearity, International Mathematial Research Notices 5 (1998) 253–283].     

Asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term

This paper deals with the asymptotic behavior of strong solutions to the 3D Navier–Stokes equations with a nonlinear damping term |u|^β−1u(β≥3)${\left|u\right|}^{\beta -1}u(\beta \ge 3)$. First, we establish an upper bound for the difference between the solution of our equation and the heat equation in L²$L^2$ space. Then, we optimize the upper bound of decay for the solutions and obtain their algebraic lower bound by using Fourier Splitting method.     

Renormalized solutions for nonlinear elliptic equations with variable exponents and L data

We prove existence and uniqueness of a renormalized solution to nonlinear elliptic equations with variable exponents and L¹$L^1$ data. The functional setting involves Lebesgue–Sobolev space with variable exponents W^1,p(⋅)(Ω)$W^{1,p(\cdot )}\left(\Omega \right)$.     

Compact imbeddings between variable exponent spaces with unbounded underlying domain

In this paper, we have established some compact imbedding theorems for some subspaces of W^1,p(x)(U)$W^{1,p\left(x\right)}\left(U\right)$ when the underlying domain U$U$ is unbounded. The domain we consider is mainly of type R^N(N≥2)$R^N(N\ge 2)$ or R^L×Ω(L≥2)$R^L\times \Omega (L\ge 2)$, where Ω⊂R^M$\Omega \subset R^M$ is a bounded domain with smooth boundary.     

Global solutions to the eikonal equation

We study structural stability of smoothness of the maximal solution to the geometric eikonal equation on (R^d,G)$({\mathbb{R}}^d,G)$, d?2$d?2$. This is within the framework of order zero metrics G. For a subclass of these metrics we show existence, stability as well as precise asymptotics for derivatives of the solution. These results are applicable to examples arising in Schrödinger operator theory.     

Asymptotic profiles for wave equations with strong damping

We consider the Cauchy problem in Rⁿ${\mathbf{R}}^n$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted L^1,1(Rⁿ)$L^{1,1}({\mathbf{R}}^n)$ data by using a method introduced in [9] and/or [10].     

Existence and nonlinear stability of stationary solutions to the full compressible Navier–Stokes–Korteweg system

This paper is concerned with the existence, uniqueness, and nonlinear stability of stationary solutions to the Cauchy problem of the full compressible Navier–Stokes–Korteweg system effected by the given mass source, the external force of general form, and the energy source in R³${\mathbb{R}}^3$. Based on the weighted L²$L^2$-method and some delicate L^∞$L^{\infty }$ estimates on solutions to the linearized problem, the existence and uniqueness of stationary solution are obtained by the contraction mapping principle. The proof of the stability result is given by an elementary energy method and relies on some intrinsic properties of the full compressible Navier–Stokes–Korteweg system.     

Soliton solutions for quasilinear Schrödinger equations: The critical exponential case

Quasilinear elliptic equations in R²${\mathbb{R}}^2$ of second order with critical exponential growth are considered. By using a change of variable, the quasilinear equations are reduced to semilinear equations, whose respective associated functionals are well defined in H¹(R²)$H^1\left({\mathbb{R}}^2\right)$ and satisfy the geometric hypotheses of the mountain pass theorem. Using this fact, we obtain a Cerami sequence converging weakly to a solution v$v$. In the proof that v$v$ is nontrivial, the main tool is the concentration–compactness principle [P.L. Lions, The concentration compactness principle in the calculus of variations. The locally compact case. Part I and II, Ann. Inst. H. Poincaré Anal. Non. Linéaire 1 (1984) 109–145, 223–283] combined with test functions connected with optimal Trudinger–Moser inequality.     

Existence and optimal decay rates of the compressible non-isentropic Navier–Stokes–Poisson models with external forces

In this paper, we consider the three dimensional compressible non-isentropic Navier–Stokes–Poisson equations with the potential external force. Under the smallness assumption of the external force in some Sobolev space, the existence of the stationary solution is established by solving a nonlinear elliptic system. Next, we show global well-posedness of the initial value problem for the three dimensional compressible non-isentropic Navier–Stokes–Poisson equations, provided the prescribed initial data is close to the stationary solution. Finally, based on the elaborate energy estimates for the nonlinear system and L²$L^2$-decay estimates for the semigroup generated by the linearized equation, we give the optimal L²$L^2$-convergence rates of the solutions toward the stationary solution.     

Nonlinear Schrödinger equation with a point defect

We study the nonlinear Schrödinger equation with a delta-function impurity in one space dimension. Local well-posedness is verified for the Cauchy problem in H¹(R)$H^1(\mathbb{R})$. In case of attractive delta-function, orbital stability and instability of the ground state is proved in H¹(R)$H^1(\mathbb{R})$.