Inviscid limit for the energy-critical complex Ginzburg-Landau equation

In this paper, we consider the limit behavior for the solution of the Cauchy problem of the energy-critical complex Ginzburg-Landau equation in Rn, n?3. In lower dimension case (3?n?6), we show that its solution converges to that of the energy-critical nonlinear Schrödinger equation in , T>0, s=0,1, as a by-product, we get the regularity of solutions in H³ for the nonlinear Schrödinger equation. In higher dimension case (n>6), we get the similar convergent behavior in C(0,T,L²(Rn)). In both cases we obtain the optimal convergent rate.     

Long-time extinction of solutions of some semilinear parabolic equations

We study the long-time behavior of solutions of semilinear parabolic equation of the following type t_∂u−Δu+a₀(x)uq=0 where , d₀>0, 1>q>0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.     

On a paper by Gesztesy, Simon, and Teschl concerning isospectral deformations of ordinary Schrödinger operators

Starting from a selfadjoint Schrödinger operator A=−d²/dx²+q with a gap G in its spectrum F. Gesztesy, B. Simon, G. Teschl [J. Analyse Math. 70 (1996) 267-324] succeed in constructing another Schrödinger operator that is unitarily equivalent (and thus isospectral) to A. As the means they apply come from the Weyl-Titchmarsh theory the connections prove to be intricate, in particular the relation between A and . We show that a central assertion in GST's paper rests substantially on factorizations of the form

     

A refinement of the Strichartz inequality on the saddle and applications

We consider the nonlinear nonelliptic Schrödinger equation defined by with initial datum in L²(R²). We show that if the solution blows up in finite time, then there is a mass concentration phenomenon near the blow-up time. The key ingredient is a refinement of the Strichartz inequality on the saddle.     

Global existence for some radial, low regularity nonlinear Schrödinger equations

We prove the nonlinear Schrödinger equation has a local solution for any energy - subcritical nonlinearity when u₀ is the characteristic function of a ball in Rn. Additionally, we establish the existence of a global solution for n?3 when and α?2. Finally, we establish the existence of a global solution when the initial function is radial, the nonlinear Schrödinger equation has an energy subcritical nonlinearity, and the initial function lies in Hρ+?(Rn)∩H1/2+?(Rn)∩H1/2+?,1(Rn).     

Quantum Hamiltonians with quasi-ballistic dynamics and point spectrum

Consider the family of Schrödinger operators (and also its Dirac version) on ?²(Z) or ?²(N)

     

Local smoothing property and Strichartz inequality for Schrödinger equations with potentials superquadratic at infinity

We study smoothing properties for time-dependent Schrödinger equations , , with potentials which satisfy V(x)=O(|x|^m) at infinity, m?2. We show that the solution u(t,x) is 1/m times differentiable with respect to x at almost all , and explain that this is the result of the fact that the sojourn time of classical particles with energy λ in arbitrary compact set is less than CTλ^−1/m during [0,T] when λ is very large. We also show Strichartz's inequality with derivative loss for such potentials and give its application to nonlinear Schrödinger equations.     

Schrödinger operators and unique continuation. Towards an optimal result

In this article we prove the property of unique continuation (also known for C^∞ functions as quasianalyticity) for solutions of the differential inequality |Δu|?|Vu| for V from a wide class of potentials (including class) and u in a space of solutions YV containing all eigenfunctions of the corresponding self-adjoint Schrödinger operator. Motivating question: is it true that for potentials V, for which self-adjoint Schrödinger operator is well defined, the property of unique continuation holds?     

Well-posedness of the fourth-order perturbed Schrödinger type equation in non-isotropic Sobolev spaces

In this paper we study the Cauchy problem of the non-isotropically perturbed fourth-order nonlinear Schrödinger type equation: ((x₁,x₂,…,xn)∈Rn, t?0), where a is a real constant, 1?d<n is an integer, g(x,|u|)u is a nonlinear function which behaves like α|u|u for some constant α>0. By using Kato method, we prove that this perturbed fourth-order Schrödinger type equation is locally well-posed with initial data belonging to the non-isotropic Sobolev spaces provided that s₁,s₂ satisfy the conditions: s₁?0, s₂?0 for or for with some additional conditions. Furthermore, by using non-isotropic Sobolev inequality and energy method, we obtain some global well-posedness results for initial data belonging to non-isotropic Sobolev spaces .     

On the Cauchy problem of 3-D energy-critical Schrödinger equations with subcritical perturbations

We investigate the global well-posedness, scattering and blow up phenomena when the 3-D quintic nonlinear Schrödinger equation, which is energy-critical, is perturbed by a subcritical nonlinearity λ₁p|u|u. We find when the quintic term is defocussing, then the solution is always global no matter what the sign of λ₁ is. Scattering will occur either when the perturbation is defocussing and or when the mass of the solution is small enough and . When the quintic term is focusing, we show the blow up for certain solutions.     

Exact boundary controllability of the nonlinear Schrödinger equation

This paper studies the exact boundary controllability of the semi-linear Schrödinger equation posed on a bounded domain Ω⊂Rn with either the Dirichlet boundary conditions or the Neumann boundary conditions. It is shown that if

     

Asymptotic dynamics of nonlinear Schrödinger equations with many bound states

We consider a nonlinear Schrödinger equation with a bounded localized potential in . The linear Hamiltonian is assumed to have three or more bound states with the eigenvalues satisfying some resonance conditions. Suppose that the initial data is localized and small of order n in H¹, and that its ground state component is larger than n^3−ε with ε>0 small. We prove that the solution will converge locally to a nonlinear ground state as the time tends to infinity.     

Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case

We consider the focusing energy-critical nonlinear Schrödinger equation of fourth order , d?5. We prove that if a maximal-lifespan radial solution obeys supt∈I‖Δu(t)_‖2<‖ΔW_‖2, then it is global and scatters both forward and backward in time. Here W denotes the ground state, which is a stationary solution of the equation. In particular, if a solution has both energy and kinetic energy less than those of the ground state W at some point in time, then the solution is global and scatters.