Related papers: On the Local Smoothing for a class of conformally …
We prove a family of identities that involve the solutions to the free Schreodinger equation. As a consequence of these identities we shall deduce a lower bound for the local smoothing estimate and a uniqueness criterion.
We construct a local in time, exponentially decaying solution of the one-dimensional variable coefficient Schrodinger equation by solving a nonstandard boundary value problem. A main ingredient in the proof is a new commutator estimate…
We prove a local smoothing result for the Schr\"odinger equation on a class of surfaces of revolution which have infinitely many trapped geodesics. Our main result is a local smoothing estimate with loss (compared to \cite{ChMe-lsm})…
We consider a family of surfaces of revolution, each with a single periodic geodesic which is degenerately unstable. We prove a local smoothing estimate for solutions to the linear Schr\"odinger equation with a loss that depends on the…
We prove a local in time smoothing estimate for a magnetic Schrodinger equation with coefficients growing polynomially at spatial infinity. The assumptions on the magnetic field are gauge invariant and involve only the first two…
The results of this paper are twofold. One is that we show the local existence and uniqueness of very regular or smooth solution to the initial-Neumann boundary value problem of the Schr\"{o}dinger flow for maps from a smooth bounded domain…
We consider nonlocal initial boundary value problems with integral boundary conditions for integro-differential first order hyperbolic systems. We prove a general regularity result stating that the $L^2$-generalized solutions become…
In this work we shall review some of our recent results concerning unique continuation properties of solutions of Schr\"odinger equations. In this equations we include linear ones with a time depending potential and semi-linear ones.
This paper concerns local gradient estimates to solutions of general conformally invariant fully nonlinear elliptic equations of second order.
Using first and second order supersymmetry formalism we obtain a class of exactly solvable potentials subject to moving boundary conditions.
We study solutions to conformally invariant equations with isolated singularties.
This work is devoted to the study of non-Newtonian fluids of grade three on two-dimensional and three-dimensional bounded domains, driven by a nonlinear multiplicative Wiener noise. More precisely, we establish the existence and uniqueness…
We study local, analytic solutions for a class of initial value problems for singular ODEs. We prove existence and uniqueness of such solutions under a certain non-resonance condition. Our proof translates the singular initial value problem…
We relax the regularity condition on potentials of the Schr\"odinger equation in uniqueness results on the inverse boundary value problem which were recently proved in [11] and [5].
We prove the existence of C^{\infty} local solutions to a class of mixed type Monge-Ampere equations in the plane. More precisely, the equation changes type to finite order across two smooth curves intersecting transversely at a point.…
We investigate the decay estimates of global solutions for a class of one-dimensional inhomogeneous nonlinear Schr\"odinger equations. While most existing results focus on spatial dimensions $d\geq2$, the decay properties in one dimension…
We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local…
In this paper we study microlocal singularities of solutions to Schrodinger equations on scattering manifolds, i.e., noncompact Riemannian manifolds with asymptotically conic ends. We characterize the wave front set of the solutions in…
We describe a certain "self-similar" family of solutions to the free Schroedinger equation in all dimensions, and derive some consequences of such solutions for two specific problems.
We present some lower bounds for regular solutions of Schr\"odinger equations with bounded and time dependent complex potentials. Assuming that the solution has some positive mass at time zero within a ball of certain radius, we prove that…