Related papers: Low regularity solutions to the logarithmic Schrod…

In this note we reprove the Lipschitz stability for the inverse problem for the Schr\"odinger operator with finite-dimensional potentials by using quantitative Runge approximation results. This provides a quantification of the Schr\"odinger…

We prove inverse Strichartz theorems at $L^2$ regularity for a family of Schr\"{o}dinger evolutions in one space dimension. Prior results rely on spacetime Fourier analysis and are limited to the translation-invariant equation $i\partial_t…

We study the reducibility of a Linear Schr\"odinger equation subject to a small unbounded almost-periodic perturbation which is analytic in time and space. Under appropriate assumptions on the smallness, analiticity and on the frequency of…

We consider the logarithmic Schr{\"o}dinger equations with damping, also called Schr{\"o}dinger-Langevin equation. On a periodic domain, this equation possesses plane wave solutions that are explicit. We prove that these solutions are…

In this paper, we first establish regularity of the heat flow of biharmonic maps into the unit sphere $S^L\subset\mathbb R^{L+1}$ under a smallness condition of renormalized total energy. For the class of such solutions to the heat flow of…

In several contexts the defining invariant structures of a hyperbolic dynamical system are smooth only in systems of algebraic origin (smooth rigidity), and we prove new results of this type for a class of flows. For a compact Riemannian…

The Riemann-Hilbert approach is extended to discuss the well-posedness of the nonlinear Schr\"odinger-Gerdjikov-Ivanon equation. The Lipschitz continuity of potential in $H^{2}(\mathbb{R})\cap H^{1,1}(\mathbb{R})$ to scattering data is…

We solve the Schr{\"o}dinger equation with logarithmic nonlinearity and multiplicative spatial white noise on R d with d $\le$ 2. Because of the nonlinearity, the regularity structures and the paracontrolled calculus can not be used. To…

Given any continuous, lower bounded and $\kappa$-convex function $V$ on a metric measure space $(X,d,m)$ which is infinitesimally Hilbertian and satisfies some synthetic lower bound for the Ricci curvature in the sense of…

In 1997, J. Jost [27] and F. H. Lin [39], independently proved that every energy minimizing harmonic map from an Alexandrov space with curvature bounded from below to an Alexandrov space with non-positive curvature is locally H\"older…

Two notions of "having a derivative of logarithmic order" have been studied. They come from the study of regularity of flows and renormalized solutions for the transport and continuity equation associated to weakly differentiable drifts.

We prove a Lipschitz extension lemma in which the extension procedure simultaneously preserves the Lipschitz continuity for two non-equivalent distances. The two distances under consideration are the Euclidean distance and, roughly…

In this paper, we introduce a new notion named as Schr\"odinger soliton. So-called Schr\"odinger solitons are defined as a class of special solutions to the Schr\"odinger flow equation from a Riemannian manifold or a Lorentzian manifold $M$…

The nonlinear wave equation $u_{tt}-c(u)(c(u)u_x)_x=0$ determines a flow of conservative solutions taking values in the space $H^1(\mathbb{R})$. However, this flow is not continuous w.r.t. the natural $H^1$ distance. Aim of this paper is to…

We establish the sharp rate of continuity of extensions of $\mathbb{R}^m$-valued $1$-Lipschitz maps from a subset $A$ of $\mathbb{R}^n$ to a $1$-Lipschitz maps on $\mathbb{R}^n$. We consider several cases when there exists a $1$-Lipschitz…

We study the gradient flow of the length functional on the space of planar immersed closed curves, where the gradient is taken with respect to a family of homogeneous Sobolev $H^1$-type Riemannian metrics depending on parameters $\lambda>0$…

For some class of geometric flows, we obtain the (logarithmic) Sobolev inequalities and their equivalence up to different factors directly and also obtain the long time non-collapsing and non-inflated properties, which generalize the…

Using a rough path formulation, we investigate existence, uniqueness and regularity for the stochastic Landau-Lifshitz-Gilbert equation with Stratonovich noise on the one dimensional torus. As a main result we show the continuity of the…

This paper studies the regularity of solutions to the Zakharov and Klein-Gordon-Schr\"{o}dinger systems at low regularity levels. The main result is that the nonlinear part of the solution flow falls in a smoother space than the initial…

The dynamics of Schr\"odinger equation with time dependent potentials of general time dependence is considered. It is shown that for localized in space potentials, there is propagation of regularity which is uniformly bounded in higher…