Related papers: Smoothing and growth bound of periodic generalized…
We prove some smoothing effects for the 3-D Navier-Stokes equations for initial data belonging to the critical Sobolev space $H^{1/2}(\R^3)$. Asymptotic behavior of the global solution when the time goes to infinity is studied. We also…
We study the one-dimensional nonlinear Klein-Gordon (NLKG) equation with a convolution potential, and we prove that solutions with small $H^s$ norm remain small for long times. The result is uniform with respect to $c \geq 1$, which however…
We consider the semiclassical Schr\"odinger equation on $\mathbb R^d$ given by $$\mathrm{i} \hbar \partial_t \psi = \left(-\frac{\hbar^2}{2} \Delta + W_l(x) \right)\psi + V(t,x)\psi ,$$ where $W_l$ is an anharmonic trapping of the form…
We consider the cubic defocusing nonlinear Schr\"odinger equation on the two dimensional torus. We exhibit smooth solutions for which the support of the conserved energy moves to higher Fourier modes. This weakly turbulent behavior is…
We generalize the nonlinear one-dimensional equation for a fluid layer surface to any geometry and we introduce a new infinite order differential equation for its traveling solitary waves solutions. This equation can be written as a…
Under the uniform H\"{o}rmander's hypothesis we study smoothness and exponential bounds of the density of the law of the solution of a stochastic differential equation (SDE) with locally Lipschitz drift that satisfy a monotonicity…
We study smoothness of densities for the solutions of SDEs whose coefficients are smooth and nondegenerate only on an open domain $D$. We prove that a smooth density exists on $D$ and give upper bounds for this density. Under some…
We prove an abstract result giving a $\langle t \rangle^\varepsilon$ upper bound on the growth of the Sobolev norms of a time-dependent Schr\"odinger equation of the form ${i} \dot \psi = H_0 \psi + V (t)\psi$. Here $H_0$ is assumed to be…
We study the modulated Korteweg-de~Vries equation (KdV) on the circle with a time non-homogeneous modulation acting on the linear dispersion term. By adapting the normal form approach to the modulated setting, we prove sharp unconditional…
In this paper, we study a quadratic equation on the one-dimensional torus : $$i \partial_t u = 2J\Pi(|u|^2)+\bar{J}u^2, \quad u(0, \cdot)=u_0,$$ where $J=\int_\mathbb{T}|u|^2u \in\mathbb{C}$ has constant modulus, and $\Pi$ is the Szeg\H{o}…
We give a detailed description in 1-D the growth of Sobolev norms for time dependent linear generalized KdV-type equations on the circle. For most initial data, the growth of Sobolev norms is polynomial in time for fixed analytic potential…
We prove logarithmic growth bounds on Sobolev norms of the focusing mass-critical NLS and gKdV equations on the torus, which hold almost surely under the focusing Gibbs measure with optimal mass threshold constructed by Oh, Sosoe, and…
We utilize a modulation restricted normal form approach to establish local well-posedness of the periodic Korteweg-de Vries equation in $H^s(\mathbb{T})$ for $s> -\frac23$. This work creates an analogue of the mKdV result by Nakanishi,…
We consider a randomly perturbed Korteweg-de Vries equation. The perturbation is a random potential depending both on space and time, with a white noise behavior in time, and a regular, but stationary behavior in space. We investigate the…
For the solution $q(t)=(q_n(t))_{n\in\mathbb Z}$ to one-dimensional discrete Schr\"odinger equation $${\rm i}\dot{q}_n=-(q_{n+1}+q_{n-1})+ V(\theta+n\omega) q_n, \quad n\in\mathbb Z,$$ with $\omega\in\mathbb R^d$ Diophantine, and $V$ a…
We study local and global existence and smoothing properties for the initial value problem associated to a higher order nonlinear Schr\"odinger equation with constant coefficients which appears as a model for propagation of pulse in optical…
We develop an optimal regularity theory for $L^p$-viscosity solutions of fully nonlinear uniformly elliptic equations in nondivergence form whose gradient growth is described through a Hamiltonian function with measurable and possibly…
The paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type…
We construct a family of smooth initial data for the Navier-Stokes equations, bounded in $BMO^{-1}(\mathbb T^3)$, that gives rise to arbitrarily large global solutions. As a consequence, we rule out various hypothetical a priori estimates…
We consider the problem of establishing nonlinear smoothing as a general feature of nonlinear dispersive equations, i.e. the improved regularity of the integral term in Duhamel's formula, with respect to the initial data and the…