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We discuss existence, non-uniqueness and regularity of one- and two-sided solutions of initial value problems for scalar quasi-linear ordinary differential equations where the initial condition corresponds to an impasse point of the…
We study a class of nonlinear Burgers-type stochastic partial differential equations driven by additive space-time white noise in one spatial dimension. Building on the rough path framework initiated by Hairer, which provides a pathwise…
We prove local well-posedness for the initial-boundary-value problem associated to some quadratic nonlinear Schr\"odinger equations on the half-line. The results are obtained in the low regularity setting by introducing an analytic family…
We are interested in (uniformly) parabolic PDEs with a nonlinear dependance of the leading-order coefficients, driven by a rough right hand side. For simplicity, we consider a space-time periodic setting with a single spatial variable:…
Initial-boundary value problems for integrable nonlinear partial differential equations have become tractable in recent years due to the development of so-called unified transform techniques. The main obstruction to applying these methods…
This paper is concerned with the initial-boundary value problem \; for stochastic transport equations in bounded domains. For a given stochastic perturbation of the drift vector field, we prove existence and uniqueness of weak solutions…
For a one-dimensional mildly quasilinear wave equation given in the upper half-plane, we consider the Cauchy problem. The initial conditions have discontinuity of the first kind at one point. We construct the solution using the method of…
Two important cases, where boundary conditions and solutions of the well-known integrable equations on a semi-strip are uniquely determined by the initial conditions, are rigorously studied in detail. First, the case of rectangular matrix…
We study an initial value problem for the one-dimensional non-stationary linear Schr\"odinger equation with a point singular potential. In our approach, the problem is considered as a system of coupled initial-boundary value (IBV) problems…
In this article we consider the initial value problem of the binormal flow with initial data given by curves that are regular except at one point where they have a corner. We prove that under suitable conditions on the initial data a unique…
In this paper, we study the initial-boundary value problem of the Navier-Stokes system in the half space. We prove the unique solvability of the weak solution on some short time interval (0, T) with the velocity in $C^{\alpha, \frac12…
An initial-boundary value problem with one boundary condition is considered for the higher order nonlinear Schr\"odinger equation. It is assumed that either the boundary condition is homogeneous or the nonlinearity in the equation is…
We present a method to solve initial-boundary value problems for linear and integrable nonlinear differential-difference evolution equations. The method is the discrete version of the one developed by A. S. Fokas to solve initial-boundary…
We review our work on initial value problems in Quantum Field Theory which is based on using Schwinger's Closed Time Path formalism and a large-N expansion of the Path Integral.
We consider initial-boundary problems for general linear first-order strictly hyperbolic systems with local or nonlocal nonlinear boundary conditions. While boundary data are supposed to be smooth, initial conditions can contain…
Evolution PDEs for dispersive waves are considered in both linear and nonlinear integrable cases, and initial-boundary value problems associated with them are formulated in spectral space. A method of solution is presented, which is based…
We establish local well-posedness for the higher-order nonlinear Schr\"odinger equation, formulated on the half-line. We consider the scenario of associated coefficients such that only one boundary condition is required, which is assumed to…
In the first of two papers, we study the initial boundary-value problem that underlies the theory of the Boltzmann equation for general non-spherical hard particles. In this work, for two congruent ellipses and for a large class of…
In a previous work, we prove the existence of weak solutions to an initial-boundary value problem, with $H^1(\Omega)$ initial data, for a system of partial differential equations, which consists of the equations of linear elasticity and a…
We prove the existence and uniqueness of solution of the obstacle problem for quasilinear stochastic partial differential equations (OSPDEs for short) with Neumann boundary condition. Our method is based on the analytical technics coming…