Related papers: The Cauchy problem for the Pavlov equation
We study the Cauchy problem for the generalized KdV and one-dimensional fourth-order derivative nonlinear Schr\"odinger equations, for which the global well-posedness of solutions with the small rough data in certain scaling limit of…
In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on $\T \times \R$ and $\T \times \R^2$. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the…
In this paper we present some open problems pertaining to the approximation theory involved in the solution of the important class of Nonlinear Partial Differential Equations (NPDEs) of integrable type. For this class of NPDEs, any Initial…
We investigate the Cauchy problem for linear, constant-coefficient evolution PDEs on the real line with discontinuous initial conditions (ICs) in the small-time limit. The small-time behavior of the solution near discontinuities is…
In mathematical physics it is of interest to study Schr\"odinger equations with friction and possessing an invariant measure. The focus of this paper is the Cauchy problem for the Schr\"odinger equation $\p_t f - i \mathscr L f = 0$, where…
In this paper we consider a one-dimensional diffusion equation on the interval $[0,1]$ satisfying non-Feller boundary conditions. As a consequence, the initial value Cauchy problem fails to preserve nonnegativity or boundedness.…
In this work, we study the Cauchy problem for a class of dispersive PDEs where a rough time coefficient is present in front of the dispersion. Under minimal assumptions on the occupation measure of this coefficient, we show that for the…
We analyze the system of equations describing the flow of a dilute particle system coupled with an incompressible non-Newtonian fluid in a bounded domain. In this setting, both PDEs are connected via a drag force, or the friction force. We…
As is known, the problems for the differential equations with continuously changing order of the derivatives are not considered completely. In this paper we consider the initial and boundary value problems for this type of linear ordinary…
The global existence of strong solutions to the compressible viscous magnetohydrodynamic (MHD) equations in $\mathbb{R}^3$ remains a significant open problem. When there is no magnetic diffusion, even small data global well-posedness is…
Using the continuous limit approximation in the dynamical system we study a nonlinear partial differential equation which corresponds to the generalization of both the Fermi-Pasta-Ulam and the Frenkel-Kontorova models. This generalized…
We consider inverse boundary value problems for general real principal type differential operators. The first results state that the Cauchy data set uniquely determines the scattering relation of the operator and bicharacteristic ray…
We consider the Cauchy problem to the 3D fractional Schr\"odinger equation with quadratic interaction of $u\bar u$ type. We prove the global existence of solutions and scattering properties for small initial data. For the proof, one novelty…
We study the problem of existence, uniqueness and regularity of probabilistic solutions of the Cauchy problem for nonlinear stochastic partial differential equations involving operators corresponding to regular (nonsymmetric) Dirichlet…
The Cauchy problem for the inelastic Boltzmann equation is studied for small data. Existence and uniqueness of mild and weak solutions is obtained for sufficiently small data that lies in the space of functions bounded by Maxwellians. The…
In this paper, we consider the Cauchy problem for semi-linear wave equations with structural damping term $\nu (-\Delta)^2 u_t$, where $\nu >0$ is a constant. As being mentioned in [8,10], the linear principal part brings both the diffusion…
We consider a Cauchy problem for an overdetermined system of PDEs, and give necessary and sufficient conditions for solvability of this Cauchy problem for all data. As an application, we find all real tube hypersurfaces in complex space…
The forward problem here is the Cauchy problem for a 1D hyperbolic PDE with a variable coefficient in the principal part of the operator. That coefficient is the spatially distributed dielectric constant. The inverse problem consists of the…
This paper focuses on Cauchy problem for the three-dimensional two-fluid type model, in which the presence of vacuum is permitted. Under some assumptions that the initial data satisfy appropriate regularity conditions and a compatibility…
The recent significant enrichment of the Order Completion Method for nonlinear Systems of PDEs resulted in the global existence of generalized solutions to a large class of such equations. In this paper we investigate the existence and…