Consider an operator equation (*) B(u) − f = 0 in a real Hilbert space. Let us call this equation ill-posed if the operator B (u) is not boundedly invertible, and well-posed otherwise. The DSM (dynamical systems method) for solving equation (*) consists of a construction of a Cauchy problem, which has the following properties: 1) it has a global solution for an arbitrary initial data, 2) this s...

In this article, we prove that the Kolmogorov operator associated to the Burgers equation driven by a space-time white noise is m-dissipative. This implies several properties on the Kolmogorov equation. This result is obtained thanks to the introduction of a modified Kolmogorov operator. New a priori estimates on the solutions of the Burgers equation and on the invariant measure are obtained. T...

This paper is devoted to the study of solvability of an evolution equation obtained as a hydrodynamic limit of a linear Boltzmann equation with inelastic scattering term. An interesting feature of this equation is that it combines the diiusion operator and a singular operator of a kinetic type. The study is carried out in the L 1 space which is natural from physical point of view and allows to ...

This report is concerned with a control problem associated to a semilinear second order ordinary diierential equation with pointwise state constraints. The control acts as a coeecient of the state equation. The nonlinear part of the equation is governed by a Nemytskij operator deened by a nonsmooth function. We prove the existence of optimal controls and obtain a necessary optimality condition ...

The spinless Salpeter equation presents a rather particular differential operator. In this paper we rewrite this equation into integral and integro-differential equations. This kind of equations are well known and can be more easily handled. We also present some analytical results concerning the spinless Salpeter equation and the action of the square-root operator. 03.65.Pm, 02.30.Rz Typeset us...

We prove that a free relativistic rotor (or rotator) Klein-Gordon field is massless. To this end, the magnitude of the centrifugal force and its associated quantum operator for a free particle with a circular motion are determined in Special Relativity. A partial differential equation including the centrifugal force operator is derived. Finally, from the conjunction of the above equation and th...

We show that Fokker-Planck equation for chordal SLE process under a simple rescaling of the probability density can be traced to the minisuperspace Wheeler-de Witt equation for boundary operator in 2d Liouville gravity. Insertion of an operator, calculating SLE critical exponent, corresponds to adding matter contribution to WdW equation. This observation may be useful for understanding of why S...

By the recursion operator of the Kaup–Newell hierarchy we construct the relativistic derivative NLS (RDNLS) equation and the corresponding Lax pair. In the nonrelativistic limit c→∞ it reduces to DNLS equation and preserves integrability at any order of relativistic corrections. The compact explicit representation of the linear problem for this equation becomes possible due to notions of the q-...

A proof of high-order convergence of three deterministic particle methods for the convectiondiffusion equation in two dimensions is presented. The methods are based on discretizations of an integro-differential equation in which an integral operator approximates the diffusion operator. The methods differ in the discretization of this operator. The conditions for convergence imposed on the kerne...