Related papers: Solving pseudo-differential equations
For a principal type pseudodifferential operator, we prove that condition (psi) implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker's paper on the proof of the Nirenberg-Treves conjecture and we provide…
In this paper we give a proof of the Nirenberg-Treves conjecture: that local solvability of principal type pseudo-differential operators is equivalent to condition ($\Psi$). This condition rules out certain sign changes of the imaginary…
The paper studies the solvability for square systems of pseudodifferential operators. We assume that the system is of principal type, i.e., the principal symbol vanishes of first order on the kernel. We shall also assume that the…
In this paper we show that condition $\operatorname{Sub_r}(\Psi)$ on the subprincipal symbol is sufficient for local solvability of linear pseudodifferential operators of real subprincipal type. These are the operators having real principal…
The paper studies the local solvability and subellipticity for square systems of principal type. These are the systems for which the principal symbol vanishes of first order on its kernel. For systems of principal type having constant…
We consider the derivative nonlinear Schr\"odinger equation on the real line, with a background function $\psi(t,x)\in L^\infty(\mathbb{R}^2)$ that satisfies suitable conditions. Such a function may, for example, be a non-decaying solution…
Spatial differentiability of solutions of stochastic differential equations (SDEs) is a classical question in stochastic analysis. The case of coefficients with globally Lipschitz continuous derivatives is well understood in the literature.…
We provide sufficient conditions of local solvability for partial differential operators with variable Colombeau coefficients. We mainly concentrate on operators which admit a right generalized pseudodifferential parametrix and on operators…
We prove results on solvability of nonlinear elliptic partial differential systems of principle type of second order. They are consequences of existence of non-radial solutions for nonlinear partial differential systems of Poisson type. As…
One of the simplified solutions of the Dirac equations with the pure Coulomb potential given in a paper published in 1985 is pseudo. The original paper solved the Dirac equations by introducing a transformation of functions with two strange…
This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic differential operators $L,$ defined, say, in an open set $\Om\subset \RR^n.$ Suppose the principal symbol $p_k$ of $L$…
The purpose of this paper is to obtain microlocal analogues of results by L. H \"ormander about inclusion relations between the ranges of first order differential operators with coefficients in $C^\infty$ which fail to be locally solvable.…
Olver and Rosenau studied group-invariant solutions of (generally nonlinear) partial differential equations through the imposition of a side condition. We apply a similar idea to the special case of finite-dimensional Hamiltonian systems,…
An astonishing fact was established by Lee A. Rubel (1981): there exists a fixed non-trivial fourth-order polynomial differential algebraic equation (DAE) such that for any positive continuous function $\varphi$ on the reals, and for any…
A solution $\psi $ to Schr\"odinger's equation needs some degree of regularity in order to allow the construction of a Bohmian mechanics from the integral curves of the velocity field $\hbar \Im \left( \bigtriangledown \psi /m\psi \right)…
In this paper, the successive approximation method is applied to investigate the existence and uniqueness of solutions to the stochastic differential equations (SDEs) driven by L\'evy noise under non-Lipschitz condition which is a much…
This is a the first in a series of two articles devoted to the question of local solvability of doubly characteristic second order differential operators. For a large class of such operators, we show that local solvability at a given point…
We review $H^{1}$-well-posedness for initial value problems of ordinary differential equations with state-dependent right-hand side. We streamline known approaches to infer existence and uniqueness of solutions for small times given a…
The work considers a system of fractional order partial differential equations. The existence and uniqueness theorems for the classical solution of initial-boundary value problems are proved in two cases: 1) the right-hand side of the…
It is shown that the Cauchy problem for the DNLS equation in the spatially periodic setting is locally well-posed in Sobolev spaces H^s(T) for s \geq 1/2. Moreover, global well-posedness is shown for s \geq 1 and data with small L^2 norm.