Related papers: Generalized Cauchy-Riemann equations and relevant …
This article is devoted to the Cauchy problem for the 2D gravity-capillary water waves in fluid domains with general bottoms. We prove that the Cauchy problem in Sobolev spaces is uniquely solvable for data $\frac{1}{4}$ derivatives less…
We give here results on the existence of nonclassical solutions of the Hilbert boundary value problem in terms of the so-called angular limits (along nontangent curves to the boundary) for Beltrami equations with sources in Jordan domains…
A recent analysis of real general relativity based on multisymplectic techniques has shown that boundary terms may occur in the constraint equations, unless some boundary conditions are imposed. This paper studies the corresponding form of…
We prove global existence of a derivative bi-harmonic wave equation with a non-generic quadratic nonlinearity and small initial data in the scaling critical space $\dot{B}^{2,1}_{\frac{d}{2}}(\mathbb{R}^d) \times…
This paper investigates functional equations arising from perturbations of Cauchy differences. We study equations of the form \[ f(x+y)-f(x)-f(y)=B(x,y) \quad \text{or} \quad f(xy)-f(x)f(y) = B(x,y) \] where $B$ is a biadditive mapping, and…
We study the Cauchy problem for a general homogeneous linear partial differential equation in two complex variables with constant coefficients and with divergent initial data. We state necessary and sufficient conditions for the summability…
In this paper we revisit the classical Cauchy problem for Laplace's equation as well as two further related problems in the light of regularisation of this highly ill-conditioned problem by replacing integer derivatives with fractional…
In this paper, we study the Cauchy problem for a generalized integrable Camassa-Holm equation with both quadratic and cubic nonlinearity. By overcoming the difficulties caused by the complicated mixed nonlinear structure, we firstly…
We consider the global Cauchy problem for the generalized incompressible Navier- Stokes system in 3D whole space $$ u_t+u\cdot\nabla u+\nabla p=\mathcal{A}_h u, $$ \begin{equation}\label{main0} \nabla\cdot u=0, \end{equation} $$…
We prove multipolar Hardy inequalities on complete Riemannian manifolds, providing various curved counterparts of some Euclidean multipolar inequalities due to Cazacu and Zuazua [Improved multipolar Hardy inequalities, 2013]. We notice that…
We prove maximal Schauder regularity for solutions to elliptic systems and Cauchy problems, in the space $C_b(\mathbb{R}^d;\mathbb{R}^m)$ of bounded and continuous functions, associated to a class of nonautonomous weakly coupled…
In this paper, we study a class of variable coefficient Schr\"{o}dinger equations with a linear potential \[i\partial_tu+\nabla\cdot(|x|^b\nabla u)-V(x)u=-|x|^c|u|^pu,\] where $2-n<b\leq0,\ c\geq b-2$ and $0<\textbf{p}_c\leq(2-b)(p+2)$,…
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, regularity properties, Strichartz type estimates to solutions of multipo{\i}nt Cauchy problem for linear and nonlinear abstract wave equations in vector-valued function spaces are obtained. The equation includes a linear…
The paper studies some ill-posed boundary value problems on semi-plane for parabolic equations with homogenuous Cauchy condition at initial time and with the second order Cauchy condition on the boundary of the semi-plane. A class of inputs…
In the present manuscript we consider the Boltzmann equation that models a polyatomic gas by introducing one additional continuous variable, referred to as microscopic internal energy. We establish existence and uniqueness theory in the…
In this paper we investigate some algebraic and geometric consequences which arise from an extremal bound on the Hilbert function of the general hyperplane section of a variety (Green's Hyperplane Restriction Theorem). These geometric…
In this note, we study the potential algebra for several models arising out of quantum mechanics with generalized uncertainty principle. We first show that the eigenvalue equation corresponding to the momentum-space Hamiltonian…
[1] investigates advanced connotations of Hardy and Rellich-type inequalities on complete noncompact Riemannian manifolds, delving on deriving inequalities that incorporate poignant weight functions. These inequalities prolongate classical…
A new approach to the solution of quasilinear nonelliptic first-order systems of inhomogeneous PDEs in many dimensions is presented. It is based on a version of the conditional symmetry and Riemann invariant methods. We discuss in detail…