Related papers: Smooth local solutions to weingarten equations and…

We consider wave equations on Lorentzian manifolds in case of low regularity. We first extend the classical solution theory to prove global unique solvability of the Cauchy problem for distributional data and right hand side on smooth…

The study of the $k$-th elementary symmetric function of the Weyl-Schouten curvature tensor of a Riemannian metric, the so called $\sigma_k$ curvature, has produced many fruitful results in conformal geometry in recent years, especially…

We show that for any uniformly parabolic fully nonlinear second-order equation with bounded measurable "coefficients" and bounded "free" term in any cylindrical smooth domain with smooth boundary data one can find an approximating equation…

We consider the Cahn-Hilliard equation on manifolds with conical singularities. For appropriate initial data, we show that the solution exists in the maximal $L^q$-regularity space for all times and becomes instantaneously smooth in space…

We prove existence of smooth solutions to linear degenerate parabolic equations on bounded domains assuming a structure condition of Fichera. We use this to give a proof of a smooth short time existence result for the porous medium equation…

For any positive integer $k$, we prove the existence of nontrivial $C^k$-smooth uniformly rotating solutions to the 2D incompressible Euler equations with compact spatial support. These solutions, which can be chosen to be small…

We prove that small smooth solutions of semi-linear Klein-Gordon equations with quadratic potential exist over a longer interval than the one given by local existence theory, for almost every value of mass. We use normal form for the…

We prove the existence of non-smooth solutions to fully nonlinear uniformly elliptic equations.

In this work,we show the long time existence of smooth solu- tions to semigeostrophic equations on a torus when the initial dual density is bounded between two positive constants and smooth.The key ingredient is a more precise estimate on C…

We study smoothness of generalized solutions of nonlocal elliptic problems in plane bounded domains with piecewise smooth boundary. The case where the support of nonlocal terms can intersect the boundary is considered. We announce…

We propose a probabilistic definition of solutions of semilinear elliptic equations with (possibly nonlocal) operators associated with regular Dirichlet forms and with measure data. Using the theory of backward stochastic differential…

Let $M^n$ be either a simply connected space form or a rank-one symmetric space of noncompact type. We consider Weingarten hypersurfaces of $M\times\mathbb R$, which are those whose principal curvatures $k_1,\dots ,k_n$ and angle function…

In this paper, we give a sufficient condition to guarantee the existence of a smooth solution of the Navier-Stokes Equation with the nice decreasing properties at infinity. In this way, we prove the existence of smooth physically reasonable…

We study the porous medium equation on manifolds with conical singularities. Given strictly positive initial values, we show that the solution exists in the maximal $L^{q}$-regularity space for all times and is instantaneously smooth in…

Smooth K-functors are introduced and the smooth K-theory of locally convex algebras is developed. It is proved that the algebraic and smooth K-functors are isomorphic on the category of quasi stable real (or complex) Frechet algebras.

The local existence of solutions to nonhomogeneous Navier-Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the…

In the present paper, we investigate the regularity and symmetry properties of weak solutions to semilinear elliptic equations which are locally stable.

We consider a system coupling the parabolic-parabolic Keller-Segel equations to the in- compressible Navier-Stokes equations in spatial dimensions two and three. We establish the local existence of regular solutions and present some blow-up…

In this paper, we show the existence of a family of compactly supported smooth vorticities, which are solutions of the 2D incompressible Euler equation and rotate uniformly in time and space.

It is proved that the local smoothing conjecture for wave equations implies certain improvements on Stein's analytic family of maximal spherical means. Some related problems are also discussed.