We consider a non degenerate quasilinear parabolic stochastic partial differential equation with a uniformly elliptic diffusion matrix. It is driven by a nonlinear noise. We study regularity properties of its weak solution satisfying classical a priori estimates. In particular, we determine conditions on coefficients and initial data under which the weak solution is Hölder continuous in time an...

For sufficiently small perturbations with continuous derivative, we show how to establish the optimal regularity of invariant center manifolds when the linear equation admits a nonuniform exponential trichotomy. We also consider the general case of exponential growth rates given by an arbitrary function. This includes the usual exponential behavior as a very special case. Our proof uses the fib...

The linear quadratic optimal regulator problem on a wave equation in a bounded subset of R for n = 2; 3 with boundary damping and pointwise control input is formulated as a linear quadratic control problem with unbounded input/output operators using a variational approach. A dual control system is de ned which allows us to investigate the regularity of the solution of the generalized wave equat...

Abstract. We give dimension-free regularity conditions for a class of possibly degenerate sub-elliptic equations in the Heisenberg group exhibiting super-quadratic growth in the horizontal gradient; this solves an issue raised in [40], where only dimension dependent bounds for the growth exponent are given. We also obtain explicit a priori local regularity estimates, and cover the case of the h...

For a word S, let f(S) be the largest integer m such that there are two disjoint identical (scattered) subwords of length m. Let f(n,Σ) = min{f(S) : S is of length n, over alphabet Σ}. Here, it is shown that 2f(n, {0, 1}) = n− o(n) using the regularity lemma for words. In other words, any binary word of length n can be split into two identical subwords (referred to as twins) and, perhaps, a rem...

We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces. We obtain our results by proving the global invertibility of linear operators with coefficients in high regularity L2-based function spaces and using iterative arguments for the nonlinear problems....

Recently, a lot of progress has been made on the study for the spatially homogeneous Boltzmann equation without angular cutoff, cf. [2, 3, 8, 22] and references therein, which shows that the singularity of collision cross-section yields some gain of regularity in the Sobolev space frame on weak solutions for Cauchy problem. That means, this gives the C regularity of weak solution for the spatia...