Stabilization via Homogenization

In this short note we treat a 1+1-dimensional system of changing type. On different spatial domains the system is of hyperbolic and elliptic type, that is, formally, $\partial_t^2 u_n-\partial_x^2 u_n = \partial_t f$ and $u_n-\partial_x^2 u_n= f$ on the respective spatial domains $\bigcup_{j\in \{1,\ldots,n\}} \big(\frac{j-1}{n},\frac{2j-1}{2n}\big)$ and $\bigcup_{j\in \{1,\ldots,n\}} \big(\frac{2j-1}{2n},\frac{j}{n}\big)$. We show that $(u_n)_n$ converges weakly to $u$, which solves the exponentially stable limit equation $\partial_t^2 u +2\partial_t u + u -4\partial_x^2 u = 2(f+\partial_t f)$ on $[0,1]$. If the elliptic equation is replaced by a parabolic one, the limit equation is \emph{not} exponentially stable.Comment: 8 pages; some typos corrected; referee's comments incorporate

Well-posedness via Monotonicity. An Overview

The idea of monotonicity (or positive-definiteness in the linear case) is shown to be the central theme of the solution theories associated with problems of mathematical physics. A "grand unified" setting is surveyed covering a comprehensive class of such problems. We elaborate the applicability of our scheme with a number examples. A brief discussion of stability and homogenization issues is also provided.Comment: Thoroughly revised version. Examples correcte

A solution theory for a general class of SPDEs

In this article we present a way of treating stochastic partial differential equations with multiplicative noise by rewriting them as stochastically perturbed evolutionary equations in the sense of Picard and McGhee (Partial differential equations: a unified Hilbert space approach, DeGruyter, Berlin, 2011), where a general solution theory for deterministic evolutionary equations has been developed. This allows us to present a unified solution theory for a general class of stochastic partial differential equations (SPDEs) which we believe has great potential for further generalizations. We will show that many standard stochastic PDEs fit into this class as well as many other SPDEs such as the stochastic Maxwell equation and time-fractional stochastic PDEs with multiplicative noise on sub-domains of RdRd. The approach is in spirit similar to the approach in DaPrato and Zabczyk (Stochastic equations in infinite dimensions, Cambridge University Press, Cambridge, 2008), but complementing it in the sense that it does not involve semi-group theory and allows for an effective treatment of coupled systems of SPDEs. In particular, the existence of a (regular) fundamental solution or Green’s function is not required