Maximal regularity

Abstract

In this chapter, we address the issue of maximal regularity. More precisely, we provide a criterion on the ‘structure’ of the evolutionary equation (∂t,νM(∂t,ν)+A¯)U=F $$\displaystyle \left (\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}\right )U=F $$ in question and the right-hand side F in order to obtain U∈ dom (∂t,νM(∂t,ν) ) ∩ dom (A). If F∈ L2,ν(ℝ; H), U∈ dom (∂t,νM(∂t,ν) ) ∩ dom (A) is the optimal regularity one could hope for. However, one cannot expect U to be as regular since (∂t,νM(∂t,ν) + A) is simply not closed in general. Hence, in all the cases where (∂t,νM(∂t,ν) + A) is not closed, the desired regularity property does not hold for F∈ L2,ν(ℝ; H). However, note that by Picard’s theorem, F∈ dom (∂t,ν) implies the desired regularity property for U given the positive definiteness condition for the material law is satisfied and A is skew-selfadjoint. In this case, one even has U∈ dom (∂t,ν) ∩ dom (A), which is more regular than expected. Thus, in the general case of an unbounded, skew-selfadjoint operator A neither the condition F∈ dom (∂t,ν) nor F∈ L2,ν(ℝ; H) yields precisely the regularity U∈ dom (∂t,νM(∂t,ν) ) ∩ dom (A) since dom(∂t,ν)∩dom(A)⊆dom(∂t,νM(∂t,ν))∩dom(A)⊆dom(∂t,νM(∂t,ν)+A¯), $$\displaystyle \operatorname {dom}(\partial _{t,\nu })\cap \operatorname {dom}(A)\subseteq \operatorname {dom}(\partial _{t,\nu }M(\partial _{t,\nu }))\cap \operatorname {dom}(A)\subseteq \operatorname {dom}(\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}), $$ where the inclusions are proper in general. It is the aim of this chapter to provide an example case, where less regularity of F actually yields more regularity for U. If one focusses on time-regularity only, this improvement of regularity is in stark contrast to the general theory developed in the previous chapters. Indeed, in this regard, one can coin the (time) regularity asserted in Picard’s theorem as “U is as regular as F”. For a more detailed account on the usual perspective of maximal regularity (predominantly) for parabolic equations, we refer to the Comments section of this chapter.