Maximal Regularity for Non-autonomous Evolutionary Equations

Trostorff, Sascha; Waurick, Marcus

doi:10.1007/s00020-021-02645-5

Cited by 4 publications

(1 citation statement)

References 27 publications

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“…Second, the formulation in spaces of higher spatial regularity allows for a wider class of nonlinearities and complements previous work on spatial regularity for evolutionary equations, see [PTW17] and also [TW21].…”

Section: Introductionmentioning

confidence: 53%

Well-Posedness and Exponential Stability of Nonlinear Maxwell Equations for Dispersive Materials with Interface

Dohnal¹,

Ionescu-Tira²,

Waurick³

2023

Preprint

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In this paper we consider an abstract Cauchy problem for a Maxwell system modelling electromagnetic fields in the presence of an interface between optical media. The electric polarization is in general time-delayed and nonlinear, turning the macroscopic Maxwell equations into a system of nonlinear integro-differential equations. Within the framework of evolutionary equations, we obtain well-posedness in function spaces exponentially weighted in time and of different spatial regularity and formulate various conditions on the material functions, leading to exponential stability on a bounded domain.

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Section: Introductionmentioning

confidence: 53%

Well-Posedness and Exponential Stability of Nonlinear Maxwell Equations for Dispersive Materials with Interface

Dohnal¹,

Ionescu-Tira²,

Waurick³

2023

Preprint

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Maximal Regularity

Seifert

Trostorff

Waurick

2021

Evolutionary Equations

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In this chapter, we address the issue of maximal regularity. More precisely, we provide a criterion on the ‘structure’ of the evolutionary equation $$\displaystyle \left (\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}\right )U=F $$ ∂ t , ν M ( ∂ t , ν ) + A ¯ U = F in question and the right-hand side F in order to obtain $$U\in \operatorname {dom}(\partial _{t,\nu }M(\partial _{t,\nu }))\cap \operatorname {dom}(A)$$ U ∈ dom ( ∂ t , ν M ( ∂ t , ν ) ) ∩ dom ( A ) . If $$F\in L_{2,\nu }(\mathbb {R};H)$$ F ∈ L 2 , ν ( ℝ ; H ) , $$U\in \operatorname {dom}(\partial _{t,\nu }M(\partial _{t,\nu }))\cap \operatorname {dom}(A)$$ U ∈ dom ( ∂ t , ν M ( ∂ t , ν ) ) ∩ dom ( A ) is the optimal regularity one could hope for. However, one cannot expect U to be as regular since $$\left (\partial _{t,\nu }M(\partial _{t,\nu })+A\right )$$ ∂ t , ν M ( ∂ t , ν ) + A is simply not closed in general. Hence, in all the cases where $$\left (\partial _{t,\nu }M(\partial _{t,\nu })+A\right )$$ ∂ t , ν M ( ∂ t , ν ) + A is not closed, the desired regularity property does not hold for $$F\in L_{2,\nu }(\mathbb {R};H)$$ F ∈ L 2 , ν ( ℝ ; H ) . However, note that by Picard’s theorem, $$F\in \operatorname {dom}(\partial _{t,\nu })$$ 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 $$U\in \operatorname {dom}(\partial _{t,\nu })\cap \operatorname {dom}(A)$$ 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 $$F\in \operatorname {dom}(\partial _{t,\nu })$$ F ∈ dom ( ∂ t , ν ) nor $$F\in L_{2,\nu }(\mathbb {R};H)$$ F ∈ L 2 , ν ( ℝ ; H ) yields precisely the regularity $$U\in \operatorname {dom}(\partial _{t,\nu }M(\partial _{t,\nu }))\cap \operatorname {dom}(A)$$ U ∈ dom ( ∂ t , ν M ( ∂ t , ν ) ) ∩ dom ( A ) since $$\displaystyle \operatorname {dom}(\partial _{t,\nu })\cap \operatorname {dom}(A)\subseteq \operatorname {dom}(\partial _{t,\nu }M(\partial _{t,\nu }))\cap \operatorname {dom}(A)\subseteq \operatorname {dom}(\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}), $$ dom ( ∂ t , ν ) ∩ dom ( A ) ⊆ dom ( ∂ t , ν M ( ∂ t , ν ) ) ∩ dom ( A ) ⊆ dom ( ∂ t , ν M ( ∂ t , ν ) + 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.

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Maximal Regularity for Non-autonomous Evolutionary Equations

Trostorff¹,

Waurick

2021

Integr. Equ. Oper. Theory

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We discuss the issue of maximal regularity for evolutionary equations with non-autonomous coefficients. Here evolutionary equations are abstract partial-differential algebraic equations considered in Hilbert spaces. The catch is to consider time-dependent partial differential equations in an exponentially weighted Hilbert space. In passing, one establishes the time derivative as a continuously invertible, normal operator admitting a functional calculus with the Fourier–Laplace transformation providing the spectral representation. Here, the main result is then a regularity result for well-posed evolutionary equations solely based on an assumed parabolic-type structure of the equation and estimates of the commutator of the coefficients with the square root of the time derivative. We thus simultaneously generalise available results in the literature for non-smooth domains. Examples for equations in divergence form, integro-differential equations, perturbations with non-autonomous and rough coefficients as well as non-autonomous equations of eddy current type are considered.

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Maximal Regularity for Non-autonomous Evolutionary Equations

Cited by 4 publications

References 27 publications

Well-Posedness and Exponential Stability of Nonlinear Maxwell Equations for Dispersive Materials with Interface

Well-Posedness and Exponential Stability of Nonlinear Maxwell Equations for Dispersive Materials with Interface

Maximal Regularity

Maximal Regularity for Non-autonomous Evolutionary Equations

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