Well-posedness for a general class of differential inclusions

Trostorff, Sascha

doi:10.1016/j.jde.2019.11.045

Cited by 11 publications

(10 citation statements)

References 33 publications

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“…Hence, A reg j, is monotone. Finally, maximality can now be shown by noting that 1 + A reg j, is surjective and invoking Theorem 1.6 from [37].…”

Section: Mixed-dimensional Poromechanical Modelsmentioning

confidence: 98%

Mixed-dimensional poromechanical models of fractured porous media

Boon¹,

Nordbotten²

2021

Preprint

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We combine classical continuum mechanics with the recently developed calculus for mixed-dimensional problems to obtain governing equations for flow in, and deformation of, fractured materials. We present models both in the context of finite and infinitesimal strain, and discuss non-linear (and non-differentiable) constitutive laws such as friction models and contact mechanics in the fracture. Using the theory of well-posedness for evolutionary equations with maximal monotone operators, we show well-posedness of the model in the case of infinitesimal strain and under certain assumptions on the model parameters.

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“…Hence, A reg j, is monotone. Finally, maximality can now be shown by noting that 1 + A reg j, is surjective and invoking Theorem 1.6 from [37].…”

Section: Mixed-dimensional Poromechanical Modelsmentioning

confidence: 98%

Mixed-dimensional poromechanical models of fractured porous media

Boon¹,

Nordbotten²

2021

Preprint

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“…Due to the flexibility of the choice of the operators M and N , which act in space-time, the problem class comprises many different types of differential equations, like delay equations, fractional differential equations, integro-differential equations and coupled problems thereof (see e.g. [18,22] for some survey in the autonomous case and [20,24,28] for some non-autonomous and/or nonlinear examples).…”

Section: Infimum Taken Over All Appropriate G Yieldsmentioning

confidence: 99%

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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“…Theorem 16.3.1 also has a nonlinear analogue. This can be found in [122]. For an autonomous well-posedness result for nonlinear evolutionary inclusions we also refer to Chap.…”

Section: Commentsmentioning

confidence: 99%

Non-Autonomous Evolutionary Equations

Seifert

Trostorff

Waurick

2021

Evolutionary Equations

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Previously, we focussed on evolutionary equations of the form $$\displaystyle \left (\overline {\partial _{t,\nu }M(\partial _{t,\nu })+A}\right )U=F. $$ ∂ t , ν M ( ∂ t , ν ) + A ¯ U = F . In this chapter, where we turn back to well-posedness issues, we replace the material law operator M(∂t,ν), which is invariant under translations in time, by an operator of the form $$\displaystyle \mathcal {M}+\partial _{t,\nu }^{-1}\mathcal {N}, $$ ℳ + ∂ t , ν − 1 N , where both $$\mathcal {M}$$ ℳ and $$\mathcal {N}$$ N are bounded linear operators in $$L_{2,\nu }(\mathbb {R};H)$$ L 2 , ν ( ℝ ; H ) . Thus, it is the aim in the following to provide criteria on $$\mathcal {M}$$ ℳ and $$\mathcal {N}$$ N under which the operator $$\displaystyle \partial _{t,\nu }\mathcal {M}+\mathcal {N}+A $$ ∂ t , ν ℳ + N + A is closable with continuous invertible closure in $$L_{2,\nu }(\mathbb {R};H)$$ L 2 , ν ( ℝ ; H ) . In passing, we shall also replace the skew-selfadjointness of A by a suitable real part condition. Under additional conditions on $$\mathcal {M}$$ ℳ and $$\mathcal {N}$$ N , we will also see that the solution operator is causal. Finally, we will put the autonomous version of Picard’s theorem into perspective of the non-autonomous variant developed here.

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Well-posedness for a general class of differential inclusions

Cited by 11 publications

References 33 publications

Mixed-dimensional poromechanical models of fractured porous media

Mixed-dimensional poromechanical models of fractured porous media

Maximal Regularity for Non-autonomous Evolutionary Equations

Non-Autonomous Evolutionary Equations

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