R-sectoriality of Cylindrical Boundary Value Problems

Nau, Tobias; Saal, Jürgen

doi:10.1007/978-3-0348-0075-4_25

Cited by 11 publications

(14 citation statements)

References 14 publications

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“…This proves to be a strong tool for the treatment of related elliptic and parabolic, as well as hyperbolic problems. Operators in cylindrical domains with a similar splitting property as in the present paper were, in the case of an infinite cylinder, also considered by Nau et al in [9,[19][20][21][22].…”

Section: Introductionmentioning

confidence: 68%

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$L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

Lizama

Murillo‐Arcila

2020

Adv Differ Equ

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We consider the maximal regularity problem for a PDE of linear acoustics, named the Van Wijngaarden–Eringen equation, that models the propagation of linear acoustic waves in isothermal bubbly liquids, wherein the bubbles are of uniform radius. If the dimensionless bubble radius is greater than one, we prove that the inhomogeneous version of the Van Wijngaarden–Eringen equation, in a cylindrical domain, admits maximal regularity in Lebesgue spaces. Our methods are based on the theory of operator-valued Fourier multipliers.

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Section: Introductionmentioning

confidence: 68%

“…It is well known that many situations in applied sciences naturally lead to problems in cylindrical domains . We refer, e.g., to the textbooks [5] and [6] and the references [9,20,22] for a demonstration of the significance of problems on such .…”

Section: Introductionmentioning

confidence: 99%

$L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

Lizama

Murillo‐Arcila

2020

Adv Differ Equ

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“…This proves to be a strong tool for the treatment of related elliptic and parabolic (see [15] and [16]), as well as hyperbolic (see [16]), problems. Operators in cylindrical domains with a similar splitting property to those in the present paper were, in the case of an infinite cylinder, also considered in [22].…”

Section: ⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬mentioning

confidence: 83%

“…, n)}. However, in order to apply perturbation results similar to [10] or [22], we assume that Q ≡ 1, i.e. we consider A(x, D) :…”

Section: Non-constant Coefficients Of Pmentioning

confidence: 99%

Discrete Fourier multipliers and cylindrical boundary-value problems

Denk

Nau

2013

Proceedings of the Royal Society of Edinburgh: Section A Mathem

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We consider operator-valued boundary value problems in (0, 2π) n with periodic or, more generally, ν-periodic boundary conditions. Using the concept of discrete vector-valued Fourier multipliers, we give equivalent conditions for the unique solvability of the boundary value problem. As an application, we study vector-valued parabolic initial boundary value problems in cylindrical domains (0, 2π) n × V with ν-periodic boundary conditions in the cylindrical directions. We show that under suitable assumptions on the coefficients, we obtain maximal L q -regularity for such problems.2010 Mathematics Subject Classification. 35J40, 35K46.

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“…in models of coagulation-fragmentation processes where an additional parameter (the cluster size) appears in the model, see [Am00]. Another example is given by boundary value problems in cylindrical domains which can be treated by a Fourier multiplier approach which leads to operator-valued symbols, too (see [NS11]). Therefore, during the last decade, the investigation of vector-valued function spaces and differential equations with operator-valued coefficients has gained increasing interest.…”

Section: Introductionmentioning

confidence: 99%

Pseudodifferential operators with non-regular operator-valued symbols

2013

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In this paper, we consider pseudodifferential operators with operatorvalued symbols and their mapping properties, without assumptions on the underlying Banach space E. We show that, under suitable parabolicity assumptions, the W k p (R n , E)-realization of the operator generates an analytic semigroup. An application to non-autonomous pseudodifferential Cauchy problems gives the existence and uniqueness of a classical solution. Our approach is based on oscillatory integrals and kernel estimates for them.

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R-sectoriality of Cylindrical Boundary Value Problems

Cited by 11 publications

References 14 publications

$L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

$L^{p}-L^{q}$-Maximal regularity of the Van Wijngaarden–Eringen equation in a cylindrical domain

Discrete Fourier multipliers and cylindrical boundary-value problems

Pseudodifferential operators with non-regular operator-valued symbols

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