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

Advances in Difference Equations ◽

10.1186/s13662-020-03054-5 ◽

2020 ◽

Vol 2020 (1) ◽

Author(s):

Carlos Lizama ◽

Marina Murillo-Arcila

Keyword(s):

Acoustic Waves ◽

Maximal Regularity ◽

Bubble Radius ◽

Fourier Multipliers ◽

Cylindrical Domain ◽

Lebesgue Spaces ◽

Regularity Problem ◽

Bubbly Liquids

Abstract 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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Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces

Forum Mathematicum ◽

10.1515/forum-2019-0319 ◽

2020 ◽

Vol 32 (4) ◽

pp. 919-936 ◽

Cited By ~ 2

Author(s):

Jiao Chen ◽

Wei Ding ◽

Guozhen Lu

Keyword(s):

Hardy Spaces ◽

Differential Operators ◽

Integral Operators ◽

Fourier Multipliers ◽

Regularity Of Solutions ◽

Local Version ◽

Fourier Integral Operators ◽

Pseudo Differential Operators ◽

Local Hardy Spaces ◽

The One

AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].

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