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.
Highlights
1 Introduction In this paper, we study the following model arising in acoustics propagation in viscous, isothermal bubbly liquids known as the Van Wijngaarden–Eringen (VWE) equation [14, p. 1121]:
The constant a0 > 0 is a Knudsen number that corresponds to the dimensionless bubble radius
In the case N = 1, equation (1.1) was obtained by Van Wijngaarden [26] to describe the propagation of linear acoustic waves in isothermal bubbly liquids
Summary
We study the following model arising in acoustics propagation in viscous, isothermal bubbly liquids known as the Van Wijngaarden–Eringen (VWE) equation [14, p. 1121]:. In the present paper we are concerned with the Lp – Lq-maximal regularity problem in a cylindrical domain = U × V ⊂ Rn+d for the following inhomogeneous version of the VWE equation subject to Dirichlet boundary conditions:. The Lp – Lq-maximal regularity problem consists of obtaining conditions on the parameters a20, (Red)–1 in order to conclude that the solution u of (1.2) has the same behavior as f and the following estimate u Lp(T,Lq( )) + u W 1,p(T,Lq( )) + u Wp2e,pr (T,Lq( )) + u Lp(T,[D( )]). Replacing in (1.2) the negative Laplacian operator – by a closed linear operator A with domain D(A) defined on a Banach space X, one of the main difficulties we are faced with in order to analyze maximal regularity for (1.2) relies in the unbounded operator M := I + a20A in front of the second order term ∂tt which, for general A, produces a kind of degenerate second order problem.
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