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

Abstract

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.

Keywords

• Maximal regularity
• Van Wijngaarden–Eringen equation
• Degenerate evolution equations
• R-boundedness