On the Evolution of Compressible and Incompressible Viscous Fluids with a Sharp Interface

• Main Authors: Takayuki Kubo, Yoshihiro Shibata
• Format: Article
• Language: English
• Published: MDPI AG 2021-03-01
• Series: Mathematics
• Subjects: Navier–Stokes equations; two phase problem; local in time unique existence theorem; R-bounded operator
• Online Access: https://www.mdpi.com/2227-7390/9/6/621

In this paper, we consider some two phase problems of compressible and incompressible viscous fluids’ flow without surface tension under the assumption that the initial domain is a uniform <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msubsup><mi>W</mi><mi>q</mi><mrow><mn>2</mn><mo>−</mo><mn>1</mn><mo>/</mo><mi>q</mi></mrow></msubsup></semantics></math></inline-formula> domain in <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mi mathvariant="double-struck">R</mi><mi>N</mi></msup></semantics></math></inline-formula> (<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo>≥</mo><mn>2</mn></mrow></semantics></math></inline-formula>). We prove the local in the time unique existence theorem for our problem in the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula> in time and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>q</mi></msub></semantics></math></inline-formula> in space framework with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>2</mn><mo><</mo><mi>p</mi><mo><</mo><mi>∞</mi></mrow></semantics></math></inline-formula> and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>N</mi><mo><</mo><mi>q</mi><mo><</mo><mi>∞</mi></mrow></semantics></math></inline-formula> under our assumption. In our proof, we first transform an unknown time-dependent domain into the initial domain by using the Lagrangian transformation. Secondly, we solve the problem by the contraction mapping theorem with the maximal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>q</mi></msub></semantics></math></inline-formula> regularity of the generalized Stokes operator for the compressible and incompressible viscous fluids’ flow with the free boundary condition. The key step of our proof is to prove the existence of an <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>-bounded solution operator to resolve the corresponding linearized problem. The Weis operator-valued Fourier multiplier theorem with <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi mathvariant="script">R</mi></semantics></math></inline-formula>-boundedness implies the generation of a continuous analytic semigroup and the maximal <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>p</mi></msub></semantics></math></inline-formula>-<inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi>L</mi><mi>q</mi></msub></semantics></math></inline-formula> regularity theorem.