Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping
Abstract In this article, we consider the well-posedness of the local smooth solutions to the physical vacuum free boundary problem of the cylindrical symmetric Euler equations with time-dependent damping <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mo>−</m:mo> <m:mfrac> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>λ</m:mi> </m:mrow> </m:msup> </m:mrow> </m:mfrac> <m:mi>ρ</m:mi> <m:mi mathvariant="bold">u</m:mi> </m:math> -\frac{\mu }{{(1+t)}^{\lambda }}\rho {\bf{u}} , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> \mu \gt 0 , and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>></m:mo> <m:mn>0</m:mn> </m:math> \lambda \gt 0 . In this case, the free boundary is moving in the radial direction and the radial velocity will affect the angular velocity, but the axial velocity can be explicitly expressed by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>ω</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>μ</m:mi> </m:mrow> </m:msup> </m:math> \omega ={\omega }_{0}{\left(1+t)}^{-\mu } when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> \lambda =1 and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> <m:mo>=</m:mo> <m:msub> <m:mrow> <m:mi>ω</m:mi> </m:mrow> <m:mrow> <m:mn>0</m:mn> </m:mrow> </m:msub> <m:msup> <m:mrow> <m:mi>e</m:mi> </m:mrow> <m:mrow> <m:mstyle displaystyle="false"> <m:mfrac> <m:mrow> <m:mi>μ</m:mi> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>λ</m:mi> </m:mrow> </m:mfrac> </m:mstyle> <m:mrow> <m:mo>[</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:msup> <m:mrow> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>+</m:mo> <m:mi>t</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mn>1</m:mn> <m:mo>−</m:mo> <m:mi>λ</m:mi> </m:mrow> </m:msup> </m:mrow> <m:mo>]</m:mo> </m:mrow> </m:mrow> </m:msup> </m:math> \omega ={\omega }_{0}{e}^{\tfrac{\mu }{1-\lambda }\left[1-{\left(1+t)}^{1-\lambda }]} when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>λ</m:mi> <m:mo>≠</m:mo> <m:mn>1</m:mn> </m:math> \lambda \ne 1 . Thus, when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mn>0</m:mn> <m:mo><</m:mo> <m:mi>λ</m:mi> <m:mo><</m:mo> <m:mn>1</m:mn> </m:
Authors
- Haitong Li (ORCID: https://orcid.org/0000-0003-3393-9252)
- La-Su Mai (ORCID: https://orcid.org/0000-0003-0045-5980)
- Shiyu Wang (ORCID: https://orcid.org/0000-0001-8263-9956)
Institutions
- Inner Mongolia University (CN)
- Changchun University of Technology (CN)
Publication Details
- Journal
- Advances in Nonlinear Analysis
- Published
- 2025-01-01
- DOI
- https://doi.org/10.1515/anona-2025-0102
- Primary Topic
- Navier-Stokes equation solutions
- Type
- article
- Field-Weighted Citation Impact
- 0.00