Well-posedness for physical vacuum free boundary problem of compressible Euler equations with time-dependent damping

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> -{μ }{{(1+t)}λ}ρ {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> μ > 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> λ > 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> ω ={ω }₀{(1+t)}-μ 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> λ =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> ω ={ω }₀{e}^{μ /1-λ [1-{(1+t)}1-λ]} 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> λ ≠ 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:math> 0< λ < 1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> ω converges to 0 as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>t</m:mi> <m:mo>→</m:mo> <m:mo>+</m:mo> <m:mi>∞</m:mi> </m:math> t→ +∞ , and 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> λ > 1 , <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ω</m:mi> </m:math> ω converges to <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <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:msup> </m:math> {ω }₀{e}μ /1-λ as <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>t</m:mi> <m:mo>→</m:mo> <m:mo>+</m:mo> <m:mi>∞</m:mi> </m:math> t→ +∞ . This is essentially different from the case of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>μ</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:math> μ =1 , <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> λ =0 in (Meng-Mai-Mei, JDE, 2022), whose axial velocity is only 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:mi>e</m:mi> </m:mrow> <m:mrow> <m:mo>−</m:mo> <m:mi>t</m:mi> </m:mrow> </m:msup> </m:math> ω ={ω }₀{e}⁻ᵗ . Moreover, at the moving boundary, the compressible Euler equations with time-dependent damping become a degenerate hyperbolic system. Thanks to the Hardy-type inequality and cut-off functions, we choose a suitable weighted Sobolev spaces to construct a priori estimates to overcome the degeneracy of the system in the Lagrangian coordinates, which confirms the existence of local smooth solution. Gronwall’s inequality guarantees the uniqueness of local smooth solution.