<p>In this paper, we consider the vacuum free boundary problem of a viscous two-phase model with spherical symmetry, which consists of two mass equations and a momentum equation with viscosity coefficients of the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\mu (n,\rho )=(n+\rho )^{\theta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>μ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mi>θ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\lambda (n,\rho )=(\theta -1)(n+\rho )^{\theta }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>λ</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <mrow> <mo stretchy="false">(</mo> <mi>θ</mi> <mo>-</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>+</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mi>θ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation>. We take the pressure function as <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$P(n,\rho )=n^{\alpha }+\rho ^{\gamma }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>P</mi> <mrow> <mo stretchy="false">(</mo> <mi>n</mi> <mo>,</mo> <mi>ρ</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mi>n</mi> <mi>α</mi> </msup> <mo>+</mo> <msup> <mi>ρ</mi> <mi>γ</mi> </msup> </mrow> </math></EquationSource> </InlineEquation> (<InlineEquation ID="IEq4"> <EquationSource Format="TEX">$\alpha \ge 1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>≥</mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>, <InlineEquation ID="IEq5"> <EquationSource Format="TEX">$\gamma >1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>γ</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation>), where <i>n</i> and <InlineEquation ID="IEq6"> <EquationSource Format="TEX">$\rho $</EquationSource> <EquationSource Format="MATHML"><math> <mi>ρ</mi> </math></EquationSource> </InlineEquation> are the densities of two phases. We provide a class of global analytical solutions to this problem when <InlineEquation ID="IEq7"> <EquationSource Format="TEX">$\alpha =\gamma =\theta =2$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>α</mi> <mo>=</mo> <mi>γ</mi> <mo>=</mo> <mi>θ</mi> <mo>=</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>. Moreover, we prove that the free boundary spreads out at least sub-linearly in time with the rate of <InlineEquation ID="IEq8"> <EquationSource Format="TEX">$\mathcal {O}((1+t)^{1/3})$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="script">O</mi> <mo stretchy="false">(</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mn>1</mn> <mo stretchy="false">/</mo> <mn>3</mn> </mrow> </msup> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> and not more than linearly in time for the constructed solutions, by using the averaged quantities method.</p>

A Class of Analytical Solutions to the Vacuum Free Boundary Problem of a Viscous Two-Phase Model with Spherical Symmetry

期刊论文
发表日期：

2026年3月31日

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Hongjing Jiang,
Jianwei Dong

摘要

In this paper, we consider the vacuum free boundary problem of a viscous two-phase model with spherical symmetry, which consists of two mass equations and a momentum equation with viscosity coefficients of the form $\mu (n,\rho )=(n+\rho )^{\theta }$ $μ (n, ρ) = {(n + ρ)}^{θ}$ and $\lambda (n,\rho )=(\theta -1)(n+\rho )^{\theta }$ $λ (n, ρ) = (θ - 1) {(n + ρ)}^{θ}$ . We take the pressure function as $P(n,\rho )=n^{\alpha }+\rho ^{\gamma }$ $P (n, ρ) = n^{α} + ρ^{γ}$ ( $\alpha \ge 1$ $α \geq 1$ , $\gamma >1$ $γ > 1$ ), where n and $\rho $ $ρ$ are the densities of two phases. We provide a class of global analytical solutions to this problem when $\alpha =\gamma =\theta =2$ $α = γ = θ = 2$ . Moreover, we prove that the free boundary spreads out at least sub-linearly in time with the rate of $\mathcal {O}((1+t)^{1/3})$ $O ({(1 + t)}^{1 / 3})$ and not more than linearly in time for the constructed solutions, by using the averaged quantities method.