<p>This work investigates the spherically symmetric solutions for the <i>N</i>-dimensional pressureless Navier-Stokes equations with density-dependent viscosity and damping term. By employing the method of variable separation, we systematically derive a set of reduced equations characterizing such solutions with spherical symmetry. Our analysis yields several new classes of exact spherically symmetric solutions featuring velocity <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\varvec{u}$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="bold-italic">u</mi> </mrow> </math></EquationSource> </InlineEquation> with nonlinear form of spatial variable, particularly demonstrating solutions of the form <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$\varvec{u}(\varvec{x}, t)=c(t) |\varvec{x}|^\alpha \varvec{x}\ (\alpha \ne 0) $</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mrow> <mi mathvariant="bold-italic">u</mi> </mrow> <mrow> <mo stretchy="false">(</mo> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo>,</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mi>c</mi> <mrow> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> </mrow> <mo stretchy="false">|</mo> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mo stretchy="false">|</mo> </mrow> <mi>α</mi> </msup> <mrow> <mi mathvariant="bold-italic">x</mi> </mrow> <mspace width="4pt" /> <mrow> <mo stretchy="false">(</mo> <mi>α</mi> <mo>≠</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Notably, these solutions exhibit nonlinear form of the spatial variable - representing, to our knowledge, the first nonlinear-form solutions construction for this class of pressureless Navier-Stokes systems with density-dependent viscous and damping effects. The developed methodology demonstrates potential applicability for investigating exact spherically symmetric solutions across other nonlinear partial differential equations, suggesting promising avenues for extending this analytical approach.</p>

Studies on Solutions to Pressureless Navier-Stokes Equations with Density-Dependent Viscosity and Damping. Part I: Nonlinear Exact Spherically Symmetric Solutions

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发表日期：

2026年2月6日

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Guangxuan He,
Manwai Yuen,
Lijun Zhang

摘要

This work investigates the spherically symmetric solutions for the N-dimensional pressureless Navier-Stokes equations with density-dependent viscosity and damping term. By employing the method of variable separation, we systematically derive a set of reduced equations characterizing such solutions with spherical symmetry. Our analysis yields several new classes of exact spherically symmetric solutions featuring velocity $\varvec{u}$ $u$ with nonlinear form of spatial variable, particularly demonstrating solutions of the form $\varvec{u}(\varvec{x}, t)=c(t) |\varvec{x}|^\alpha \varvec{x}\ (\alpha \ne 0) $ $u (x, t) = {c (t) | x |}^{α} x (α \neq 0)$ . Notably, these solutions exhibit nonlinear form of the spatial variable - representing, to our knowledge, the first nonlinear-form solutions construction for this class of pressureless Navier-Stokes systems with density-dependent viscous and damping effects. The developed methodology demonstrates potential applicability for investigating exact spherically symmetric solutions across other nonlinear partial differential equations, suggesting promising avenues for extending this analytical approach.