The objective of this article is to consider the initial-value problem derived from an extended quasilinear May-Nowak system in a two-dimensional smoothly bounded domain, encompassing viral kinetics with particular emphasis on scenarios captured by two dominant mechanisms: the cross-diffusive behavior of healthy individuals toward the orientation of the infected populations, and the nonlinear diffusion process in the form <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$\nabla \cdot (D(r)\nabla r)$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="normal">∇</mi> <mo>·</mo> <mo stretchy="false">(</mo> <mi>D</mi> <mo stretchy="false">(</mo> <mi>r</mi> <mo stretchy="false">)</mo> <mi mathvariant="normal">∇</mi> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> </math></EquationSource> </InlineEquation> among fluid environment. Here, the diffusion term D(r) represents a slight generalization of the prototypical expression <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$(1+r)^{m-1}$</EquationSource> <EquationSource Format="MATHML"><math> <msup> <mrow> <mo stretchy="false">(</mo> <mn>1</mn> <mo>+</mo> <mi>r</mi> <mo stretchy="false">)</mo> </mrow> <mrow> <mi>m</mi> <mo>-</mo> <mn>1</mn> </mrow> </msup> </math></EquationSource> </InlineEquation> with <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$r\ge 0$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>r</mi> <mo>≥</mo> <mn>0</mn> </mrow> </math></EquationSource> </InlineEquation>. Under the optimal assumption when <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$m>1$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>m</mi> <mo>></mo> <mn>1</mn> </mrow> </math></EquationSource> </InlineEquation> in parabolic-parabolic-elliptic framework, then for arbitrary choice of the initial datum, the corresponding problem possesses at least one globally bounded weak solution. Insofar as we are aware, this is the first result to reveal the complex interplay between cross-diffusion dynamic, nonlinear diffusion process and fluid coupling mechanism of such system, therefore effectively improving the regularity of solutions without compromising global well-posedness.

An Optimal Result on Global Well-Posedness of Bounded Weak Solution for Quasilinear May-Nowak-fluid System with Nonlinear Diffusion Term and Immune Chemokine in Two Dimensions

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发表日期：

2026年2月16日

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Jiashan Zheng,
Yuying Wang

摘要

The objective of this article is to consider the initial-value problem derived from an extended quasilinear May-Nowak system in a two-dimensional smoothly bounded domain, encompassing viral kinetics with particular emphasis on scenarios captured by two dominant mechanisms: the cross-diffusive behavior of healthy individuals toward the orientation of the infected populations, and the nonlinear diffusion process in the form $\nabla \cdot (D(r)\nabla r)$ $\nabla \cdot (D (r) \nabla r)$ among fluid environment. Here, the diffusion term D(r) represents a slight generalization of the prototypical expression $(1+r)^{m-1}$ ${(1 + r)}^{m - 1}$ with $r\ge 0$ $r \geq 0$ . Under the optimal assumption when $m>1$ $m > 1$ in parabolic-parabolic-elliptic framework, then for arbitrary choice of the initial datum, the corresponding problem possesses at least one globally bounded weak solution. Insofar as we are aware, this is the first result to reveal the complex interplay between cross-diffusion dynamic, nonlinear diffusion process and fluid coupling mechanism of such system, therefore effectively improving the regularity of solutions without compromising global well-posedness.