双曲平衡律系统解的扩散接触波收敛性

IF 0.5 4区 数学 Q4 MATHEMATICS, APPLIED Journal of Hyperbolic Differential Equations Pub Date : 2023-03-01 DOI:10.1142/s0219891623500078
Yanni Zeng
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引用次数: 1

摘要

我们考虑一个[公式:见文本]双曲平衡律系统,它是Keller-Segel型趋化模型在逆Hopf-Cole变换下的转换形式。当柯西数据连接两个不同的终态时,我们研究柯西问题[公式:见文]。背景波是约简系统的扩散接触波。建立了解的整体存在性,并研究了解的时间渐近性。在胞群初始接近稳定平衡值的特殊情况下,我们得到了小假设下扩散接触波的非线性稳定性。在一般情况下,种群最初不接近其稳定的平衡值,至少在一个远场,我们使用一个修正函数在时间渐近的方差分析,并表明群体接近逻辑稳定的平衡值。与有阻尼的欧拉方程相比,我们的结果有两个显著的不同。第一个是在时间渐近解中存在二次波。这意味着我们的解收敛到扩散接触波的速度比有阻尼的欧拉方程慢。第二个是修正函数逻辑增长而不是指数衰减。
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Convergence to a diffusive contact wave for solutions to a system of hyperbolic balance laws
We consider a [Formula: see text] system of hyperbolic balance laws that is the converted form under inverse Hopf–Cole transformation of a Keller–Segel type chemotaxis model. We study Cauchy problem when Cauchy data connect two different end-states as [Formula: see text]. The background wave is a diffusive contact wave of the reduced system. We establish global existence of solution and study the time asymptotic behavior. In the special case where the cellular population initially approaches its stable equilibrium value as [Formula: see text], we obtain nonlinear stability of the diffusive contact wave under smallness assumption. In the general case where the population initially does not approach to its stable equilibrium value at least at one of the far fields, we use a correction function in the time asymptotic ansatz, and show that the population approaches logistically to its stable equilibrium value. Our result shows two significant differences when comparing to Euler equations with damping. The first one is the existence of a secondary wave in the time asymptotic ansatz. This implies that our solutions converge to the diffusive contact wave slower than those of Euler equations with damping. The second one is that the correction function logistically grows rather than exponentially decays.
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来源期刊
Journal of Hyperbolic Differential Equations
Journal of Hyperbolic Differential Equations 数学-物理:数学物理
CiteScore
1.10
自引率
0.00%
发文量
15
审稿时长
24 months
期刊介绍: This journal publishes original research papers on nonlinear hyperbolic problems and related topics, of mathematical and/or physical interest. Specifically, it invites papers on the theory and numerical analysis of hyperbolic conservation laws and of hyperbolic partial differential equations arising in mathematical physics. The Journal welcomes contributions in: Theory of nonlinear hyperbolic systems of conservation laws, addressing the issues of well-posedness and qualitative behavior of solutions, in one or several space dimensions. Hyperbolic differential equations of mathematical physics, such as the Einstein equations of general relativity, Dirac equations, Maxwell equations, relativistic fluid models, etc. Lorentzian geometry, particularly global geometric and causal theoretic aspects of spacetimes satisfying the Einstein equations. Nonlinear hyperbolic systems arising in continuum physics such as: hyperbolic models of fluid dynamics, mixed models of transonic flows, etc. General problems that are dominated (but not exclusively driven) by finite speed phenomena, such as dissipative and dispersive perturbations of hyperbolic systems, and models from statistical mechanics and other probabilistic models relevant to the derivation of fluid dynamical equations. Convergence analysis of numerical methods for hyperbolic equations: finite difference schemes, finite volumes schemes, etc.
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