非局部速度守恒定律:奇异极限问题

IF 1.9 4区 数学 Q1 MATHEMATICS, APPLIED
Jan Friedrich, Simone Göttlich, Alexander Keimer, Lukas Pflug
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引用次数: 0

摘要

SIAM 应用数学杂志》,第 84 卷第 2 期,第 497-522 页,2024 年 4 月。 摘要。我们考虑了具有非局部速度的守恒定律,并证明对于指数型非局部权重,当非局部权重接近狄拉克分布时,唯一解在弱或强意义上(取决于速度的正则性)收敛于局部守恒定律的熵解。为此,我们首先建立了非局部速度的均匀总变化约束,它可用于通过弱解的极限。对于所需的熵容许性,我们使用了定制的熵通量对,并利用了一个众所周知的结果,即在速度上有一些额外约束的情况下,单个严格凸熵通量对足以保证唯一性。对于一般权重,我们证明初始基准的单调性会随着时间的推移而保持不变,这使我们能够证明,如果初始基准是单调的,那么对于相当一般的内核,可以收敛到局部熵解。这种情况涵盖了局部守恒定律的原型:冲击波和稀有效应。这些结果表明,"非局部速度 "近似可能比非局部解近似更适合近似局部守恒定律,因为在非局部解近似中,这种单调性只适用于特定速度。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Conservation Laws with Nonlocal Velocity: The Singular Limit Problem
SIAM Journal on Applied Mathematics, Volume 84, Issue 2, Page 497-522, April 2024.
Abstract. We consider conservation laws with nonlocal velocity and show, for nonlocal weights of exponential type, that the unique solutions converge in a weak or strong sense (dependent on the regularity of the velocity) to the entropy solution of the local conservation law when the nonlocal weight approaches a Dirac distribution. To this end, we first establish a uniform total variation bound on the nonlocal velocity, which can be used to pass to the limit in the weak solution. For the required entropy admissibility, we use a tailored entropy-flux pair and take advantage of a well-known result that a single strictly convex entropy-flux pair is sufficient for uniqueness, given some additional constraints on the velocity. For general weights, we show that the monotonicity of the initial datum is preserved over time, which enables us to prove convergence to the local entropy solution for rather general kernels if the initial datum is monotone. This case covers the archetypes of local conservation laws: shock waves and rarefactions. These results suggest that a “nonlocal in the velocity” approximation might be better suited to approximating local conservation laws than a nonlocal in the solution approximation, in which such monotonicity only holds for specific velocities.
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来源期刊
CiteScore
3.60
自引率
0.00%
发文量
79
审稿时长
12 months
期刊介绍: SIAM Journal on Applied Mathematics (SIAP) is an interdisciplinary journal containing research articles that treat scientific problems using methods that are of mathematical interest. Appropriate subject areas include the physical, engineering, financial, and life sciences. Examples are problems in fluid mechanics, including reaction-diffusion problems, sedimentation, combustion, and transport theory; solid mechanics; elasticity; electromagnetic theory and optics; materials science; mathematical biology, including population dynamics, biomechanics, and physiology; linear and nonlinear wave propagation, including scattering theory and wave propagation in random media; inverse problems; nonlinear dynamics; and stochastic processes, including queueing theory. Mathematical techniques of interest include asymptotic methods, bifurcation theory, dynamical systems theory, complex network theory, computational methods, and probabilistic and statistical methods.
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