Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1612.08533

The Riemann solution to the Chaplygin pressure Aw-Rascle model with Coulomb-like friction and its vanishing pressure limit

Qingling Zhang
View original source ↗Read source document ↗Export metadata ↓

Abstract

The Riemann solution to the Chaplygin pressure Aw-Rascle model with Coulomb-like friction is constructed explicitly and its vanishing pressure limit is analyzed precisely. It is shown that the delta shock wave appears in the Riemann solutions in some certain situations. The generalized Rankine-Hugoniot conditions of the delta shock wave are established and the exact position, propagation speed and strength of the delta shock wave are given explicitly, which enables us to see the influence of the Coulomb-like friction on the Riemann solution to the Chaplygin pressure Aw-Rascle model clearly. It is shown that the Coulomb-like friction term makes contact discontinuities and delta shock waves bend into parabolic shapes and the Riemann solutions are not self-similar anymore. Finally, the occurrence mechanism on the phenomenon of concentration and cavitation and the formation of delta shock wave and vacuum in the process of vanishing pressure limit are analyzed and identified in detail. Moreover, we show the Riemann solutions to the nonhomogeneous Chaplygin pressure Aw-Rascle model converge to the Riemann solutions to the transportation equations with the same source term as the pressure vanishes. These two results generalize those obtained in [7,38] for homogeneous equations to nonhomogeneous equations and are also applicable to the nonsymmetric system of Keyfitz-Kranzer type with the same Chaplygin pressure and Coulomb-like friction.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Qingling Zhang. 2016-12-27. The Riemann solution to the Chaplygin pressure Aw-Rascle model with Coulomb-like friction and its vanishing pressure limit. https://arxiv.org/abs/1612.08533

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

A linear test approach to global controllability of third- and fifth-order nonlinear dispersive equations

We investigate third- and fifth-order nonlinear dispersive equations of KdV type on the torus and establishes approximate controllability by a fixed four-dimensional control; rather than relying solely on the saturation machinery, the analysis exploits the finite-dimensional controllability of the inviscid Burgers equation linearized around a carefully constructed return trajectory, with the trajectory itself obtained from an observable family. This ``linear test" strategy, yields more information about the structure of the control than the standard approach. In particular, the constructed control is shown to depend continuously on the initial and target states, a property that is by no means automatic in nonlinear control problems, and to decompose as a bounded linear operator applied to the data plus a fixed remainder, with the operator part interestingly independent of the order of dispersion.

math.AP↗
arXiv2026

A conditional Lagrangian clock barrier at the $C^{1,\frac{1}{3}}$ threshold for axisymmetric Euler without swirl

We consider axisymmetric no-swirl solutions to the three-dimensional incompressible Euler equations, with initial velocity in $C^{1,α}\cap L^2$, where $α\in\left[\frac{1}{3},1\right)$. In a major breakthrough, Shkoller introduced a clock-and-driver framework that he used in order to prove finite-time type I blow-up below the $C^{1,\frac{1}{3}}$ threshold in this setting. Motivated by this, we define Lagrangian classes of coherent conditional solutions for which the same mechanism yields a supercritical-critical barrier to blow-up when $α\geq\frac{1}{3}$. When $α>\frac{1}{3}$, the aforementioned barrier is genuinely depleted, whereas at the critical endpoint $α=\frac{1}{3}$, we obtain an exponential bound preventing blow-up. In the general case, we formulate a matrix-clock criterion in terms of the smallest singular value of the deformation gradient and show that, under transverse cusp-tail, longitudinal, off-clock, Dini, and suitable geometric coherence hypotheses, this singular value cannot collapse in finite time. In particular, we also show that the class of such coherent solutions includes the smooth ones locally in time. In the on-axis case, the criterion reduces to the scalar clock inequality $\displaystyle \dot{J}(t)\gtrsim -B(t)J(t)-CJ(t)^{3α}$, which rules out Shkoller-type clock collapse for $α\geq\frac{1}{3}$. These results do not enlarge the known Lorentz-space global regularity classes. Rather, they in particular identify the supercritical Lagrangian obstruction dual to Shkoller's subcritical blow-up mechanism in the case $α>\frac{1}{3}$.

math.AP↗