arXiv subjects

Jiayun Meng

Publications and source records attributed to Jiayun Meng.

5 recordsLinked to original sources

arXiv2026

$a$-contraction applied to the time-asymptotic stability of weak shocks for a general 1-D viscous system

The method of $a$-contraction with shifts builds on the relative entropy method of Dafermos and DiPerna and allows both viscous shocks and rarefaction waves to be studied directly at the level of the solution. This unified framework led to the resolution of Matsumura's long-standing conjecture on the time-asymptotic stability of a viscous shock in composition with a rarefaction. Kang, Vasseur, and Wang established this stability for the one-dimensional compressible barotropic Navier-Stokes system. Their result was subsequently extended to composite planar waves for general $n\times n$ viscous systems in arbitrary spatial dimensions. The present paper is a companion to that work. Here we prove the estimates on viscous shock profiles needed to apply $a$-contraction with shifts, including derivative estimates that go beyond the standard bounds. We also give a streamlined account of the main ideas of the method by focusing on a single viscous shock for a general one-dimensional system.

math.AP↗

arXiv2026

Weak-BV Stability and Inviscid Limits for Barotropic Flows with Outflow Boundary Conditions

Boundary value problems for systems of conservation laws, including inviscid compressible flows, arise naturally in physical models such as shallow water flow in a canal, where fluid may leave the domain through the boundary. In contrast to the Cauchy problem, the inviscid limit is highly sensitive to the underlying viscous model: different viscous regularizations may select different effective boundary conditions for the limiting system. In this paper, we show that small BV solutions of the one-dimensional compressible barotropic Euler equations arise as inviscid limits of the barotropic Navier-Stokes system for the outflow boundary problem. We work with the physically meaningful effective boundary condition selected by the Navier-Stokes system for the limiting Euler solutions. Moreover, going beyond smooth flows or a fixed class of Riemann solutions, our stability results apply to small BV solutions exhibiting general wave patterns and complicated configurations generated by interactions with the outflow boundary, including reflected shocks.

math.AP↗

arXiv2026

Weak-BV stability of inflow and outflow problems for the isentropic Euler system

We study the well-posedness of small BV solutions to the one-dimensional isentropic Euler system on the half-line under inflow and outflow boundary conditions. These boundary conditions are formulated in terms of admissible trace sets determined by Navier--Stokes boundary layers and zero-speed shocks. For both the inflow and outflow problems, we construct small BV solutions taking values in the subsonic region. These solutions are unique and satisfy the quantitative stability estimate \begin{align*} \|U(\cdot, t)-V(\cdot, t)\|_{L^2} \lesssim \sqrt{\|U(\cdot, 0) - V(\cdot, 0)\|_{L^2}} \, , \end{align*} which holds for any such small BV solution $V$ and any $L^\infty$ entropy solution $U$ satisfying the strong trace property and the inflow/outflow boundary conditions. In particular, we emphasize that $U$ may take values outside the subsonic region and need not have bounded variation or be constructed by the front tracking scheme. This establishes the first weak-BV stability and uniqueness theory for inflow and outflow initial-boundary value problems. A key difficulty lies in verifying that the boundary set is compatible with the $a$-contraction framework, and in proving that the front-tracking limit satisfies the prescribed inflow/outflow conditions.

math.AP↗

arXiv2025

Time-asymptotic stability of composite weak planar waves for a general $n\times n$ multi-D viscous system

We prove the time-asymptotic stability of the superposition of a weak planar viscous 1-shock and either a weak planar n-rarefaction or a weak planar viscous n-shock for a general $n\times n$ multi-D viscous system. In 2023, Kang-Vasseur-Wang [11] showed the stability of the superposition of a viscous shock and a rarefaction for 1-D compressible barotropic Navier-Stokes equations and solved a long-standing open problem officially introduced by Matsumura-Nishihara [23] in 1992. Our work is an extension of [11], where a general $n\times n$ multi-D viscous system is studied. Same as in [11], we apply the $a$-contraction method with shifts, an energy based method invented by Kang and Vasseur in [9], for both viscous shock and rarefaction at the level of the solution. In such a way, we can work with general perturbations and compositions of waves. Finally, a technique to classify and control higher-order terms is developed to work in multi-D.

math.AP↗

arXiv2022

Counting compatible indexing systems for $C_{p^n}$

We count the number of compatible pairs of indexing systems for the cyclic group $C_{p^n}$. Building on work of Balchin--Barnes--Roitzheim, we show that this sequence of natural numbers is another family of Fuss--Catalan numbers. We count this two different ways: showing how the conditions of compatibility give natural recursive formulas for the number of admissible sets and using an enumeration of ways to extend indexing systems by conceptually simpler pieces.

math.AT↗