Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1407.5517

On the instability problem of a 3-D transonic oblique shock wave

Li Liang·Xu Gang·Yin Huicheng
View original source ↗Read source document ↗Export metadata ↓

Abstract

In this paper, we are concerned with the instability problem of a 3-D transonic oblique shock wave for the steady supersonic flow past an infinitely long sharp wedge. The flow is assumed to be isentropic and irrotational. It was indicated in pages 317 of [9] that if a steady supersonic flow comes from minus infinity and hits a sharp symmetric wedge, then it follows from the Rankine-Hugoniot conditions and the physical entropy condition that there possibly appear a weak shock or a strong shock attached at the edge of the sharp wedge, which corresponds to a supersonic shock or a transonic shock, respectively. The question arises which of the two actually occurs. It has frequently been stated that the strong one is unstable and that, therefore, only the weak one could occur. However, a convincing proof of this instability has apparently never been given. The aim of this paper is to understand such a longstanding open question. We will show that the attached 3-D transonic oblique shock problem is overdetermined, which implies that the 3-D transonic shock is unstable in general.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Li Liang, Xu Gang, Yin Huicheng. 2014-07-21. On the instability problem of a 3-D transonic oblique shock wave. https://arxiv.org/abs/1407.5517

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

KEEP EXPLORING

Related papers

arXiv2026

Renormalized Lambert-W Cascade and Finite-Time Amplification and Blowup for reconstructed b Dynamics on T^3 for the 3D Navier Stokes Equations

This article extracts and consolidates the renormalized Lambert-$W$ branch-point cascade, its distinguished phase reduction, the exact characteristic invariant and finite-time amplification mechanism, and the extended reconstructed $b_i$ equation on $\mathbb T^3$. Repeated historical derivations are removed while the principal proofs and terminal reconstruction estimates are retained. The presentation separates exact finite-depth statements from coupled-depth asymptotics and records the hypotheses required for the extended PDE reconstruction. This paper further supports a recent paper \cite {moschandreou2026exploration} published by the corresponding author which claims that the Navier Stokes equations lose smoothness in finite time from initial smooth data.

math.AP↗
arXiv2026

Global harmonic analysis for $Φ^4_3$ on closed Riemannian manifolds

Following Parisi \& Wu's paradigm of stochastic quantization, we constructed in \cite{BDFT} a $Φ^4$ measure on an arbitrary closed, compact Riemannian manifold of dimension $3$ as an invariant measure of a singular stochastic partial differential equation. This solves a longstanding open problem in quantum fields on curved backgrounds. In the present work, we build all the harmonic and microlocal analysis tools that are needed in \cite{BDFT}. In particular, we extend the approach of Jagannath--Perkowski to the vectorial $Φ^4_3$ model by introducing a new Cole-Hopf transform involving random bundle maps.

math.AP↗
arXiv2026

Unconditional uniqueness for the derivative nonlinear Schrödinger equation by normal form approach

We prove uniqueness of solutions to the Cauchy problem for the derivative nonlinear Schrödinger equation in $L^\infty_tH^{1/2}_x$. Our proof is based on the method of normal form reduction (NFR), which has been employed to obtain the uniqueness in $C_tH^s_x$, $s>1/2$. To overcome logarithmic divergences at the $H^{1/2}$ regularity, we exploit the $B^{0+}_{\infty,1}$ control of solutions provided by a refined Strichartz estimate. Our NFR argument consists of two stages: we first use NFR finitely many times to derive an intermediate equation in which the main cubic nonlinearity is restricted to a certain type of frequency interaction; we then apply the infinite NFR scheme to the intermediate equation. Moreover, we modify the usual NFR argument relying on continuity in time of solutions so that the uniqueness in the class $L^\infty_tH^{1/2}_x$ can be obtained directly.

math.AP↗