Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · math/0305016

Analysis of some singular solutions in fluid dynamics

Zhouping Xin
View original source ↗Read source document ↗Export metadata ↓

Abstract

Studies on singular flows in which either the velocity fields or the vorticity fields change dramatically on small regions are of considerable interests in both the mathematical theory and applications. Important examples of such flows include supersonic shock waves, boundary layers, and motions of vortex sheets, whose studies pose many outstanding challenges in both theoretical and numerical analysis. The aim of this talk is to discuss some of the key issues in studying such flows and to present some recent progress. First we deal with a supersonic flow past a perturbed cone, and prove the global existence of a shock wave for the stationary supersonic gas flow past an infinite curved and symmetric cone. For a general perturbed cone, a local existence theory for both steady and unsteady is also established. We then present a result on global existence and uniqueness of weak solutions to the 2-D Prandtl's system for unsteady boundary layers. Finally, we will discuss some new results on the analysis of the vortex sheets motions which include the existence of 2-D vortex sheets with reflection symmetry; and no energy concentration for steady 3-D axisymmetric vortex sheets.

Explore related subjects

math.AP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Zhouping Xin. 2003-05-01. Analysis of some singular solutions in fluid dynamics. https://arxiv.org/abs/math/0305016

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

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↗
arXiv2026

Equivalence between solvability of the Dirichlet and Regularity problem under an $L^1$ Carleson condition on $\partial_t A$

We study an elliptic operator $L:=-\mathrm{div}(A\nabla \cdot)$ on the upper half space. It is known that solvability of the Regularity problem in $\dot{W}^{1,p}$ implies solvability of the adjoint Dirichlet problem in $L^{p'}$. Previously, Shen (2007) established a partial reverse result. In our work, we show that if we assume a mixed \(L^1-L^\infty\) condition on only \(|\partial_t A|\), the full reverse direction holds. As a result, we obtain equivalence between solvability of the Dirichlet problem $(D)^*_{p'}$ and the Regularity problem $(R)_p$ under this condition. As a further consequence, we can extend the class of operators for which the $L^p$ Regularity problem is solvable by operators satisfying the mixed $L^1-L^\infty$ condition. Additionally in the case of the upper half plane, this class includes operators satisfying this this mixed \(L^1-L^\infty\) condition.

math.AP↗