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In this article we consider weak solutions of the three-dimensional incompressible fluid flow equations with initial data admitting a one-dimensional symmetry group. We examine both the viscous and inviscid cases. For the case of viscous…

We consider the one-dimensional dynamics of nonlinear non-dispersive waves. The problem can be mapped onto a linear one by means of the hodograph transform. We propose an approximate scheme for solving the corresponding Euler-Poisson…

Pattern Formation and Solitons · Physics 2020-06-09 M. Isoard , N. Pavloff , A. M. Kamchatnov

Considering the isentropic Euler equations of compressible fluid dynamics with geometric effects included, we establish the existence of entropy solutions for a large class of initial data. We cover fluid flows in a nozzle or in spherical…

Analysis of PDEs · Mathematics 2008-12-16 Philippe G. LeFloch , Michael Westdickenberg

The existence and uniqueness of two dimensional steady compressible Euler flows past a wall or a symmetric body are established. More precisely, given positive convex horizontal veloicty in the upstream, there exists a critical value…

Analysis of PDEs · Mathematics 2014-10-09 Chao Chen , Lili Du , Chunjing Xie , Zhouping Xin

This paper deals with isothermal Euler-Poisson system which is used to model collapse of self-gravitating Newtonian star. Density dependent viscosity term is added on the right-hand side of momentum equation and it has been proved that…

Analysis of PDEs · Mathematics 2025-08-19 Marko Nedeljkov , Sanja Ružičić

This paper is concerned with the well-posedness theory of the impact of a subsonic axially symmetric jet emerging from a semi-infinitely long nozzle, onto a rigid wall. The fluid motion is described by the steady isentropic Euler system. We…

Analysis of PDEs · Mathematics 2020-12-16 Jianfeng Cheng , Lili Du , Qin Zhang

In this paper, we investigate Childress's conjecture proposed in [Phys.D 237(14-17):1921-1925, 2008] on the growth rate of the vorticity maximum for axisymmetric swirl-free Euler flows in three and higher dimensions. We consider the setting…

Analysis of PDEs · Mathematics 2025-11-07 Daomin Cao , Junhong Fan , Guolin Qin

A nontrivial smooth steady incompressible Euler flow in three dimensions with compact support is constructed. Another uncommon property of this solution is the dependence between the Bernoulli function and the pressure.

Differential Geometry · Mathematics 2018-10-19 A. V. Gavrilov

We are concerned with the structural stability of conical shocks in the three-dimensional steady supersonic flows past Lipschitz perturbed cones whose vertex angles are less than the critical angle. The flows under consideration are…

Analysis of PDEs · Mathematics 2021-05-25 Gui-Qiang G. Chen , Jie Kuang , Yongqian Zhang

We study the uniqueness of solutions with a transonic shock in a two-dimensional Riemannian manifold with a special metric, which can be regarded as an approximate model of the general physical nozzles, within a class of transonic shock…

Analysis of PDEs · Mathematics 2025-09-01 Minghong Han , Bingsong Long , Hairong Yuan

We established the existence, uniqueness and stability of subsonic flows past an airfoil with a vortex line at the trailing edge. Such a flow pattern is governed by the two dimensional steady compressible Euler equations. The vortex line…

Analysis of PDEs · Mathematics 2020-08-18 Jun Chen , Zhouping Xin , Aibin Zang

In this paper, we consider the one-dimensional isentropic compressible Euler equations with source term $\beta(t,x)\rho|u|^{\alpha}u$ in a bounded domain, which can be used to describe gas transmission in a nozzle.~The model is imposed a…

Analysis of PDEs · Mathematics 2022-07-19 Xiaomin Zhang , Jiawei Sun , Huimin Yu

We consider admissible weak solutions to the compressible Euler system with source terms, which include rotating shallow water system and the Euler system with damping as special examples. In the case of anti-symmetric sources such as…

Analysis of PDEs · Mathematics 2015-06-04 Tianwen Luo , Chunjing Xie , Zhouping Xin

In this paper, a novel immersed boundary method is developed, validated, and applied. Through devising a second-order three-step flow reconstruction scheme, the proposed method is able to enforce the Dirichlet, Neumann, Robin, and Cauchy…

Fluid Dynamics · Physics 2018-09-10 Huangrui Mo , Fue-Sang Lien , Fan Zhang , Duane S. Cronin

We investigate the Cauchy problem of three-dimensional compressible non-isothermal nematic liquid crystal flows in $\mathbb{R}^3$. We derive the global existence and uniqueness of strong solutions with both interior and far field vacuum…

Analysis of PDEs · Mathematics 2021-08-24 Yang Liu , Xin Zhong

In this paper, we consider the well-posedness theory of two-dimensional compressible subsonic jet flows for steady full Euler system with general vorticity. Inspired by the analysis in arXiv:2006.05672, we show that the stream function…

Analysis of PDEs · Mathematics 2024-04-26 Yan Li

The Euler-Poisson system is a fundamental two-fluid model to describe the dynamics of the plasma consisting of compressible electrons and a uniform ion background. In the 3D case Guo first constructed a global smooth irrotational solution…

Mathematical Physics · Physics 2013-11-25 Dong Li , Yifei Wu

We construct a supersonic-sonic smooth patch solution for the two dimensional steady Euler equations in gas dynamics. This patch is extracted from the Frankl problem in the study of transonic flow with local supersonic bubble over an…

Analysis of PDEs · Mathematics 2021-01-05 Yanbo Hu , Jiequan Li

We consider the problem of supersonic flow of a Chaplygin gas past a delta wing with a shock or rarefaction wave attached to the leading edges. The flow under study is described by the three-dimensional steady Euler system. In conical…

Analysis of PDEs · Mathematics 2020-12-29 Bingsong Long , Chao Yi

We study the motion of isentropic gas in nozzles. This is a major subject in fluid dynamics. In fact, the nozzle is utilized to increase the thrust of rocket engines. Moreover, the nozzle flow is closely related to astrophysics. These…

Analysis of PDEs · Mathematics 2017-02-21 Naoki Tsuge