Table of Content

20 September 2023, Volume 5 Issue 3

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Preface to the Focused Issue in Honor of Professor Tong Zhang on the Occasion of His 90th Birthday

Jiequan Li, Wancheng Sheng, Chi-Wang Shu, Ping Zhang, Yuxi Zheng

2023, 5(3):  965-966.  doi:10.1007/s42967-022-00230-0

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On the Vortex Sheets of Compressible Flows

Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan

2023, 5(3):  967-986.  doi:10.1007/s42967-022-00191-4

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This paper provides a review of the recent results on the stability of vortex sheets in compressible flows. Vortex sheets are contact discontinuities of the underlying flows. The vortex sheet problem is a free boundary problem with a characteristic boundary and is challenging in analysis. The formulation of the vortex sheet problem will be introduced. The linear stability and nonlinear stability for both the two-dimensional two-phase compressible flows and the two-dimensional elastic flows are summarized. The linear stability of vortex sheets for the three-dimensional elastic flows is also presented. The difficulties of the vortex sheet problems and the ideas of proofs are discussed.

Recent Progress on Outflow/Inflow Problem for Viscous Multi-phase Flow

Fangfang Hao, Hai-Liang Li, Luyao Shang, Shuang Zhao

2023, 5(3):  987-1014.  doi:10.1007/s42967-022-00194-1

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According to the boundary condition with the zero, negative, or positive velocity, the initial boundary problem for compressible multi-phase flow with the Dirichlet-type boundary condition can be classified into three cases: impermeable problem, inflow problem, or outflow problem. In this paper, we review the recent progress on the existence and nonlinear stability of the stationary solution to the outflow/inflow problems for viscous multi-phase flow.

Two-Dimensional Riemann Problems: Transonic Shock Waves and Free Boundary Problems

Gui-Qiang G. Chen

2023, 5(3):  1015-1052.  doi:10.1007/s42967-022-00210-4

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We are concerned with global solutions of multidimensional (M-D) Riemann problems for nonlinear hyperbolic systems of conservation laws, focusing on their global configurations and structures. We present some recent developments in the rigorous analysis of two-dimensional (2-D) Riemann problems involving transonic shock waves through several prototypes of hyperbolic systems of conservation laws and discuss some further M-D Riemann problems and related problems for nonlinear partial differential equations. In particular, we present four different 2-D Riemann problems through these prototypes of hyperbolic systems and show how these Riemann problems can be reformulated/solved as free boundary problems with transonic shock waves as free boundaries for the corresponding nonlinear conservation laws of mixed elliptic-hyperbolic type and related nonlinear partial differential equations.

ORIGINAL PAPER

Conical Sonic-Supersonic Solutions for the 3-D Steady Full Euler Equations

Yanbo Hu, Xingxing Li

2023, 5(3):  1053-1096.  doi:10.1007/s42967-021-00185-8

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This paper concerns the sonic-supersonic structures of the transonic crossflow generated by the steady supersonic flow past an infinite cone of arbitrary cross section. Under the conical assumption, the three-dimensional (3-D) steady Euler equations can be projected onto the unit sphere and the state of fluid can be characterized by the polar and azimuthal angles. Given a segment smooth curve as a conical-sonic line in the polar-azimuthal angle plane, we construct a classical conical-supersonic solution near the curve under some reasonable assumptions. To overcome the difficulty caused by the parabolic degeneracy, we apply the characteristic decomposition technique to transform the Euler equations into a new degenerate hyperbolic system in a partial hodograph plane. The singular terms are isolated from the highly nonlinear complicated system and then can be handled successfully. We establish a smooth local solution to the new system in a suitable weighted metric space and then express the solution in terms of the original variables.

Radon Measure Solutions to Riemann Problems for Isentropic Compressible Euler Equations of Polytropic Gases

Yunjuan Jin, Aifang Qu, Hairong Yuan

2023, 5(3):  1097-1129.  doi:10.1007/s42967-022-00187-0

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We solve the Riemann problems for isentropic compressible Euler equations of polytropic gases in the class of Radon measures, and the solutions admit the concentration of mass. It is found that under the requirement of satisfying the over-compressing entropy condition: (i) there is a unique delta shock solution, corresponding to the case that has two strong classical Lax shocks; (ii) for the initial data that the classical Riemann solution contains a shock wave and a rarefaction wave, or two shocks with one being weak, there are infinitely many solutions, each consists of a delta shock and a rarefaction wave; (iii) there are no delta shocks for the case that the classical entropy weak solutions consist only of rarefaction waves. These solutions are self-similar. Furthermore, for the generalized Riemann problem with mass concentrated initially at the discontinuous point of initial data, there always exists a unique delta shock for at least a short time. It could be prolonged to a global solution. Not all the solutions are self-similar due to the initial velocity of the concentrated point-mass (particle). Whether the delta shock solutions constructed satisfy the over-compressing entropy condition is clarified. This is the first result on the construction of singular measure solutions to the compressible Euler system of polytropic gases, that is strictly hyperbolic, and whose characteristics are both genuinely nonlinear. We also discuss possible physical interpretations and applications of these new solutions.

Singularity Formation for the General Poiseuille Flow of Nematic Liquid Crystals

Geng Chen, Majed Sofiani

2023, 5(3):  1130-1147.  doi:10.1007/s42967-022-00190-5

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We consider the Poiseuille flow of nematic liquid crystals via the full Ericksen-Leslie model. The model is described by a coupled system consisting of a heat equation and a quasilinear wave equation. In this paper, we will construct an example with a finite time cusp singularity due to the quasilinearity of the wave equation, extended from an earlier result on a special case.

The Perturbed Riemann Problem for a Geometrical Optics System

Shiwei Li, Hanchun Yang

2023, 5(3):  1148-1179.  doi:10.1007/s42967-022-00192-3

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The perturbed Riemann problem for a hyperbolic system of conservation laws arising in geometrical optics with three constant initial states is solved. By studying the interactions among of the delta-shock, vacuum, and contact discontinuity, fourteen kinds of structures of Riemann solutions are obtained. The compound wave solutions consisting of delta-shocks, vacuums, and contact discontinuities are found. The single and double closed vacuum cavitations develop in solutions. Furthermore, it is shown that the solutions of the Riemann problem for the geometrical optics system are stable under certain perturbation of the initial data. Finally, the numerical results completely coinciding with theoretical analysis are presented.

Envelope Method and More General New Global Structures of Solutions for Multi-dimensional Conservation Law

Gui-Qin Qiu, Gao-Wei Cao, Xiao-Zhou Yang, Yuan-An Zhao

2023, 5(3):  1180-1234.  doi:10.1007/s42967-022-00245-7

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For the two-dimensional (2D) scalar conservation law, when the initial data contain two different constant states and the initial discontinuous curve is a general curve, then complex structures of wave interactions will be generated. In this paper, by proposing and investigating the plus envelope, the minus envelope, and the mixed envelope of 2D non-selfsimilar rarefaction wave surfaces, we obtain and the prove the new structures and classifications of interactions between the 2D non-selfsimilar shock wave and the rarefaction wave. For the cases of the plus envelope and the minus envelope, we get and prove the necessary and sufficient criterion to judge these two envelopes and correspondingly get more general new structures of 2D solutions.

Global Existence of Smooth Solutions for the One-Dimensional Full Euler System for a Dusty Gas

Geng Lai, Yingchun Shi

2023, 5(3):  1235-1246.  doi:10.1007/s42967-022-00197-y

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We study the existence of global-in-time classical solutions for the one-dimensional nonisentropic compressible Euler system for a dusty gas with large initial data. Using the characteristic decomposition method proposed by Li et al. (Commun Math Phys 267: 1–12, 2006), we derive a group of characteristic decompositions for the system. Using these characteristic decompositions, we find a sufficient condition on the initial data to ensure the existence of global-in-time classical solutions.

Global Existence and Stability of Solutions to River Flow System

Xian-ting Wang, Yun-guang Lu, Naoki Tsuge

2023, 5(3):  1247-1255.  doi:10.1007/s42967-022-00198-x

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In this short note, we are concerned with the global existence and stability of solutions to the river flow system. We introduce a new technique to set up a relation between the Riemann invariants and the finite mass to obtain a time-independent, bounded solution for any adiabatic exponent. The global existence of solutions was known long ago [Klingenberg and Lu in Commun. Math. Phys. 187: 327–340, 1997]. However, since the uncertainty of the function b(x), which corresponds physically to the slope of the topography, the L^∞ estimates growed larger with respect to the time variable. As a result, it does not guarantee the stability of solutions. By employing a suitable mathematical transformation to control the slope of the topography by the friction and the finite mass, we prove the uniformly bounded estimate with respect to the time variable. This means that our solutions are stable.

Configurations of Shock Regular Reflection by Straight Wedges

Qin Wang, Junhe Zhou

2023, 5(3):  1256-1273.  doi:10.1007/s42967-022-00207-z

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We are concerned with the shock regular reflection configurations of unsteady global solutions for a plane shock hitting a symmetric straight wedge. It has been known that patterns of the shock reflection are various and complicated, including the regular and the Mach reflection. Most of the fundamental issues for the shock reflection have not been understood. Recently, there are great progress on the mathematical theory of the shock regular reflection problem, especially for the global existence, uniqueness, and structural stability of solutions. In this paper, we show that there are two more possible configurations of the shock regular reflection besides known four configurations. We also give a brief proof of the global existence of solutions.

Global Well-Posedness for Aggregation Equation with Time-Space Nonlocal Operator and Shear Flow

Binbin Shi, Weike Wang

2023, 5(3):  1274-1288.  doi:10.1007/s42967-022-00214-0

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In this paper, we consider the two-dimensional aggregation equation with the shear flow and time-space nonlocal attractive operator. Without the advection, the solution of the aggregation equation may blow up in finite time. We show that the shear flow can suppress the blow-up.

Regularity of Fluxes in Nonlinear Hyperbolic Balance Laws

Matania Ben-Artzi, Jiequan Li

2023, 5(3):  1289-1298.  doi:10.1007/s42967-022-00224-y

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This paper addresses the issue of the formulation of weak solutions to systems of nonlinear hyperbolic conservation laws as integral balance laws. The basic idea is that the “meaningful objects” are the fluxes, evaluated across domain boundaries over time intervals. The fundamental result in this treatment is the regularity of the flux trace in the multi-dimensional setting. It implies that a weak solution indeed satisfies the balance law. In fact, it is shown that the flux is Lipschitz continuous with respect to suitable perturbations of the boundary. It should be emphasized that the weak solutions considered here need not be entropy solutions. Furthermore, the assumption imposed on the flux f(u) is quite minimal—just that it is locally bounded.