Yanbo Hu

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Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations

2012-03-08

Yanbo Hu , Jiequan Li , Wancheng Sheng

The two-dimensional Riemann problem for Chaplygin gas dynamics with three constant states

2012-04-21

Guo‐Dong Wang , Bangkao Chen , Yanbo Hu

Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases

2014-06-02

Yanbo Hu , Guodong Wang

Characteristic decomposition of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" display="inline" overflow="scroll"><mml:mn>2</mml:mn><mml:mo>×</mml:mo><mml:mn>2</mml:mn></mml:math> quasilinear strictly hyperbolic systems

2011-08-29

Yanbo Hu , Wancheng Sheng

The Riemann problem of conservation laws in magnetogasdynamics

2012-09-01

Yanbo Hu , Wancheng Sheng

In this paper, we study the Riemann problem for a simplified modelof one dimensional ideal gas in magnetogasdynamics. By using thecharacteristic analysis method, we prove the global existence of solutions … In this paper, we study the Riemann problem for a simplified modelof one dimensional ideal gas in magnetogasdynamics. By using thecharacteristic analysis method, we prove the global existence of solutions to the Riemann problem constructively underthe Lax entropy condition. The image of contact discontinuity inmagnetogasdynamics is a curve with $u=Const.$ in the $(\tau,p,u)$ space. Itsprojection on the $(p,u)$ plane is a straight line that parallels tothe $p$-axis. In contrast with the problem in gas dynamics, theresult causes more complicated and difficult than that in gasdynamics.

On a degenerate boundary value problem to the two-dimensional self-similar nonlinear wave system

2019-01-05

Jiajia Liu , Yanbo Hu , Tie‐Hong Zhao

This paper focuses on a degenerate boundary value problem arising from the study of the two-dimensional Riemann problem to the nonlinear wave system. In order to deal with the parabolic … This paper focuses on a degenerate boundary value problem arising from the study of the two-dimensional Riemann problem to the nonlinear wave system. In order to deal with the parabolic degeneracy, we introduce a partial hodograph transformation to transform the nonlinear wave system into a new system, which displays a clear regularity–singularity structure. The local existence of classical solutions for the new system is established in a weighted metric space. Returning the solution to the original variables, we obtain the existence of classical solutions to the degenerate boundary value problem for the nonlinear wave system.

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Conservative solutions to a one-dimensional nonlinear variational wave equation

2015-02-19

Yanbo Hu

On a degenerate mixed-type boundary value problem to the 2-D steady Euler equation

2019-06-28

Fengyan Li , Yanbo Hu

Asymptotic nonlinear stability of traveling waves to a system of coupled Burgers equations

2012-07-23

Yanbo Hu

Well-posedness of the initial-boundary value problem for 1D degenerate quasilinear wave equations

2024-11-27

Yanbo Hu , Yuusuke Sugiyama

Conservative solutions to a system of variational wave equations

2011-12-19

Yanbo Hu

The interaction of rarefaction waves of a two-dimensional nonlinear wave system

2014-08-05

Yanbo Hu , Guodong Wang

Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations

2019-01-01

Yanbo Hu , Tong Li

This paper is focused on the existence of classical sonic-supersonic solutions near sonic curves for the two-dimensional pseudo-steady full Euler equations in gas dynamics. By introducing a novel set of … This paper is focused on the existence of classical sonic-supersonic solutions near sonic curves for the two-dimensional pseudo-steady full Euler equations in gas dynamics. By introducing a novel set of change variables and using the idea of characteristic decomposition, the Euler system is transformed into a new system which displays a transparent singularity-regularity structure. With a choice of weighted metric space, we establish the local existence of smooth solutions for the new system by the fixed-point method. Finally, we obtain a local classical solution for the pseudo-steady full Euler equations by converting the solution from the partial hodograph variables to the original variables.

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On a global supersonic-sonic patch characterized by 2-D steady full Euler equations

2020-05-01

Yanbo Hu , Jiequan Li

Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and … Supersonic-sonic patches are ubiquitous in regions of transonic flows and they boil down to a family of degenerate hyperbolic problems in regions surrounded by a streamline, a characteristic curve and a possible sonic curve. This paper establishes the global existence of solutions in a whole supersonic-sonic patch characterized by the two-dimensional full system of steady Euler equations and studies solution behaviors near sonic curves, depending on the proper choice of boundary data extracted from the airfoil problem and related contexts. New characteristic decompositions are developed for the full system and a delicate local partial hodograph transformation is introduced for the solution estimates. It is shown that the solution is uniformly $C^{1,\frac{1}{6}}$ continuous up to the sonic curve and the sonic curve is also $C^{1,\frac{1}{6}}$ continuous.

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Sonic-supersonic Solutions to a Mixed-type Boundary Value Problem for the Two-Dimensional Full Euler Equations

2021-01-01

Yanbo Hu , Jianjun Chen

We are concerned with the sonic-supersonic structures extracted from the transonic flow problems in gas dynamics. A local classical supersonic solution for the two-dimensional steady full Euler equations is established … We are concerned with the sonic-supersonic structures extracted from the transonic flow problems in gas dynamics. A local classical supersonic solution for the two-dimensional steady full Euler equations is established in an angular region bounded by the sonic curve and the characteristic curve. In order to overcome the challenges caused by the coupling of nonlinearity and degeneracy at the corner point, we develop a new iteration pattern to show the convergence of the iterative sequence generated by the Euler equations in terms of a partial hodograph coordinate system. The pattern developed here will be useful for studying the degenerate mixed-type boundary value problems for other related nonlinear hyperbolic systems.

Sonic-Supersonic Solutions for the Two-Dimensional Steady Full Euler Equations

2019-10-10

Yanbo Hu , Jiequan Li

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Simple waves and characteristic decompositions of quasilinear hyperbolic systems in two independent variables

2014-05-08

Yanbo Hu , Wancheng Sheng

This paper is concerned with the simple waves for the general quasilinear strictly hyperbolic systems in two independent variables. By using the method of characteristic decomposition, we first establish a … This paper is concerned with the simple waves for the general quasilinear strictly hyperbolic systems in two independent variables. By using the method of characteristic decomposition, we first establish a more general sufficient condition for the existence of characteristic decompositions. These decompositions allow us to extend a well-known result on reducible equations by Courant and Friedrichs to the non-reducible equations. Then we construct a simple wave solution, in which the characteristics may be a set of non-straight curves, around a given curve under the obtained sufficient condition. Copyright © 2014 John Wiley & Sons, Ltd.

An improved regularity result of semi-hyperbolic patch problems for the 2-D isentropic Euler equations

2018-07-30

Yanbo Hu , Tong Li

On the Cauchy problem for a nonlinear variational wave equation with degenerate initial data

2018-07-07

Yanbo Hu , Guodong Wang

Global existence and stability to the polytropic gas dynamics with an outer force

2019-03-19

Yanbo Hu , Yun‐guang Lu , Naoki Tsuge

On a global supersonic-sonic patch characterized by 2-D steady full Euler equations

2019-01-01

Yanbo Hu , Jiequan Li

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Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors

2020-08-25

Yanbo Hu , Christian Klingenberg , Yun‐guang Lu

Axisymmetric solutions of the pressure-gradient system

2012-07-01

Yanbo Hu

A family of self-similar and global bounded weak solutions are constructed rigorously for all positive time to the two-dimensional pressure-gradient system in this paper. By using the axisymmetric and self-similar … A family of self-similar and global bounded weak solutions are constructed rigorously for all positive time to the two-dimensional pressure-gradient system in this paper. By using the axisymmetric and self-similar assumptions, the equations are reduced to a system of ordinary differential equations, from which we obtain detailed structures of solutions as well as their existence. On this basis, we explore the behavior of characteristics near the sonic boundary.

Axisymmetric solutions of the two-dimensional Euler equations with a two-constant equation of state

2013-07-01

Yanbo Hu

Weak solutions to a nonlinear variational sine–Gordon equation

2013-01-07

Yanbo Hu , Guo‐Dong Wang

On the existence of solutions to a one-dimensional degenerate nonlinear wave equation

2018-02-26

Yanbo Hu

Singularity and existence for a multidimensional variational wave equation arising from nematic liquid crystals

2020-03-09

Wenhui Duan , Yanbo Hu , Guodong Wang

On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry

2020-10-18

Yanbo Hu , Fengyan Li

Abstract The transonic channel flow problem is one of the most important problems in mathematical fluid dynamics. The structure of solutions near the sonic curve is a key part of … Abstract The transonic channel flow problem is one of the most important problems in mathematical fluid dynamics. The structure of solutions near the sonic curve is a key part of the whole transonic flow problem. This paper constructs a local classical hyperbolic solution for the 3-D axisymmetric steady compressible full Euler equations with boundary data given on the degenerate hyperbolic curve. By introducing a novel set of dependent and independent variables, we use the idea of characteristic decomposition to transform the axisymmetric Euler equations as a new system which has explicitly singularity-regularity structures. We first establish a local classical solution for the new system in a weighted metric space and then convert the solution in terms of the original variables.

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Sonic–supersonic solutions to a degenerate Cauchy–Goursat problem for 2D relativistic Euler equations

2021-12-20

Yongqiang Fan , Lihui Guo , Yanbo Hu +1 more

Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals

2022-11-19

Yanbo Hu

Abstract This article focuses on the singularity formation of smooth solutions for a one-dimensional nonlinear degenerate hyperbolic-parabolic coupled system originating from the Poiseuille flow of nematic liquid crystals. Without assuming … Abstract This article focuses on the singularity formation of smooth solutions for a one-dimensional nonlinear degenerate hyperbolic-parabolic coupled system originating from the Poiseuille flow of nematic liquid crystals. Without assuming that the wave speed of the hyperbolic equation is a positive function, we show that its smooth solution will break down in finite time even for an arbitrarily small initial energy. Based on an estimate of the solution for the heat equation, we use the method of characteristics to control the wave speed and its derivative so that the wave speed does not degenerate and its derivative does not change sign in a period of time.

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Singularity for the one-dimensional rotating Euler equations of Chaplygin gases

2022-11-19

Ping Lv , Yanbo Hu

Global energy conservative solutions to a system of variational wave equations

2012-09-18

Yanbo Hu

The Roe-type interface flux for conservation laws with discontinuous flux function

2017-07-08

Guodong Wang , Yanbo Hu

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Existence of smooth solutions to a one-dimensional nonlinear degenerate variational wave equation

2017-10-16

Yanbo Hu , Guodong Wang

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Sonic-supersonic solutions for the two dimensional steady relativistic Euler equations

2022-03-04

Yongqiang Fan , Lihui Guo , Yanbo Hu +1 more

On a Supersonic-Sonic Patch in the Three-Dimensional Steady Axisymmetric Transonic Flows

2022-03-07

Yanbo Hu

This paper focuses on the structure of solutions in a supersonic bubble arising from the three-dimensional transonic flows. Given a velocity distribution on a streamline, we construct a small smooth … This paper focuses on the structure of solutions in a supersonic bubble arising from the three-dimensional transonic flows. Given a velocity distribution on a streamline, we construct a small smooth supersonic-sonic patch for the three-dimensional axisymmetric steady isentropic irrotational Euler equations. This patch can be regarded as the region near the upstream vertex of a supersonic bubble. The main difficulty is the coupling of nonhomogeneous terms and sonic degeneracy. To overcome it, we adopt the idea of characteristic decompositions to solve the problem in a partial hodograph coordinate system composed by the Mach angle and the potential function. By converting to the physical plane, a smooth solution for the original problem is established and the uniform regularity of solution up to the sonic curve is also discussed.

Conservative solutions to a nonlinear variational sine-Gordon equation

2011-07-25

Yanbo Hu

The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations

2019-01-01

Yanbo Hu , Tong Li

We study the regularity of solution and of sonic boundary to a degenerate Goursat problem originated from the two-dimensional Riemann problem of the compressible isothermal Euler equations. By using the … We study the regularity of solution and of sonic boundary to a degenerate Goursat problem originated from the two-dimensional Riemann problem of the compressible isothermal Euler equations. By using the ideas of characteristic decomposition and the bootstrap method, we show that the solution is uniformly ${C^{1,\frac{1}{6}}}$ up to the degenerate sonic boundary and that the sonic curve is ${C^{1,\frac{1}{6}}}$.

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On a supersonic-sonic patch arising from the frankl problem in transonic flows

2021-01-01

Yanbo Hu , Jiequan Li

<p style="text-indent:20px;">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 … <p style="text-indent:20px;">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 airfoil. Based on the methodology of characteristic decompositions, we establish the global existence and regularity of solutions in a partial hodograph coordinate system in terms of angle variables. The original problem is solved by transforming the solution in the partial hodograph plane back to that in the physical plane. Moreover, the uniform regularity of the solution and the regularity of an associated sonic curve are also verified.</p>

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The interaction of rarefaction waves for a system of granular flow equations

2022-10-14

Binyu Zhang , Yanbo Hu

An initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations

2023-05-18

Jueling Ye , Houbin Guo , Yanbo Hu

Global solutions to a nonlinear wave system arising from cholesteric liquid crystals

2017-03-01

Yanbo Hu

We investigate a one-dimensional nonlinear wave system which is the Euler–Lagrange equations of a variational principle modeling a type of cholesteric liquid crystals. Due to the chiral effect, the Lagrangian … We investigate a one-dimensional nonlinear wave system which is the Euler–Lagrange equations of a variational principle modeling a type of cholesteric liquid crystals. Due to the chiral effect, the Lagrangian density of the variational principle includes linear terms as well as quadratic terms in derivatives of the field, which is different from the nematic liquid crystals. Moreover, the linear terms and zero term in derivatives of the field might cause the energy to be negative in some cases. By using the method of energy-dependent coordinates and the Young measure theory, we establish the global existence of weak solutions to its Cauchy problem under some conditions on the coefficients.

Self-Similar Solutions to the Spherically-Symmetric Euler Equations with a Two-Constant Equation of State

2019-03-01

Qitao Zhang , Yanbo Hu

On a regularity result for the transonic pressure-gradient system in gas dynamics

2019-09-11

Huijuan Song , Yanbo Hu

The resonance behavior for the coupling of two Aw–Rascle traffic models

2020-02-07

Yanbo Hu , Guodong Wang

Abstract By studying the Riemann problem for the Aw–Rascle traffic model with different pressure laws, which is the coupling of two one-dimensional hyperbolic systems, we investigate the resonance phenomena. The … Abstract By studying the Riemann problem for the Aw–Rascle traffic model with different pressure laws, which is the coupling of two one-dimensional hyperbolic systems, we investigate the resonance phenomena. The main difficulty arises from the possible resonance behavior which may result in multiple solutions. We discover a new and interesting phenomenon showing that there exist infinitely many solutions for some certain initial data, which is quite different compared to earlier studies for the isentropic model of a fluid flow in a nozzle with variable cross-section and the shallow water equations with discontinuous topography. In order to overcome this difficulty, we impose the so-called TV-condition to obtain the uniqueness of solution to the Riemann problem.

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On the degenerate Cauchy problem for a nonlinear variational wave system

2022-11-29

Yanbo Hu , Huijuan Song

An initial-boundary value problem for the two-dimensional rotating shallow water equations with axisymmetry

2022-09-01

Yanbo Hu , Yating Qian

This paper is concerned with an initial-boundary value problem for the two-dimensional axisymmetric rotating shallow water equations. The Dirichlet boundary conditions are imposed only on the radial velocity, while no … This paper is concerned with an initial-boundary value problem for the two-dimensional axisymmetric rotating shallow water equations. The Dirichlet boundary conditions are imposed only on the radial velocity, while no boundary condition is imposed on the height of the fluid or the angular velocity. A series of a priori estimates for the solution of the approximate linear problem are derived and the strong convergence of the approximate solution sequences is verified. Consequently, we establish the local well-posedness in time of strong solutions for the initial-boundary value problem of the model.

Semi-hyperbolic patch characterized by 2D steady relativistic Euler equations

2023-01-27

Yongqiang Fan , Lihui Guo , Yanbo Hu +1 more

The solvability for a nonlinear degenerate hyperbolic–parabolic coupled system arising from nematic liquid crystals

2023-10-13

Yanbo Hu

Abstract This paper focuses on the Cauchy problem for a one-dimensional quasilinear hyperbolic–parabolic coupled system with initial data given on a line of parabolicity. The coupled system is derived from … Abstract This paper focuses on the Cauchy problem for a one-dimensional quasilinear hyperbolic–parabolic coupled system with initial data given on a line of parabolicity. The coupled system is derived from the Poiseuille flow of full Ericksen–Leslie model in the theory of nematic liquid crystals, which incorporates the crystal and liquid properties of the materials. The main difficulty comes from the degeneracy of the hyperbolic equation, which makes that the system is not continuously differentiable and then the classical methods for the strictly hyperbolic–parabolic coupled systems are invalid. With a choice of a suitable space for the unknown variable of the parabolic equation, we first solve the degenerate hyperbolic problem in a partial hodograph plane and express the smooth solution in terms of the original variables. Based on the smooth solution of the hyperbolic equation, we then construct an iterative sequence for the unknown variable of the parabolic equation by the fundamental solution of the heat equation. Finally, we verify the uniform convergence of the iterative sequence in the selected function space and establish the local existence and uniqueness of classical solutions to the degenerate coupled problem.

Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations

2023-10-20

Yanbo Hu

Lie symmetry analysis and solutions for a fractional Riccati–Bernoulli equation

2025-05-01

Shaoru Liu , Yanbo Hu

Well-Posedness of Degenerate Initial-Boundary Value Problems to a Hyperbolic-Parabolic Coupled System Arising from Nematic Liquid Crystals

2025-02-28

Yanbo Hu , Yuusuke Sugiyama

A class of generalized Liénard equations with high power damping and its applications to the KP-Burgers type equation

2025-01-31

Yanbo Hu , Shaoru Liu

Abstract In this paper, a class of generalized Liénard equations with high power damping, which can describe the dynamic behavior of many physical phenomena, is considered. The property of integrating … Abstract In this paper, a class of generalized Liénard equations with high power damping, which can describe the dynamic behavior of many physical phenomena, is considered. The property of integrating factors of the equations is investigated, and the corresponding first integral can be derived. Specially, the explicit expressions of integrating factors of several families of the equations with n=2 are obtained. The linearizable family of the equations via the certain non-local transformation is given, and an explicit expression that connects integrating factors of the linearizable equations and that of linear equations is provided. Finally, the applications to a class of (2+1)-dimensional KP-Burgers type equation are proposed, and the linearization condition of traveling wave reduction of the equation is obtained, therefore, the corresponding traveling wave solutions of the original partial differential equation can be deduced. Furthermore, the three-dimensional images of the traveling wave solutions are provided for a better understanding of the behavior of the solutions.

Global existence of solutions to a one dimension nonlinear wave equation

2024-12-01

Ying Zeng , Yanbo Hu

Well-posedness of the initial-boundary value problem for 1D degenerate quasilinear wave equations

2024-11-27

Yanbo Hu , Yuusuke Sugiyama

On the analytical construction of radially symmetric solutions for the relativistic Euler equations

2024-10-17

Yanbo Hu , Binyu Zhang

Abstract This paper is concerned with the analytical construction of piecewise smooth solutions containing a single shock wave for the radially symmetric relativistic Euler equations with polytropic gases. We derive … Abstract This paper is concerned with the analytical construction of piecewise smooth solutions containing a single shock wave for the radially symmetric relativistic Euler equations with polytropic gases. We derive meticulously the a priori ‐estimates on the Riemann invariants of the governing system under some assumptions on the piecewise initial data. Based on these estimates, we show that the long time of existence of smooth solutions in the angular region bounded by a characteristic curve and a shock curve. The piecewise smooth initial conditions ensured the existence of smooth solutions in the angular region are discussed. Moreover, it is verified that the existence time is proportional to the initial discontinuous position.

On a supersonic-sonic patch arising from the two-dimensional Riemann problem of the compressible Euler equations

2024-09-18

Yanbo Hu , Guodong Wang

We are interested in the two-dimensional four-constant Riemann problem to the isentropic compressible Euler equations. In terms of the self-similar variables, the governing system is of nonlinear mixed-type and the … We are interested in the two-dimensional four-constant Riemann problem to the isentropic compressible Euler equations. In terms of the self-similar variables, the governing system is of nonlinear mixed-type and the solution configuration typically contains transonic and small-scale structures. We construct a supersonic-sonic patch along a pseudo-streamline from the supersonic part to a sonic point. This kind of patch appears frequently in the two-dimensional Riemann problem and is a building block for constructing a global solution. To overcome the difficulty caused by the sonic degeneracy, we apply the characteristic decomposition technique to handle the problem in a partial hodograph plane. We establish a regular supersonic solution for the original problem by showing the global one-to-one property of the partial hodograph transformation. The uniform regularity of the solution and the regularity of an associated sonic curve are also discussed.

The classical solvability for a one-dimensional nonlinear thermoelasticity system with the far field degeneracy

2024-06-14

Yanbo Hu , Yuusuke Sugiyama

Abstract We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation … Abstract We study the local classical solvability of the Cauchy problem to the equations of one-dimensional nonlinear thermoelasticity. The governing model is a coupled system of a nonlinear hyperbolic equation for the displacement and a parabolic equation for the temperature of the elastic material. We allow the hyperbolic equation to degenerate at spacial infinity, which results that the coefficients of the coupled system are not uniformly bounded and then the previous methods for the strictly hyperbolic-parabolic coupled systems are invalid. We introduce a suitable weighted norm to establish the local existence and uniqueness of classical solutions by the contraction mapping principle. The existence time T of the solution is independent of the spatial variable.

Continuous/smooth transonic solutions to the quasi-one-dimensional steady relativistic Euler equations

2024-05-07

Yanbo Hu

Global solution and singularity formation for the supersonic expanding wave of compressible Euler equations with radial symmetry

2024-04-11

Geng Chen , Faris A. El-Katri , Yanbo Hu +1 more

In this paper, we define the rarefaction and compression characters for the supersonic expanding wave of the compressible Euler equations with radial symmetry. Under this new definition, we show that … In this paper, we define the rarefaction and compression characters for the supersonic expanding wave of the compressible Euler equations with radial symmetry. Under this new definition, we show that solutions with rarefaction initial data will not form shock in finite time, i.e. exist global-in-time as classical solutions. On the other hand, singularity forms in finite time when the initial data include strong compression somewhere. Several useful invariant domains will be also given.

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Singularity formation for the cylindrically symmetric rotating relativistic Euler equations of Chaplygin gases

2024-03-28

Yanbo Hu , Houbin Guo

Abstract This paper studies the formation of singularities in smooth solutions of the relativistic Euler equations of Chaplygin gases with cylindrically symmetric rotating structures. This is a nonhomogeneous hyperbolic system … Abstract This paper studies the formation of singularities in smooth solutions of the relativistic Euler equations of Chaplygin gases with cylindrically symmetric rotating structures. This is a nonhomogeneous hyperbolic system with highly nonlinear structures and fully linearly degenerating characteristic fields. We introduce a pair of auxiliary functions and use the characteristic decomposition technique to overcome the influence of the rotating structures in the system. It is verified that smooth solutions develop into a singularity in finite time and the mass-energy density tends to infinity at the blowup point for a type of rotating initial data.

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Initial-Boundary Value Problems for Poiseuille Flow of Nematic Liquid Crystal via Full Ericksen–Leslie Model

2024-03-01

Geng Chen , Yanbo Hu , Qingtian Zhang

.In this paper, we study the initial-boundary value problem for the Poiseuille flow of a hyperbolic-parabolic Ericksen–Leslie model of nematic liquid crystals in one space dimension. We consider a simplified … .In this paper, we study the initial-boundary value problem for the Poiseuille flow of a hyperbolic-parabolic Ericksen–Leslie model of nematic liquid crystals in one space dimension. We consider a simplified system by restricting the Leslie coefficients to special cases such that some quantities are constants. Due to the quasilinearity, the solution of this model in general forms cusp singularity. We prove the global existence of a Hölder continuous solution, which may include cusp singularity, for initial-boundary value problems with different types of boundary conditions.Keywordsliquid crystalEricksen–LesliePoiseuille flowglobal existenceinitial-boundary value problemMSC codes35M3335L5376D03

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The Riemann problem for two-layer shallow water equations with bottom topography

2024-01-01

Fan Wang , Wancheng Sheng , Yanbo Hu +1 more

Abstract In this article, we propose a model describing laminar shallow water flows with two velocities to handle an uneven bottom. The model is established by taking the different velocities … Abstract In this article, we propose a model describing laminar shallow water flows with two velocities to handle an uneven bottom. The model is established by taking the different velocities into account. The source terms generated from the bottom topography prevent us from solving the Riemann problem directly. We first derive the elementary waves including the stationary wave where the global entropy condition is used to ensure uniqueness. Then, we analyze the resonance phenomenon and coalescence of waves by classifying the initial data into different regions. Finally, the Riemann problem is resolved explicitly on a case-by-case basis.

Sonic-supersonic solutions for the two-dimensional steady compressible multiphase flow equations

2023-10-20

Yanbo Hu

The solvability for a nonlinear degenerate hyperbolic–parabolic coupled system arising from nematic liquid crystals

2023-10-13

Yanbo Hu

An initial-boundary value problem for the one-dimensional rotating shallow water magnetohydrodynamic equations

2023-05-18

Jueling Ye , Houbin Guo , Yanbo Hu

Semi-hyperbolic patch characterized by 2D steady relativistic Euler equations

2023-01-27

Yongqiang Fan , Lihui Guo , Yanbo Hu +1 more

On a degenerate mixed-type boundary value problem for the two-dimensional self-similar Euler equations

2023-01-01

Yanbo Hu

Initial-boundary value problems for Poiseuille flow of nematic liquid crystal via full Ericksen-Leslie model

2023-01-01

Geng Chen , Yanbo Hu , Qingtian Zhang

In this paper, we study the initial-boundary value problem for the Poiseuille flow of hyperbolic-parabolic Ericksen-Leslie model of nematic liquid crystals in one space dimension. Due to the quasilinearity, the … In this paper, we study the initial-boundary value problem for the Poiseuille flow of hyperbolic-parabolic Ericksen-Leslie model of nematic liquid crystals in one space dimension. Due to the quasilinearity, the solution of this model in general forms cusp singularity. We prove the global existence of H\"older continuous solution, which may include cusp singularity, for initial-boundary value problems with different types of boundary conditions.

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Global solutions to nonlinear wave equations arising from a variational principle

2023-01-01

Ying Zeng , Yanbo Hu

In this paper, we establish the global existence of weak solutions to the initial-boundary value and initial value problems for two classes of nonlinear wave equations which are the Euler-Lagrange … In this paper, we establish the global existence of weak solutions to the initial-boundary value and initial value problems for two classes of nonlinear wave equations which are the Euler-Lagrange equation of a variational principle.We use the method of energy-dependent coordinates to rewrite these equations as semilinear systems and resolve all singularities by introducing a new set of dependent and independent variables.The global weak solutions can be constructed by expressing the solutions of these semilinear systems in terms of the original variables.

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On the degenerate Cauchy problem for a nonlinear variational wave system

2022-11-29

Yanbo Hu , Huijuan Song

Singularity for a nonlinear degenerate hyperbolic-parabolic coupled system arising from nematic liquid crystals

2022-11-19

Yanbo Hu

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Singularity for the one-dimensional rotating Euler equations of Chaplygin gases

2022-11-19

Ping Lv , Yanbo Hu

The interaction of rarefaction waves for a system of granular flow equations

2022-10-14

Binyu Zhang , Yanbo Hu

An initial-boundary value problem for the two-dimensional rotating shallow water equations with axisymmetry

2022-09-01

Yanbo Hu , Yating Qian

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

2022-04-20

Yanbo Hu , Xingxing Li

On a Supersonic-Sonic Patch in the Three-Dimensional Steady Axisymmetric Transonic Flows

2022-03-07

Yanbo Hu

Sonic-supersonic solutions for the two dimensional steady relativistic Euler equations

2022-03-04

Yongqiang Fan , Lihui Guo , Yanbo Hu +1 more

Sonic–supersonic solutions to a degenerate Cauchy–Goursat problem for 2D relativistic Euler equations

2021-12-20

Yongqiang Fan , Lihui Guo , Yanbo Hu +1 more

Sonic-supersonic solutions for a reactive transonic small disturbance model

2021-05-31

Ping Lv , Yanbo Hu

On a supersonic-sonic patch arising from the Frankl problem in transonic flows

2021-01-01

Yanbo Hu , Jiequan Li

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 … 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 airfoil. Based on the methodology of characteristic decompositions, we establish the global existence and regularity of solutions in a partial hodograph coordinate system in terms of angle variables. The original problem is solved by transforming the solution in the partial hodograph plane back to that in the physical plane. Moreover, the uniform regularity of the solution and the regularity of an associated sonic curve are also verified.

On a supersonic-sonic patch arising from the frankl problem in transonic flows

2021-01-01

Yanbo Hu , Jiequan Li

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Sonic-supersonic Solutions to a Mixed-type Boundary Value Problem for the Two-Dimensional Full Euler Equations

2021-01-01

Yanbo Hu , Jianjun Chen

On a degenerate hyperbolic problem for the 3-D steady full Euler equations with axial-symmetry

2020-10-18

Yanbo Hu , Fengyan Li

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Zero relaxation time limits to a hydrodynamic model of two carrier types for semiconductors

2020-08-25

Yanbo Hu , Christian Klingenberg , Yun‐guang Lu

On a global supersonic-sonic patch characterized by 2-D steady full Euler equations

2020-05-01

Yanbo Hu , Jiequan Li

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Singularity and existence for a multidimensional variational wave equation arising from nematic liquid crystals

2020-03-09

Wenhui Duan , Yanbo Hu , Guodong Wang

The resonance behavior for the coupling of two Aw–Rascle traffic models

2020-02-07

Yanbo Hu , Guodong Wang

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Sonic-Supersonic Solutions for the Two-Dimensional Steady Full Euler Equations

2019-10-10

Yanbo Hu , Jiequan Li

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On a regularity result for the transonic pressure-gradient system in gas dynamics

2019-09-11

Huijuan Song , Yanbo Hu

On a degenerate mixed-type boundary value problem to the 2-D steady Euler equation

2019-06-28

Fengyan Li , Yanbo Hu

Global existence and stability to the polytropic gas dynamics with an outer force

2019-03-19

Yanbo Hu , Yun‐guang Lu , Naoki Tsuge

Spherically Symmetric Solutions of the Full Compressible Euler Equations in $$R^N$$RN

2019-03-08

Yanbo Hu , Jiajia Liu

Self-Similar Solutions to the Spherically-Symmetric Euler Equations with a Two-Constant Equation of State

2019-03-01

Qitao Zhang , Yanbo Hu

On a degenerate boundary value problem to the two-dimensional self-similar nonlinear wave system

2019-01-05

Jiajia Liu , Yanbo Hu , Tie‐Hong Zhao

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The regularity of a degenerate Goursat problem for the 2-D isothermal Euler equations

2019-01-01

Yanbo Hu , Tong Li

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Sonic-supersonic solutions for the two-dimensional pseudo-steady full Euler equations

2019-01-01

Yanbo Hu , Tong Li

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Singularity for a multidimensional variational wave equation arising from nematic liquid crystals

2019-01-01

Yanbo Hu , Guodong Wang

This article is focused on a multidimensional nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle arising form the theory of nematic liquid crystals. By using … This article is focused on a multidimensional nonlinear variational wave equation which is the Euler-Lagrange equation of a variational principle arising form the theory of nematic liquid crystals. By using the method of characteristics, we show that the smooth solutions for the spherically-symmetric variational wave equation breakdown in finite time, even for the arbitrarily small initial energy.

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On the degenerate Cauchy problem for a nonlinear variational wave system, Part I: The same wave speed case

2019-01-01

Yanbo Hu , Huijuan Song

We investigate a one-dimensional nonlinear wave system which arises from a variational principle modeling a type of cholesteric liquid crystals. The problem treated here is the Cauchy problem for the … We investigate a one-dimensional nonlinear wave system which arises from a variational principle modeling a type of cholesteric liquid crystals. The problem treated here is the Cauchy problem for the same wave speed case with initial data on the parabolic degenerating line. By introducing a partial hodograph transformation, we establish the local existence of smooth solutions in a weighted metric space based on the iteration method. A classical solution of the primary problem is constructed by converting the solution in the partial hodograph variables to that in the original variables.

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On a global supersonic-sonic patch characterized by 2-D steady full Euler equations

2019-01-01

Yanbo Hu , Jiequan Li

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Coauthor	Papers Together
Jiequan Li	6
Wancheng Sheng	5
Guodong Wang	4
Guodong Wang	4
Huijuan Song	3
Yuusuke Sugiyama	3
Shouke You	3
Tong Li	3
Yongqiang Fan	3
Lihui Guo	3
Guo‐Dong Wang	2
Shaoru Liu	2
Houbin Guo	2
Guodong Wang	2
Fengyan Li	2
Yun‐guang Lu	2
Ying Zeng	2
Jiajia Liu	2
Binyu Zhang	2
Qinglong Zhang	1
Qingtian Zhang	1
Qingtian Zhang	1
Jueling Ye	1
Fan Wang	1
Huayong Liu	1
Qitao Zhang	1
Ping Lv	1
Wenhui Duan	1
Guodong Wang	1
Chaoran Zheng	1
Bangkao Chen	1
Tie‐Hong Zhao	1
Christian Klingenberg	1
Naoki Tsuge	1
Faris A. El-Katri	1
Jianjun Chen	1
Geng Chen	1
Geng Chen	1
Geng Chen	1
Xingxing Li	1
Ping Lv	1
Yating Qian	1
Yannan Shen	1

Sonic-supersonic solutions for the steady Euler equations

2014-01-01

Tianyou Zhang , Yuxi Zheng

Semi-hyperbolic Patches of Solutions to the Two-dimensional Euler Equations

2011-03-21

Mingjie Li , Yuxi Zheng

Interaction of Rarefaction Waves of the Two-Dimensional Self-Similar Euler Equations

2008-10-06

Jiequan Li , Yuxi Zheng

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Interaction of Four Rarefaction Waves in the Bi-Symmetric Class of the Two-Dimensional Euler Equations

2010-02-23

Jiequan Li , Yuxi Zheng

Supersonic Flow and Shock Waves

1949-05-01

M. J. Lighthill

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Weak Solutions to a Nonlinear Variational Wave Equation

2003-03-01

Ping Zhang , Yuxi Zheng

Conservative Solutions to a Nonlinear Variational Wave Equation

2006-05-24

Alberto Bressan , Yuxi Zheng

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Dynamics of Director Fields

1991-12-01

John K. Hunter , Ralph Saxton

An asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed. The equation is used to study a nonlinear instability in the director field of … An asymptotic equation for weakly nonlinear hyperbolic waves governed by variational principles is derived and analyzed. The equation is used to study a nonlinear instability in the director field of a nematic liquid crystal. It is shown that smooth solutions of the asymptotic equation break down in finite time. Also constructed are weak solutions of the equation that are continuous despite the fact that their spatial derivative blows up.

The Regularity of Semihyperbolic Patches Near Sonic Lines for The 2-D Euler System in Gas Dynamics

2015-01-01

Kyungwoo Song , Qin Wang , Yuxi Zheng

We study the regularity of semihyperbolic patches of self-similar solutions near sonic lines to a Riemann problem for the two-dimensional (2-D) Euler system. As a result, it is verified that … We study the regularity of semihyperbolic patches of self-similar solutions near sonic lines to a Riemann problem for the two-dimensional (2-D) Euler system. As a result, it is verified that there exists a global solution in the semihyperbolic patch up to the sonic boundary and that the sonic boundary has $C^1$-regularity. The study of the semihyperbolic patches of solutions for the Euler system was initiated by Li and Zheng [Arch. Rational Mech. Anal., 201 (2011), pp. 1069--1096]. This type of solution appears in the transonic flow over an airfoil and Guderley reflection and is common in the numerical configurations of 2-D Riemann problems.

Singularities of a Variational Wave Equation

1996-07-01

Robert T. Glassey , John K. Hunter , Yuxi Zheng

Simple Waves and a Characteristic Decomposition of the Two Dimensional Compressible Euler Equations

2006-05-04

Jiequan Li , Tong Zhang , Yuxi Zheng

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Weak solutions to a nonlinear variational wave equation with general data

2005-01-27

Ping Zhang , Yuxi Zheng

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Semi-hyperbolic patches of solutions of the pressure gradient system

2009-01-01

Kyungwoo Song , Yuxi Zheng

We construct patches of self-similarsolutions, in which one family out of two nonlinear families ofcharacteristics starts on sonic curves and ends on transonic shockwaves, to the two-dimensional pressure gradient system. … We construct patches of self-similarsolutions, in which one family out of two nonlinear families ofcharacteristics starts on sonic curves and ends on transonic shockwaves, to the two-dimensional pressure gradient system. This typeof solutions is common in the solutions of two-dimensional Riemannproblems, as seen from numerical experiments. They are notdetermined by the hyperbolic domain of determinacy in thetraditional sense. They are middle-way between the fully hyperbolic(supersonic) and elliptic region, which we call semi-hyperbolic orpartially hyperbolic. Our intention is to use the patches asbuilding tiles to construct global solutions to general Riemannproblems.

Energy Conservative Solutions to a One‐Dimensional Full Variational Wave System

2011-07-05

Ping Zhang , Yuxi Zheng

Abstract We establish the existence of a conservative weak solution to the initial value problem for a complete system of variational wave equations modeling liquid crystals in one space dimension, … Abstract We establish the existence of a conservative weak solution to the initial value problem for a complete system of variational wave equations modeling liquid crystals in one space dimension, in which the director has two degrees of freedom. The solutions exist globally in time and singularities may develop in finite time, but the energy of the solutions is conserved across singular times. The method for existence also yields continuous dependence of solutions on the initial data. © 2011 Wiley Periodicals, Inc.

Global Semigroup of Conservative Solutions of the Nonlinear Variational Wave Equation

2011-02-23

Helge Holden , Xavier Raynaud

We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u tt − c(u)(c(u)u x ) x = 0. We allow for initial … We prove the existence of a global semigroup for conservative solutions of the nonlinear variational wave equation u tt − c(u)(c(u)u x ) x = 0. We allow for initial data u|t = 0 and u t |t=0 that contain measures. We assume that $${0 < \kappa^{-1} \leqq c(u) \leqq \kappa}$$ . Solutions of this equation may experience concentration of the energy density $${(u_t^2+c(u)^2u_x^2){\rm d}x}$$ into sets of measure zero. The solution is constructed by introducing new variables related to the characteristics, whereby singularities in the energy density become manageable. Furthermore, we prove that the energy may focus only on a set of times of zero measure or at points where c′(u) vanishes. A new numerical method for constructing conservative solutions is provided and illustrated with examples.

Degenerate Goursat-type boundary value problems arising from the study of two-dimensional isothermal Euler equations

2012-03-08

Yanbo Hu , Jiequan Li , Wancheng Sheng

RAREFACTIVE SOLUTIONS TO A NONLINEAR VARIATIONAL WAVE EQUATION OF LIQUID CRYSTALS

2001-03-31

Ping Zhang , Yuxi Zheng

We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous … We study a nonlinear wave equation derived from a simplified liquid crystal model, in which the wave speed is a given function of the wave amplitude. We formulate a viscous approximation of the equation and establish the global existence of smooth solutions for the viscously perturbed equation. For a monotone wave speed function in the equation, we find an invariant region in the phase space in which we discover: (a) smooth data evolve smoothly forever; (b) both the viscous regularization and the smooth solutions obtained through data smoothing for rough initial data yield weak solutions to the Cauchy problem of the nonlinear variational wave equation. The main tool is the Young measure theory and related techniques.

Semi-hyperbolic patches of solutions to the two-dimensional nonlinear wave system for Chaplygin gases

2014-06-02

Yanbo Hu , Guodong Wang

Unique Conservative Solutions to a Variational Wave Equation

2015-02-06

Alberto Bressan , Geng Chen , Qingtian Zhang

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On the non‐existence of continuous transonic flows past profiles I

1956-02-01

Cathleen S. Morawetz

Lipschitz Metrics for a Class of Nonlinear Wave Equations

2017-07-24

Alberto Bressan , Geng Chen

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Conservative solutions to a one-dimensional nonlinear variational wave equation

2015-02-19

Yanbo Hu

Supersonic Flow and Shock Waves

1976-01-01

Richard Courant , Kurt Friedrichs

Orientation waves in a director field with rotational inertia

2009-01-01

Giuseppe Alı̀ , John K. Hunter

We study a variational system of nonlinear hyperbolic partial differential equationsthat describes the propagation of orientation waves in a director fieldwith rotational inertia and potential energy given by the Oseen-Frankenergy … We study a variational system of nonlinear hyperbolic partial differential equationsthat describes the propagation of orientation waves in a director fieldwith rotational inertia and potential energy given by the Oseen-Frankenergy from the continuum theory of nematic liquid crystals.There are two types of waves,which we call splay and twist waves, respectively.Weakly nonlinear splay waves are describedby the quadratically nonlinear Hunter-Saxton equation.In this paper, we derive a new cubically nonlinearasymptotic equation that describesweakly nonlinear twist waves. This equation providesa surprising representation of the Hunter-Saxton equation, andlike the Hunter-Saxton equation it is completely integrable.There are analogous cubically nonlinear representations of the Camassa-Holmand Degasperis-Procesi equations.Moreover, two different, but compatible, variational principles for the Hunter-Saxtonequation arise from a single variational principlefor the primitive director field equations in the two differentlimits for splay and twist waves. We also usethe asymptotic equation to analyzea one-dimensional initial value problem for thedirector-field equations with twist-wave initial data.

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Conservative solutions to a system of variational wave equations

2011-12-19

Yanbo Hu

Centered wave bubbles with sonic boundary of pseudosteady Guderley Mach reflection configurations in gas dynamics

2015-02-24

Geng Lai , Wancheng Sheng

Conjecture on the Structure of Solutions of the Riemann Problem for Two-Dimensional Gas Dynamics Systems

1990-05-01

Tong Zhang , Yu Xi Zheng

Two-dimensional flow of polytropic gas with initial data being constant in each quadrant is considered. Under the assumption that each jump in initial data outside of the origin projects exactly … Two-dimensional flow of polytropic gas with initial data being constant in each quadrant is considered. Under the assumption that each jump in initial data outside of the origin projects exactly one planar wave of shocks, centered rarefaction waves, or slip planes, it is proved that only 16 combinations of initial data are reasonable. For each combination, a conjecture on the structure of the solution in the whole space $t > 0$ is given.

On a weak solution for a transonic flow problem

1985-11-01

Cathleen S. Morawetz

Characteristic decompositions and interactions of rarefaction waves of 2-D Euler equations

2010-07-22

Jiequan Li , Zhi-Cheng Yang , Yuxi Zheng

Diffractive nonlinear geometrical optics for variational wave equations and the Einstein equations

2007-07-02

Giuseppe Alı̀ , John K. Hunter

Abstract We derive an asymptotic solution of the vacuum Einstein equations that describe the propagation and diffraction of a localized, large‐amplitude, rapidly varying gravitational wave. We compare and contrast the … Abstract We derive an asymptotic solution of the vacuum Einstein equations that describe the propagation and diffraction of a localized, large‐amplitude, rapidly varying gravitational wave. We compare and contrast the resulting theory of strongly nonlinear geometrical optics for the Einstein equations with nonlinear geometrical optics theories for variational wave equations. © 2007 Wiley Periodicals, Inc.

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Generic regularity of conservative solutions to a nonlinear wave equation

2015-12-22

Alberto Bressan , Geng Chen

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On Two-Dimensional Sonic-Subsonic Flow

2007-02-28

Gui‐Qiang Chen , Constantine M. Dafermos , Marshall Slemrod +1 more

Uniqueness and regularity of conservative solution to a wave system modeling nematic liquid crystal

2018-04-19

Hong Cai , Geng Chen , Yihong Du

Two-dimensional Riemann problems: from scalar conservation laws to compressible Euler equations

2009-05-21

Jiequan Li , Wancheng Sheng , Tong Zhang +1 more

Global subsonic and subsonic-sonic flows through infinitely long nozzles

2007-01-01

Chunjing Xie , Zhouping Xin

In this paper, we establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method, various elliptic estimates and a compensated compactness method.More … In this paper, we establish existence of global subsonic and subsonic-sonic flows through infinitely long axially symmetric nozzles by combining variational method, various elliptic estimates and a compensated compactness method.More precisely, it is shown that there exist global subsonic flows in nozzles for incoming mass flux less than a critical value; moreover, uniformly subsonic flows always approach to uniform flows at far fields when nozzle boundaries tend to be flat at far fields, and flow angles for axially symmetric flows are uniformly bounded away from π/2; finally, when the incoming mass flux tends to the critical value, subsonic-sonic flows exist globally in nozzles in the weak sense by using angle estimate in conjunction with a compensated compactness framework.

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Existence of classical sonic-supersonic solutions for the pseudo steady Euler equations

2017-06-09

Yuxi Zheng , ZHANG TianYou

We construct the sonic-supersonic solution for the two dimensional isentropic Euler equations in the self-similar plane, given the location of the sonic curve and the prescribed flow angle and speed … We construct the sonic-supersonic solution for the two dimensional isentropic Euler equations in the self-similar plane, given the location of the sonic curve and the prescribed flow angle and speed along it.The system is degenerate and loses the hyperbolicity on the sonic curve.Singular terms also come in there.With these features, we perform a novel coordinate change and set up a proper function space with weighted metric, which yield the contraction of the solution scheme and prove the existence.

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The two-dimensional Riemann problem in gas dynamics

1998-01-01

杰权李 , 树礼杨

Preface Preliminaries Geometry of Characteristics and Discontinuities Riemann Solution Geometry of Conservation Laws Scalar Conservation Laws One-Dimensional Scalar Conservation Laws The Generalized Characteristic Analysis Method The Four-Wave Riemann Problem Mach-Reflection-Like … Preface Preliminaries Geometry of Characteristics and Discontinuities Riemann Solution Geometry of Conservation Laws Scalar Conservation Laws One-Dimensional Scalar Conservation Laws The Generalized Characteristic Analysis Method The Four-Wave Riemann Problem Mach-Reflection-Like Configuration of Solutions Zero-Pressure Gas Dynamics Characteristics and Bounded Discontinuities Simultaneous Occurrence of Two Blowup Mechanisms Delta-Shocks, Generalized Rankine-Hugoniot Relations and Entropy Conditions The One-Dimensional Riemann Problem The Two-Dimensional Riemann Problem Riemann Solutions as the Limits of Solutions to Self-Similar Viscous Systems Pressure-Gradient Equations of the Euler System The Pme-Dimensional Riemann Problem Characteristics, Discontinuities, Elementary Waves, and Classifications The Existence of Solutions to a Transonic Pressure-Gradient Equation in an Elliptic Region with Degenerate Datum The Two-Dimensional Riemann Problem and Numerical Solutions The Compressible Euler Equations The Concepts of Characteristics and Discontinuities Planar Elementary Waves and Classification PSI Approach to Irrotational Isentropic Flow Analysis of Riemann Solutions and Numerical Results Two-Dimensional Riemann Solutions with Axisymmetry References Author Index

Smooth Transonic Flows of Meyer Type in De Laval Nozzles

2019-01-08

Chunpeng Wang , Zhouping Xin

Transonic shock in a nozzle I: 2D case

2004-04-30

Zhouping Xin , Huicheng Yin

Abstract In this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two‐dimensional nozzle with variable sections. The flow is governed … Abstract In this paper we establish the existence and uniqueness of a transonic shock for the steady flow through a general two‐dimensional nozzle with variable sections. The flow is governed by the inviscid potential equation, and is supersonic upstream, has no‐flow boundary conditions on the nozzle walls, and a given pressure at the exit of the exhaust section. The transonic shock is a free boundary dividing two regions of C flow in the nozzle. The potential equation is hyperbolic upstream where the flow is supersonic, and elliptic in the downstream subsonic region. In particular, our results show that there exists a solution to the corresponding free boundary problem such that the equation is always subsonic in the downstream region of the nozzle when the pressure in the exit of the exhaustion section is appropriately larger than that in the entry. This confirms exactly the conjecture of Courant and Friedrichs on the transonic phenomena in a nozzle [10]. Furthermore, the stability of the transonic shock is also proved when the upstream supersonic flow is a small steady perturbation for the uniform supersonic flow or the pressure at the exit has a small perturbation. The main ingredients of our analysis are a generalized hodograph transformation and multiplier methods for elliptic equation with mixed boundary conditions and corner singularities. © 2004 Wiley Periodicals, Inc.

Interaction of a centered simple wave and a planar rarefaction wave of the two-dimensional Euler equations for pseudo-steady compressible flow

2017-08-03

Wancheng Sheng , Shouke You

Representation of dissipative solutions to a nonlinear variational wave equation

2015-09-16

Alberto Bressan , Tao Huang

The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic … The paper introduces a new way to construct dissipative solutions to a second order variational wave equation. By a variable transformation, from the nonlinear PDE one obtains a semilinear hyperbolic system with sources. In contrast with the conservative case, here the source terms are discontinuous and the discontinuities are not always crossed transversally. Solutions to the semilinear system are obtained by an approximation argument, relying on Kolmogorov’s compactness theorem. Reverting to the original variables, one recovers a solution to the nonlinear wave equation where the total energy is a monotone decreasing function of time.

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Transonic Shock Formation in a Rarefaction Riemann Problem for the 2D Compressible Euler Equations

2008-01-01

James Glimm , Xiaomei Ji , Jiequan Li +4 more

It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical … It is perhaps surprising for a shock wave to exist in the solution of a rarefaction Riemann problem for the compressible Euler equations in two space dimensions. We present numerical evidence and generalized characteristic analysis to establish the existence of a shock wave in such a 2D Riemann problem, defined by the interaction of four rarefaction waves. We consider both the customary configuration of waves at the right angle and also an oblique configuration for the rarefaction waves. Two distinct mechanisms for the formation of a shock wave are discovered as the angle between the waves is varied.

Subsonic-Sonic Limit of Approximate Solutions to Multidimensional Steady Euler Equations

2015-07-31

Gui‐Qiang Chen , Feimin Huang , Tian-Yi Wang

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On a nonlinear hyperbolic variational equation: I. Global existence of weak solutions

1995-01-01

John K. Hunter , Yuxi Zheng

The regularity of semi-hyperbolic patches at sonic lines for the pressure gradient equation in gas dynamics

2014-01-01

Qin Wang , Yuxi Zheng

Existence of Global Steady Subsonic Euler Flows Through Infinitely Long Nozzles

2010-01-01

Chunjing Xie , Zhouping Xin

In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of … In this paper, we study the global existence of steady subsonic Euler flows through infinitely long nozzles without the assumption of irrotationality. It is shown that when the variation of Bernoulli's function in the upstream is sufficiently small and mass flux is in a suitable regime with an upper critical value, then there exists a unique global subsonic solution in a suitable class for a general variable nozzle. One of the main difficulties for the general steady Euler flows is that the governing equations are a mixed elliptic-hyperbolic system even for uniformly subsonic flows. A key point in our theory is to use a stream function formulation for compressible Euler equations. By this formulation, Euler equations are equivalent to a quasilinear second order equation for a stream function so that the hyperbolicity of the particle path is already involved. The existence of a solution to the boundary value problem for stream function is obtained with the help of the estimate for an elliptic equation of two variables. The asymptotic behavior for the stream function is obtained via a blowup argument and energy estimates. This asymptotic behavior, together with some refined estimates on the stream function, yields the consistency of the stream function formulation and thus the original Euler equations.

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