In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {(t, x) | t ≥ 0, x ≥0} is considered. A suffic...In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {(t, x) | t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.展开更多
We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corr...We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.展开更多
In this paper, we prove the existence of the global smooth solution to the Cauchy problems for a class of diagonalizable high dimensional quasilinear hyperbolic systems consisted of n-equations.
For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.
We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third ord...We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.展开更多
In this paper we study the asymptotic behavior of global classical solutions to the Cauchy problem with initial data given on a semi-bounded axis for quasilinear hyperbolic systems. Based on the existence result on th...In this paper we study the asymptotic behavior of global classical solutions to the Cauchy problem with initial data given on a semi-bounded axis for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that, when t tends to the infinity, the solution approaches a combination of C1 travelling wave solutions with the algebraic rate (1 + t)^-u, provided that the initial data decay with the rate (1 + x)^-(l+u) (resp. (1 - x)^-(1+u)) as x tends to +∞ (resp. -∞), where u is a positive constant.展开更多
This paper deals with the asymptotic behavior of global classical solutions to quasilinear hyperbolic systems of diagonal form with weakly linearly degenerate characteristic fields. On the basis of global existence an...This paper deals with the asymptotic behavior of global classical solutions to quasilinear hyperbolic systems of diagonal form with weakly linearly degenerate characteristic fields. On the basis of global existence and uniqueness of C^1 solution, we prove that the solution to the Cauchy problem approaches a combination of C^1 traveling wave solutions when t tends to the infinity.展开更多
In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotationa...In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotational degeneracy of hyperbolic systems of conservation laws(Ⅱ).sufficient conditions which guarantee the existence of gloats smooth solutions of the Cauchy problems (Ⅰ) and (Ⅱ) are obtained by employing the characteristic method.展开更多
The author studies M-D Riemann problems for a quasilinear nonstrictly hyperbolic system. The initial data are taken as three different constants in three sections divided by three rays starting from the origin. From ...The author studies M-D Riemann problems for a quasilinear nonstrictly hyperbolic system. The initial data are taken as three different constants in three sections divided by three rays starting from the origin. From each direction of these rays two waves coming from infinity are allowed. All possible local singularity structures are carefully studied and classified. Then based on such analysis,existence and global singularity structure of the solution are obtained under some assumptions.展开更多
Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.
By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigen...By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.展开更多
Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically inter...Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physi- cally interesting cases are found by using the partial Lagrangian approach.展开更多
We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens...We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t 〈 T*.Then, when T* ≤ t 〈 T', we get a new shock wave between two rarefactions, and then, when t 〉 T', another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t → ∞. The technique does not involve dimensional reduction or coordinate transformation.展开更多
In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi...In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi-global C^(2)solution to this second-order quasilinear hyperbolic system.After then,the general constructive framework is utilized to obtain the local exact boundary controllability for this second-order system.展开更多
In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilin...In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.展开更多
Let u(t,x)be the solution to the Cauchy problem of a scalar conservation law in one space dimension.It is well known that even for smooth initial data the solution can become discontinuous in finite time and global en...Let u(t,x)be the solution to the Cauchy problem of a scalar conservation law in one space dimension.It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations.It is impossible that the solutions belong to,for example,H^(1) because by Sobolev embedding theorem H^(1) functions are Holder continuous.However,the author notes that from any point(t,x),he can draw a generalized characteristic downward which meets the initial axis at y=α(t,x).If he regards u as a function of(t,y),it indeed belongs to H^(1) as a function of y if the initial data belongs to H^(1).He may call this generalized persistence(of high regularity)of the entropy weak solutions.The main purpose of this paper is to prove some kinds of generalized persistence(of high regularity)for the scalar and 2×2 Temple system of hyperbolic conservation laws in one space dimension.展开更多
The quasilinear hyperbolic equation with nonlinear damping is considered in this paper,a new asymptotic profile for the solution to the equation is obtained by suitably choosing the initial data of the corresponding p...The quasilinear hyperbolic equation with nonlinear damping is considered in this paper,a new asymptotic profile for the solution to the equation is obtained by suitably choosing the initial data of the corresponding parabolic equation,the convergence rates of the new profile are better than that obtained by Nishihara(1997,J.Differential Equations 137,384-395) and H.-J.Zhao(2000,J.Differential Equations 167,467-494).展开更多
文摘In this paper, the mixed initial-boundary value problem for general first order quasi- linear hyperbolic systems with nonlinear boundary conditions in the domain D = {(t, x) | t ≥ 0, x ≥0} is considered. A sufficient condition to guarantee the existence and uniqueness of global weakly discontinuous solution is given.
文摘We consider first order quasilinear hyperbolic systems with vertical characteristics. It was shown in [4] that such systems can be exactly controllable with the help of internal controls applied to the equations corresponding to zero eigenvalues. However, it is possible that, for physical or engineering reasons, we can not put any control on the equations corresponding to zero eigenvalues. In this paper, we will establish the exact controllability only by means of physically meaningfnl internal controls applied to the equations corresponding to non-zero eigenvalues. We also show the exact controllability for a very simplified model by means of switching controls.
文摘In this paper, we prove the existence of the global smooth solution to the Cauchy problems for a class of diagonalizable high dimensional quasilinear hyperbolic systems consisted of n-equations.
基金supported by the National Natural Science Foundation of China(Nos.12122104,11831011)Shanghai Science and Technology Programs(Nos.21ZR1406000,21JC1400600,19JC1420101)。
文摘For quasilinear hyperbolic systems on general networks with time-periodic boundary-interface conditions with a dissipative structure,the existence and stability of the time-periodic classical solutions are discussed.
基金partially supported by the National Nature Science Foundation of China(12271114)the Guangxi Natural Science Foundation(2023JJD110009,2019JJG110003,2019AC20214)+2 种基金the Innovation Project of Guangxi Graduate Education(JGY2023061)the Key Laboratory of Mathematical Model and Application(Guangxi Normal University)the Education Department of Guangxi Zhuang Autonomous Region。
文摘We are concerned with the large-time behavior of 3D quasilinear hyperbolic equations with nonlinear damping.The main novelty of this paper is two-fold.First,we prove the optimal decay rates of the second and third order spatial derivatives of the solution,which are the same as those of the heat equation,and in particular,are faster than ones of previous related works.Second,for well-chosen initial data,we also show that the lower optimal L^(2) convergence rate of the k(∈[0,3])-order spatial derivatives of the solution is(1+t)^(-(2+2k)/4).Therefore,our decay rates are optimal in this sense.The proofs are based on the Fourier splitting method,low-frequency and high-frequency decomposition,and delicate energy estimates.
基金Supported by the National Natural Science Foundation of China (Grant No.10771038)
文摘In this paper we study the asymptotic behavior of global classical solutions to the Cauchy problem with initial data given on a semi-bounded axis for quasilinear hyperbolic systems. Based on the existence result on the global classical solution, we prove that, when t tends to the infinity, the solution approaches a combination of C1 travelling wave solutions with the algebraic rate (1 + t)^-u, provided that the initial data decay with the rate (1 + x)^-(l+u) (resp. (1 - x)^-(1+u)) as x tends to +∞ (resp. -∞), where u is a positive constant.
基金Supported by the Doctoral Programme Foundation of the Ministry of Education of China (Grant No.20070246-173)
文摘This paper deals with the asymptotic behavior of global classical solutions to quasilinear hyperbolic systems of diagonal form with weakly linearly degenerate characteristic fields. On the basis of global existence and uniqueness of C^1 solution, we prove that the solution to the Cauchy problem approaches a combination of C^1 traveling wave solutions when t tends to the infinity.
文摘In this paper,we study the global smooth solutions of the Cauchy problem for two important nonstrictly quasilinear hyperbolic systems.i.e.,the isentropic gas dynamics system in Euler coordinates (Ⅰ) and the rotational degeneracy of hyperbolic systems of conservation laws(Ⅱ).sufficient conditions which guarantee the existence of gloats smooth solutions of the Cauchy problems (Ⅰ) and (Ⅱ) are obtained by employing the characteristic method.
文摘The author studies M-D Riemann problems for a quasilinear nonstrictly hyperbolic system. The initial data are taken as three different constants in three sections divided by three rays starting from the origin. From each direction of these rays two waves coming from infinity are allowed. All possible local singularity structures are carefully studied and classified. Then based on such analysis,existence and global singularity structure of the solution are obtained under some assumptions.
基金This work is supported in part by NNSF of China(10571126)and in part by Program for New Century Excellent Talents in University.
文摘Sufficient conditions are obtained for the oscillation of solutions of the systems of quasilinear hyperbolic differential equation with deviating arguments under nonlinear boundary condition.
文摘By means of the existence and uniqueness of semi-global C1 solution to the mixed initial-boundary value problem with general nonlinear boundary conditions for first order quasilinear hyperbolic systems with zero eigenvalues,the local exact boundary controllability for higher order quasilinear hyperbolic equations is established.
文摘Based on the Lie group method, the potential symmetries and invariant solutions for generalized quasilinear hyperbolic equations are studied. To obtain the invariant solutions in an explicit form, the physically interesting situations with potential symmetries are focused on, and the conservation laws for these equations in three physi- cally interesting cases are found by using the partial Lagrangian approach.
基金supported by the NationalNatural Science Foundation of China(11371042,1471028,11601021)the Beijing Natural Science Foundation(1142001)
文摘We study the singular structure of a family of two dimensional non-self-similar global solutions and their interactions for quasilinear hyperbolic conservation laws. For the case when the initial discontinuity happens only on two disjoint unit circles and the initial data are two different constant states, global solutions are constructed and some new phenomena are discovered. In the analysis, we first construct the solution for 0 ≤ t 〈 T*.Then, when T* ≤ t 〈 T', we get a new shock wave between two rarefactions, and then, when t 〉 T', another shock wave between two shock waves occurs. Finally, we give the large time behavior of the solution when t → ∞. The technique does not involve dimensional reduction or coordinate transformation.
基金Supported by the Science and Technology Commission of Shanghai Municipality (Grant No.23ZR1402100)the Fundamental Research Funds for the Central Universities (Grant Nos. 2232022G-13 and 2232023G-13)
文摘In this paper,we propose a second-order quasilinear hyperbolic system.By means of the theory on semi-global C^(1)solution to first-order quasilinear hyperbolic systems,we establish the existence and uniqueness of semi-global C^(2)solution to this second-order quasilinear hyperbolic system.After then,the general constructive framework is utilized to obtain the local exact boundary controllability for this second-order system.
文摘In this paper, we construct a local supersonic flow in a 3-dimensional axis-symmetry nozzle when a uniform supersonic flow inserts the throat. We apply the local existence theory of boundary value problem for quasilinear hyperbolic system to solve this problem. The boundary value condition is set in particular to guarantee the character number condition. By this trick, the theory in quasilinear hyperbolic system can be employed to a large range of the boundary value problem.
基金supported by the National Natural Science Foundation of China(No.12171097)Key Laboratory of Mathematics for Nonlinear Sciences(Fudan University),Ministry of Education of China,Shanghai Key Laboratory for Contemporary Applied Mathematics,School of Mathematical Sciences,Fudan University and Shanghai Science and Technology Program(No.21JC1400600)。
文摘Let u(t,x)be the solution to the Cauchy problem of a scalar conservation law in one space dimension.It is well known that even for smooth initial data the solution can become discontinuous in finite time and global entropy weak solution can best lie in the space of bounded total variations.It is impossible that the solutions belong to,for example,H^(1) because by Sobolev embedding theorem H^(1) functions are Holder continuous.However,the author notes that from any point(t,x),he can draw a generalized characteristic downward which meets the initial axis at y=α(t,x).If he regards u as a function of(t,y),it indeed belongs to H^(1) as a function of y if the initial data belongs to H^(1).He may call this generalized persistence(of high regularity)of the entropy weak solutions.The main purpose of this paper is to prove some kinds of generalized persistence(of high regularity)for the scalar and 2×2 Temple system of hyperbolic conservation laws in one space dimension.
基金National Natural Science Foundation of China(No.11301443,11171340)Specialized Research Fund for the Doctoral Program of Higher Education(No.20124301120002)+1 种基金Natural Science Foundation of Hunan Provincial(No.2015JJ3125)Scientific Research Fund of Hunan Provincial Education Department(No.13C935)
文摘The quasilinear hyperbolic equation with nonlinear damping is considered in this paper,a new asymptotic profile for the solution to the equation is obtained by suitably choosing the initial data of the corresponding parabolic equation,the convergence rates of the new profile are better than that obtained by Nishihara(1997,J.Differential Equations 137,384-395) and H.-J.Zhao(2000,J.Differential Equations 167,467-494).