We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics...We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ（T ） is assumed to have a positive lower bound. We first consider the Cauchy problem （the initial value problem）, that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an ad-ditional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave （weak or strong, determined by the wedge angle at the wedge vertex） when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.展开更多
We are concerned with the uniqueness of solutions of the Cauchy problemand a(s),b(s) are appropriately smooth.Since a(s) is allowed to have zero points, we call them points of degeneracy of (1), the equation (1) does ...We are concerned with the uniqueness of solutions of the Cauchy problemand a(s),b(s) are appropriately smooth.Since a(s) is allowed to have zero points, we call them points of degeneracy of (1), the equation (1) does not admit classical solutions in general. The solutions of (1) even might be discontinuous, whenever the set E = {s : a(s) = 0} includes interior points.Equations with degeneracy arise from a wide variety of diffusive processes in nature展开更多
We analyze the 2 × 2 nonhomogeneous relativistic Euler equations for perfect fluids in special relativity. We impose appropriate conditions on the lower order source terms and establish the existence of global en...We analyze the 2 × 2 nonhomogeneous relativistic Euler equations for perfect fluids in special relativity. We impose appropriate conditions on the lower order source terms and establish the existence of global entropy solutions of the Cauchy problem under these conditions.展开更多
In this paper, an existence result of entropy solutions to some parabolic problems is established. The data belongs to L^1 and no growth assumption is made on the lower-order term in divergence form.
In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t → ∞ the entropy solutions tend to a constant equi...In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t → ∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.展开更多
In this work, we study the following nonlinear homogeneous Neumann boundary value problemβ （u） -diva （x, 7u） f in fΩ, a （x, u）. η= 0 on Ω, where Ω is a smooth bounded open domain in RN, N ≥ 3 with...In this work, we study the following nonlinear homogeneous Neumann boundary value problemβ （u） -diva （x, 7u） f in fΩ, a （x, u）. η= 0 on Ω, where Ω is a smooth bounded open domain in RN, N ≥ 3 with smooth boundary Ωand ηthe outer unit normal vector on Ω . We prove the existence and uniqueness of an entropy solution for L1-data f. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.展开更多
This paper proves the local exact one-sided boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws with characteristics with constant multiplicity. This generalizes th...This paper proves the local exact one-sided boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws with characteristics with constant multiplicity. This generalizes the results in [Li, T. and Yu, L., One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws, To appear in Journal de Mathematiques Pures et Appliquees, 2016.] for a class of strictly hyperbolic systems of conservation laws.展开更多
We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operat...We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.展开更多
We study the existence and uniqueness problem for the nonhomogeneous pressureless Euler system with the initial density being a Radon measure. Our uniqueness result is obtained in the same space as the existence theor...We study the existence and uniqueness problem for the nonhomogeneous pressureless Euler system with the initial density being a Radon measure. Our uniqueness result is obtained in the same space as the existence theorem. Besides, by counterexample we prove that Huang-Wang’s energy condition is also necessary for our nonhomogeneous system.展开更多
An analytical model is established to study the influence of lattice distortion and fraction of Hf on the yield strength of the BCC TiNbTaZrHfx multi-component high entropy alloys （HEAs）. Meanwhile, the mechanism of...An analytical model is established to study the influence of lattice distortion and fraction of Hf on the yield strength of the BCC TiNbTaZrHfx multi-component high entropy alloys （HEAs）. Meanwhile, the mechanism of solid solution strengthening caused by lattice distortion is also discussed in the HEA. The distorted unit cell is introduced to indicate the lattice distortion effects induced by the differences of the atomic size and shear modulus by doping other elements in Ti-based metal. The results show that the calculated values of the alloying yield strength considering the path of least resistance are obtained with regard to various grain sizes for the equiatomic TiNbTaZrHf HEA, which is well in line with the experimental results. Furthermore, it is predicted that the alloying yield strength is the largest value in the case of the same grain size for the Hf atomic fraction of 0.122. The meaningful modeling could provide a theoretical method to investigate the yield strength and alloying design of other BCC HEAs in the future.展开更多
Thermodynamic functions solutions of a 25 binary systems formed n-alcohols and esters of aliphatic acids by were calculated using the standards ideal solution and ideal gas. The value change regularity of the thermody...Thermodynamic functions solutions of a 25 binary systems formed n-alcohols and esters of aliphatic acids by were calculated using the standards ideal solution and ideal gas. The value change regularity of the thermodynamic functions of solutions depending on their molar mass and concentration of the solutions＇ components was determined by the thermodynamic analysis. The method of prediction of the thermodynamic properties of binary solutions was suggested on the basis of the determined regularities. The corresponding equations were obtained.展开更多
We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.
We consider a class of doubly nonlinear history-dependent problems having a convection term and a pseudomonotone nonlinear diffusion operator associated an equation of the type ?<sub>t</sub>(k * (b(v) - b(...We consider a class of doubly nonlinear history-dependent problems having a convection term and a pseudomonotone nonlinear diffusion operator associated an equation of the type ?<sub>t</sub>(k * (b(v) - b(v<sub>0</sub>))) - div(a(x,Dv) + F(v)) = f where the right hand side belongs to L<sup>1</sup>. The kernel k belongs to the large class of PC kernels. In particular, the case of fractional time derivatives of order α ∈ (0,1) is included. Assuming b nondecreasing with L<sup>1</sup>-data, we prove existence in the framework of entropy solutions. The approach adopted for the proof is based on a several step approximation method and by using a result in the case of a strictly increasing b.展开更多
The property of fluid field of one-dimensional magnetohydrodynamics (MHD) transverse flow after the appearance of singularity is discussed. By the method of iteration, the strong discontinuity (shock wave) and entropy...The property of fluid field of one-dimensional magnetohydrodynamics (MHD) transverse flow after the appearance of singularity is discussed. By the method of iteration, the strong discontinuity (shock wave) and entropy solution are constructed and the estimations on the singularity of the solution near the point of blow-up are obtained.展开更多
The existence of global BV solutions for the Aw-Rascle system with linear damping is considered.In order to get approximate solutions we consider the system in Lagrangian coordinates,then by using the wave front track...The existence of global BV solutions for the Aw-Rascle system with linear damping is considered.In order to get approximate solutions we consider the system in Lagrangian coordinates,then by using the wave front tracking method coupling with and suitable splitting algorithm and the ideas of[1]we get a sequence of approximate solutions.Finally we show the convergence of this approximate sequence to the weak entropic solution.展开更多
We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（...We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.展开更多
The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p...The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p initial value.We use the device of doubling variables and some technical analysis to prove the uniqueness result.Moreover we can prove that the L p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.展开更多
Numerical approximations of multi-dimensional shock waves sometimes ex- hibit an instability called the carbuncle phenomenon. Techniques for suppressing carbuncles are trial-and-error and lack in reliability and gener...Numerical approximations of multi-dimensional shock waves sometimes ex- hibit an instability called the carbuncle phenomenon. Techniques for suppressing carbuncles are trial-and-error and lack in reliability and generality, partly because theoretical knowledge about carbuncles is equally unsatisfactory. It is not known which numerical schemes are affected in which circumstances, what causes carbuncles to appear and whether carbuncles are purely mimerical artifacts or rather features of a continuum equation or model. This article presents evidence towards the latter： we propose that carbuncles are a special class of entropy solutions which can be physically correct in some circumstances. Using ＂filaments＂, we trigger a single carbuncle in a new and more reliable way, and compute the structure in detail in similarity coordinates. We argue that carbuncles can, in some circumstances, be valid vanishing viscosity limits. Trying to suppress them is making a physical assumption that may be false.展开更多
In this article, the author uses the compensated compactness method coupled with some basic ideas of the kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existenc...In this article, the author uses the compensated compactness method coupled with some basic ideas of the kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global entropy solutions to a system of quadratic flux. The fire-new method of reduction of Young measures is a pith of this work.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
基金Gui-Qiang CHEN was supported in part by the UK EPSRC Science and Innovation Award to the Oxford Centre for Nonlinear PDE(EP/E035027/1)the NSFC under a joint project Grant 10728101+4 种基金the Royal Society-Wolfson Research Merit Award(UK)Changguo XIAO was supported in part by the NSFC under a joint project Grant 10728101Yongqian ZHANG was supported in part by NSFC Project 11031001NSFC Project 11121101the 111 Project B08018(China)
文摘We are concerned with the global existence of entropy solutions of the two-dimensional steady Euler equations for an ideal gas, which undergoes a one-step exothermic chemical reaction under the Arrhenius-type kinetics. The reaction rate function φ（T ） is assumed to have a positive lower bound. We first consider the Cauchy problem （the initial value problem）, that is, seek a supersonic downstream reacting flow when the incoming flow is supersonic, and establish the global existence of entropy solutions when the total variation of the initial data is sufficiently small. Then we analyze the problem of steady supersonic, exothermically reacting Euler flow past a Lipschitz wedge, generating an ad-ditional detonation wave attached to the wedge vertex, which can be then formulated as an initial-boundary value problem. We establish the global existence of entropy solutions containing the additional detonation wave （weak or strong, determined by the wedge angle at the wedge vertex） when the total variation of both the slope of the wedge boundary and the incoming flow is suitably small. The downstream asymptotic behavior of the global solutions is also obtained.
基金Partially supported by NSF (19631050) of China, partially supported by the grant of Ministry of Science and Technologies of China, and partially supported by the Outstanding Young Fundation (19125107) of China.
文摘We are concerned with the uniqueness of solutions of the Cauchy problemand a(s),b(s) are appropriately smooth.Since a(s) is allowed to have zero points, we call them points of degeneracy of (1), the equation (1) does not admit classical solutions in general. The solutions of (1) even might be discontinuous, whenever the set E = {s : a(s) = 0} includes interior points.Equations with degeneracy arise from a wide variety of diffusive processes in nature
基金Project supported by the National Natural Science Foundation of China (No.10571120)the Natural Science Foundation of Shanghai (No.04ZR14090).
文摘We analyze the 2 × 2 nonhomogeneous relativistic Euler equations for perfect fluids in special relativity. We impose appropriate conditions on the lower order source terms and establish the existence of global entropy solutions of the Cauchy problem under these conditions.
文摘In this paper, an existence result of entropy solutions to some parabolic problems is established. The data belongs to L^1 and no growth assumption is made on the lower-order term in divergence form.
基金supported by the National Natural Science Foundation of China(Grant No.10901095)the Promotive Research Fund for Excellent Young and Middle-Aged Scientists of Shandong Province(Grant No.BS2010SF025)
文摘In this paper the large time behavior of the global L∞ entropy solutions to the hyperbolic system with dissipative structure is investigated. It is proved that as t → ∞ the entropy solutions tend to a constant equilibrium state in L2 norm with exponential decay even when the initial values are arbitrarily large. As an illustration, a class of 2 × 2 system is studied.
文摘In this work, we study the following nonlinear homogeneous Neumann boundary value problemβ （u） -diva （x, 7u） f in fΩ, a （x, u）. η= 0 on Ω, where Ω is a smooth bounded open domain in RN, N ≥ 3 with smooth boundary Ωand ηthe outer unit normal vector on Ω . We prove the existence and uniqueness of an entropy solution for L1-data f. The functional setting involves Lebesgue and Sobolev spaces with variable exponent.
基金supported by the National Natural Science Foundation of China(No.11501122)
文摘This paper proves the local exact one-sided boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws with characteristics with constant multiplicity. This generalizes the results in [Li, T. and Yu, L., One-sided exact boundary null controllability of entropy solutions to a class of hyperbolic systems of conservation laws, To appear in Journal de Mathematiques Pures et Appliquees, 2016.] for a class of strictly hyperbolic systems of conservation laws.
文摘We give an existence result of the obstacle parabolic equations3b（x,u） div（a（x,t,u, Vu））＋div（（x,t,u））=f in QT, 3twhere b（x,u） is bounded function ot u, the term atva,x,r,u, v u）） is a Letay type operator and the function is a nonlinear lower order and satisfy only the growth condition. The second term belongs to L1 （QT）. The proof of an existence solution is based on the penalization methods.
文摘We study the existence and uniqueness problem for the nonhomogeneous pressureless Euler system with the initial density being a Radon measure. Our uniqueness result is obtained in the same space as the existence theorem. Besides, by counterexample we prove that Huang-Wang’s energy condition is also necessary for our nonhomogeneous system.
基金support from the National Natural Science Foundation of China (No. 11372103 and 11572118)the Hunan Provincial Science Fund for Distinguished Young Scholars (No. 2015JJ1006)+1 种基金the Fok Ying-Tong Education Foundation, China (No. 141005)the project of Innovation-driven Plan of Central South University, the State Key Laboratory of Powder Metallurgy
文摘An analytical model is established to study the influence of lattice distortion and fraction of Hf on the yield strength of the BCC TiNbTaZrHfx multi-component high entropy alloys （HEAs）. Meanwhile, the mechanism of solid solution strengthening caused by lattice distortion is also discussed in the HEA. The distorted unit cell is introduced to indicate the lattice distortion effects induced by the differences of the atomic size and shear modulus by doping other elements in Ti-based metal. The results show that the calculated values of the alloying yield strength considering the path of least resistance are obtained with regard to various grain sizes for the equiatomic TiNbTaZrHf HEA, which is well in line with the experimental results. Furthermore, it is predicted that the alloying yield strength is the largest value in the case of the same grain size for the Hf atomic fraction of 0.122. The meaningful modeling could provide a theoretical method to investigate the yield strength and alloying design of other BCC HEAs in the future.
文摘Thermodynamic functions solutions of a 25 binary systems formed n-alcohols and esters of aliphatic acids by were calculated using the standards ideal solution and ideal gas. The value change regularity of the thermodynamic functions of solutions depending on their molar mass and concentration of the solutions＇ components was determined by the thermodynamic analysis. The method of prediction of the thermodynamic properties of binary solutions was suggested on the basis of the determined regularities. The corresponding equations were obtained.
文摘We consider a strongly non-linear degenerate parabolic-hyperbolic problem with p(x)-Laplacian diffusion flux function. We propose an entropy formulation and prove the existence of an entropy solution.
文摘We consider a class of doubly nonlinear history-dependent problems having a convection term and a pseudomonotone nonlinear diffusion operator associated an equation of the type ?<sub>t</sub>(k * (b(v) - b(v<sub>0</sub>))) - div(a(x,Dv) + F(v)) = f where the right hand side belongs to L<sup>1</sup>. The kernel k belongs to the large class of PC kernels. In particular, the case of fractional time derivatives of order α ∈ (0,1) is included. Assuming b nondecreasing with L<sup>1</sup>-data, we prove existence in the framework of entropy solutions. The approach adopted for the proof is based on a several step approximation method and by using a result in the case of a strictly increasing b.
文摘The property of fluid field of one-dimensional magnetohydrodynamics (MHD) transverse flow after the appearance of singularity is discussed. By the method of iteration, the strong discontinuity (shock wave) and entropy solution are constructed and the estimations on the singularity of the solution near the point of blow-up are obtained.
文摘The existence of global BV solutions for the Aw-Rascle system with linear damping is considered.In order to get approximate solutions we consider the system in Lagrangian coordinates,then by using the wave front tracking method coupling with and suitable splitting algorithm and the ideas of[1]we get a sequence of approximate solutions.Finally we show the convergence of this approximate sequence to the weak entropic solution.
文摘We give an existence result of entropy and renormalized solutions for strongly nonlinear elliptic equations in the framework of Sobolev spaces with variable exponents of the type： -div （a（x, u,▽u）＋φ（u））＋g（x, u,▽u）=μ, where the right-hand side belongs to L^1（Ω）＋W^-1,p＇（x）（Ω）, -div（a（x, u,▽u）） is a Leray-Lions operator defined from W^-1,p＇（x）（Ω） into its dual and φ∈C^0（R,R^N）. The function g（x, u,▽u） is a non linear lower order term with natural growth with respect to ｜▽u｜ satisfying the sign condition, that is, g（x, u,▽u）u ≥ 0.
基金Yachun Li’s research was supported partly by National Natural Science Foundation of China (10571120,10971135)the Program for New Century Excellent Talents of Chinese Ministry of Education (NCET-07-0546)+3 种基金Shanghai Shuguang Project 06SG11Zhigang Wang’s research was supported partly by Shanghai Jiao Tong University Innovation Fund For Postgraduates (AE071202)the University Young Teacher Sciences Foundation of Anhui Province (2010SQRL145)the Quality Project Found of Fuyang Normal College (2010JPKC07)
文摘The aim of this paper is to prove the well-posedness（existence and uniqueness） of the L p entropy solution to the homogeneous Dirichlet problems for the anisotropic degenerate parabolic-hyperbolic equations with L p initial value.We use the device of doubling variables and some technical analysis to prove the uniqueness result.Moreover we can prove that the L p entropy solution can be obtained as the limit of solutions of the corresponding regularized equations of nondegenerate parabolic type.
文摘Numerical approximations of multi-dimensional shock waves sometimes ex- hibit an instability called the carbuncle phenomenon. Techniques for suppressing carbuncles are trial-and-error and lack in reliability and generality, partly because theoretical knowledge about carbuncles is equally unsatisfactory. It is not known which numerical schemes are affected in which circumstances, what causes carbuncles to appear and whether carbuncles are purely mimerical artifacts or rather features of a continuum equation or model. This article presents evidence towards the latter： we propose that carbuncles are a special class of entropy solutions which can be physically correct in some circumstances. Using ＂filaments＂, we trigger a single carbuncle in a new and more reliable way, and compute the structure in detail in similarity coordinates. We argue that carbuncles can, in some circumstances, be valid vanishing viscosity limits. Trying to suppress them is making a physical assumption that may be false.
基金Sponsored by the Foundation of Yancheng Teachers University (07YCKL061)
文摘In this article, the author uses the compensated compactness method coupled with some basic ideas of the kinetic formulation developed by Lions, Perthame, Souganidis and Tadmor to give a refined proof for the existence of global entropy solutions to a system of quadratic flux. The fire-new method of reduction of Young measures is a pith of this work.
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.
