# Nonlinear Hyperbolic Conservation Law

Keyfitz B. and Kranzer H. (1980), A system of nonstrictly hyperbolic conservation laws. Gu Chaohao and. al. (1962), A global shock-free solution for a quasilinear hyperbolic system of equations. Math. Fudan Anthology Univ. 36–39 (in Chinese). Smoller J. and Johnson J. (1969). Global solutions for an extended class of hyperbolic conservation laws.

Arch. Rat. Mech. Anal. 32:169-189. Lax D. (1964). Development of singularities of solutions of linear hyperbolic partial differential equations.

J. Math. Phys. 5:611-613. [30] Liu T.-P. (1974), Das Riemannsche Problem für allgemeine 2×2 Erhaltungsgesetze. Dilute. ANSM 199:89-112. Yamaguti M. and Nishida T. (1968).

About some global solutions for quasilinear hyperbolic equations. Funkcialaj ekvacioj 11:51-57. Chang Tung and Hiao Ling (1977). Riemann problem and discontinuous initial value problem for a typical quasilinear hyperbolic system without convexity. Acta Math. Sinica 20:229-231 (in Chinese). Ying Lungan and Wang Chinghua (1980). Global solutions of the Cauchy problem for an inhomogeneous quasilinear hyperbolic system. Comm. PureAppl.

Math. 33:579-597. Hsiao Ling and Zhang Tong (1978). Riemann problem for 2 × 2 quasilinear hyperbolic systems without convexity. Kexue Tongbao 8:465–469 (in Chinese). We deal with non-linear conservation laws and strive to establish a realistic multidimensional framework. The main question that interests us is: “When can we use hyperbolic (simplified) models?” and to answer this, we study the limits of dispersion of zero diffusion. While our evidence establishes integrity, the focus is on reliability and error. In particular, we consider all known physically relevant solutions, both classic and non-classic. The techniques we use depend on an analytical attitude towards dimensional functional theory and are energy-based methods. Tartar L. (1979) Compensated compactness and applications to partial differential equations.

Research Notes in Mathematics, Nonlinear and Mechanical Analysis: Heriot Watt Symp., Vol. 4 Knods R.J. (ed.), New York, Pitman Press. Lax P. (1954), Weak Solutions of Nonlinear Hyperbolic Equations and their Numerical Computation. Comm. Pure Appl. Math. 7:159-193. Olienik O.A.

(1957). Discontinuous solution of nonlinear differential equations. Usp. Mat. Nauk 12:3-73 Trans. English, in bitter. Soc. Transl. Ser.

2, 26:95-172. and Wang Chinghua (1985), On a nonstrictly hyperbolic system of conservation laws, J.Diff. Equat., 57:1–14. Glimm J. (1965). Large-scale solutions for nonlinear systems of hyperbolic equations. Comm. Pure Appl. Math. 18:697–715. Lin Longwei (1979), The global solutions of Cauchy problem for quasi-linearhyperbolic equation without convexity condition, Acta Jilin Univ., 2:17–26. Wang Chinghua and Li Caizhong (1982).

Study of global solutions for nonlinear conservation laws. J. Math. Anal. Appl. 85:236-256. Dafermos C. (1979). The entropy rate eligibility criterion for hyperbolic conservation law solutions. J. Diff.

Equat. 14:202-212. Liu T.-P. (1979), Quasilinear Hyperbolic System, Comm. Math. Phys., 68:141-172. Ballou D. (1970), Solution aux problèmes de Cauchy hyperboliques nonlinéaires sans conditions de convexité, Tran. ANSM 152:441-460.

Moler C. and Smoller J. (1970). Elementary interaction in quasilinear hyperbolic systems. Arch. Rat. Mech. Anal 37:309–322. Ding Xiaxi; Chang Tung; Wang Chinghua; Hsiao Ling and Li Caizhong (1973), A study of the global solutions for a quasilinear hyperbolic system of conservation laws, Scientia Sinica 16: 317-335. Chaohao; Li Daqian; Yu Wenci and Hou Zongyi (1961-1962), Discontinuous initial value problem for hyperbolic system of quasilinear equations (I), (II), (III). Acta Math.

Sinica 11:314–323; 324-337; 12:132-143 (in Chinese). Abstract: We study the limiting behavior of solutions of a class of conservation laws with endangered nonlinear scattering and dispersion terms. We prove the convergence towards the entropic solution of the first-order problem under a condition of relative size of the terms of diffusion and dispersion. This work is motivated by the pseudoviscosity approximation introduced by Neumann in the 50s. For nonlinear hyperbolic systems of conservation laws, the initial problem of limit values is studied. Two formulations of boundary conditions are proposed: a limit inequality of entropy is derived using the viscosity method, and a second formulation is based on the Riemann problem. These two formulations are equivalent for linear systems and nonlinear scalar equations. For nonlinear systems, the second formulation leads to well-posed problems. The local nonlinear structure is studied. The p-system and the isentropic Euler equations are detailed. Li Caizhong; Hsiao Ling; Yang Shaoqi and Yuan Zuwen (1963), Riemann problem for a typical quasilinear hyperbolic system without convexity. Lax P.

(1957), Hyperbolic System of Conservation Laws II. Comm. Pure Appl. Math. 10:537–566. (1983), Asymptotic States for Hyperbolic Conservation Laws with a Mobile Source. Advances Appl. Math., 4:353-379. Wang Rouhwai and Wu Zhuoqun (1963). On the mixed initial limit problem for quasilinear hyperbolic equations in two independent variables.

ActaJilin Univ. 2:459–502 (in Chinese with English Abstract). Provided by Springer Nature SharedIt content sharing initiative Hsiao Ling (1980). The eligibility criterion of the entropy rate in gas dynamics, J.Diff. 38:226–238. These keywords were added by machine, not by the authors. This process is experimental and keywords can be updated as the learning algorithm improves. University Aix Marseille Master 2 in Mathematics Partial Differential Equations Thierry Gallouët, Raphaèle Herbin. Liapijefskee V. (1974).

On the uniqueness of the generalized solution for aerodynamic equations. Docl. Akad. Nauk. SSSR 215:535–538 (in Russian). Lin Longwei (1963). On the global existence of continuous solutions of the reducible quasilinear hyperbloiic system. Acta Jilin Univ. 4:83-96. Wu Zhuoqun (1963). Existence and uniqueness of the generalized solution of Cauchy`s problem for the first-order quasilinear differential equation without convexity conditions, Acta Math.

Sinica, 13:515–530 (in Chinese). Dafermos C. and Hsiao Ling (1982). Hyperbolic system of equilibrium laws in homogeneity and dissipation. Indiana Univ. Math. J. 331:471-491. [40]-Chang Tong and Hiao Ling (1989).

The Riemann problem and the interaction of waves in gas dynamics, Pitman`s monographs and surveys in Pure and Appl.Math. 41, Longman Scien. Tech. Liu T.P. (1975). Existenz- und Eindeutigkeitssatzen für das Riemannsche Problem. ANSM 213:3755-3782. Diperna R. (1983).

Convergence of the viscosity method for isentropic gas dynamics. Math. Phys., 91:1-30. Hopf E. (1950). The partial differential equation u t + uu x = μu xx. Comm. Pure Appl. Math. 3:201-2 Ding Xiaqi; Chen Guiqiang and Luo Peizhu (1989) Convergence of the fractional step Lax-Priedrichs and Godunov scheme for the isentropi system of gas dynamics. Math. Phys., 121:63-84.

Ding Xiaxi; Chen Guiqiang and Luo Peizhu (1987–88) Convergence of the Lax-Friedrichs scheme for isentropic flow, Acta Math. Sinica (I) 7:483–540; (II) 8:61-94. (in Chinese). Hsiao Ling and Zhang Tong (1979). Interaction of elementary waves in one-dimensional adiabatic flow. Acta Math. Sinica 22:596–619 (in Chinese with English summary). Chang Tong and Guo Yufa (1965). A class of initial value problems for a system of aerodynamic equations. Acta Math. Sinica 15:386–96; Trans English, in Chinese Math. 7(1965):90-101.

Wu Zhouqun (1963). Uniqueness for a batch solution with centered rarefaction waves for aerodynamic equations. Acta Jilin Univ., 4:35–49 (in Chinese with an English summary). Unfortunately, there are currently no shareable links available for this article. Oleinik O.A. & Kalashnikov A.C. (1960) A class of discontinuous solutions for first-order quasilinear equations. Proc.

Conf. Diff. Equat. 133–137, Yerevon (Russian). Supported in part by the National Basic Research Programme of the State Commission for Science and Technology and the Academia Sinica. Gelfand I. (1959). Some problems in the theory of quasilinear equations, Usp. Mat. Nauk.

14:87-158 (in Russian). Chen Guiqiang and Lu Yungguang (1988). Study of application route of compensated compactness theory, Kexue Tongbao, 33:641–644 (in Chinese). 1991 Classification of mathematical subjects: primary school 35L65, 35L67; Secondary 65M99 Current R. and Fwedrichs K.O. (1948). Supersonic flow and shock waves. Wiley-Interscience, New York. Wendroff B. (1972). The Riemann problem for materials with nonconvex equations of state I. Isentropic flow; II..