Global existence in $L^4(\mathbf R_+\times \mathbf R)$ for a nonstrictly hyperbolic conservation law
Author:
Huijiang Zhao
Journal:
Quart. Appl. Math. 58 (2000), 627-660
MSC:
Primary 35L65
DOI:
https://doi.org/10.1090/qam/1788422
MathSciNet review:
MR1788422
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Abstract: We study the existence problem for the following nonstrictly hyperbolic system: \[ {u_t} + \frac {1}{2}{\left ( 3{u^2} + {v^2} \right )_x} = 0\], \[ {v_t} + {\left ( uv \right )_x} = 0\], with singular initial data, i.e., \[ \left ( u\left ( t, x \right ), v\left ( t, x \right ) \right )\left | {_{t = 0}} \right . = \left ( {u_{0}}\left ( x \right ), {v_0}\left ( x \right ) \right ) \in {L^{4}}\left ( R, {R^{2}} \right )\].
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J. M. Ball, Convexity conditions and existence theorems in nonlinear elasticity, Arch. Rational Mech. Anal. 63, 337–403 (1977)
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G. Q. Chen and P. T. Kan, Hyperbolic conservation laws with umbilic degeneracy I, Preprint, 1993
K. N. Chueh, C. C. Conley, and J. A. Smoller, Positively invariant regions for systems of nonlinear diffusion equations, Indiana University Math. Journal 26 (2), 372–411 (1977)
R. Courant and K. O. Friedrichs, Supersonic flow and shock waves, Intersciences Publishers, New York, 1948
R. Courant and D. Hilbert, Methods of Mathematical Physics, Vols. 1, 2, Wiley Intersciences, New York, 1962
B. Dacorogna, Weak continuity and weak lower semicontinuity of nonlinear functionals, Lecture Notes in Mathematics, Vol. 922, Springer-Verlag, New York and Berlin, 1982
C. M. Dafermos, Estimates for conservation laws with little viscosity, SIAM J. Math. Anal. 18, 409–421 (1987)
C. M. Dafermos, Hyperbolic systems of conservation laws, In Systems of Nonlinear Partial Differential Equations, NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 111, 25–70 (1983)
X. X. Ding, G. Q. Chen, and P. Z. Luo, Convergence of the Lax-Friedrichs scheme for isentropic gas dynamics, I, II, Acta Mathematica Scientia 4, 485–500, 501–540 (1985)
R. J. DiPerna, Convergence of approximate solutions to conservation laws, Arch. Rational Mech. Anal. 82, 27–70 (1983)
R. J. DiPerna, Convergence of the viscosity method for isentropic gas dynamics, Comm. Math. Phys. 91, 1–30 (1983)
R. J. DiPerna, Compensated compactness and general systems of conservation laws, Trans. Amer. Math. Soc. 292, 383–420 (1985)
P. L. Floch and T. P. Liu, Existence theory for nonlinear hyperbolic systems in nonconservative form, Forum. Math. 5, 261–280 (1993)
H. Freisthüler, Rotational degeneracy of hyperbolic systems of conservation laws, Arch. Rational Mech. Anal. 113, 39–64 (1990)
H. Frid and M. M. Santos, The Cauchy problem for the systems ${\partial _{t}z} - {\partial _x}\left ( {\bar z^r} \right ) = 0$, Journal of Differential Equations 111 (2), 340–359 (1994)
H. Frid and M. M. Santos, Nonstrictly hyperbolic systems of conservation laws of the conjugate type, Comm. Partial Differential Equations 19 (1 & 2), 27–59 (1994)
J. Glimm, Solutions in the large for nonlinear hyperbolic systems of equations, Comm. Pure Appl. Math. 18, 697–715 (1965)
D. Hoff and J. A. Smoller, Solutions in the large for certain nonlinear parabolic systems, Ann. Inst. Henri Poincaré3 (2), 213–235 (1985)
L. Hormander, Nonlinear hyperbolic differential equations, Lectures 1986–1987, Department of Math., Lund, Sweden, 1987
E. Isaacson, D. Marchesin, B. Plohr, and B. Temple, The Riemann problem near a hyperbolic singularity: The classification of solutions of quadratic Riemann problems I, SIAM J. Appl. Math. 48, 1009–1032 (1988)
E. Isaacson and B. Temple, The classification of solutions of quadratic Riemann problems II, SIAM J. Appl. Math. 48, 1287–1301 (1988); III, SIAM J. Appl. Math. 48, 1302–1318 (1988)
E. Isaacson and B. Temple, Examples and classification of nonstrictly hyperbolic systems of conservation laws, Abstracts of Amer. Math. Soc., Jan., 1985. Presented in the Special Session on “Nonstrictly Hyperbolic Conservation Laws” at the Winter Meeting of the American Mathematical Society, Anaheim, Jan., 1985
P. T. Kan, Convergence of the viscosity method for a system of nonstrictly hyperbolic conservation laws, Ph. D. thesis, Courant Institute of Mathematical Sciences, New York University, 1992
T. Kato, The Cauchy problem for quasilinear symmetric hyperbolic systems, Arch. Rational Mech. Anal. 58, 181–205 (1975)
P. Lax, Hyperbolic systems of conservation laws and the mathematical theory of shock waves, SIAM Reg. Conf. Lecture 11, Philadelphia, 1973
P. Lax, Weak solutions of nonlinear hyperbolic equations and their numerical computation, Comm. Pure Appl. Math. 7, 159–193 (1954)
P. Lin, Young measures and an application of compensated compactness to one dimensional nonlinear elastodynamics, Trans. Amer. Math. Soc. 329 (1), 377–413 (1992)
P. L. Lions, B. Perthame, and P. E. Souganidis, Existence of entropy solutions for the hyperbolic systems of isentropic gas dynamics in Eulerian and Lagrangian coordinates, Preprint, 1994
Y. G. Lu, Convergence of the viscosity method for a nonstrictly hyperbolic system, Acta Mathematica Scientia 12 (1), 230–239 (1992)
Y. G. Lu and J. H. Wang, The interactions of elementary waves of nonstrictly hyperbolic system, J. Math. Anal. Appl. 166 (1), 136–169 (1992)
A. Majda, Compressible fluid flow and systems of conservation laws in several space variables, Springer-Verlag, New York, 1984
F. Murat, Compacité par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 5, 489–507 (1978)
O. Oleĭnik, Discontinuous solutions of nonlinear differential equations, Amer. Math. Soc. Transl. Ser. 2, 26, 95–172 (1963)
B. Rubino, On the vanishing viscosity approximation to the Cauchy problem for a $2 \times 2$ system of conservation laws, Ann. Inst. Henri Poincaré10 (6), 627–656 (1993)
M. M. Santos, Reduction of generalized Young measures, preprint, 1994
D. G. Schaeffer and M. Shearer, The classification of $2 \times 2$ system of nonstrictly hyperbolic conservation laws, with applications to oil recovery, Comm. Pure Appl. Math. 40, 141–178 (1987)
M. Schonbek, Convergence of solutions to nonlinear dispersive equations, Comm. Partial Differential Equations 7, 959–1000 (1982)
D. Serre, La compacité par compensation pour les systèmes hyperboliques non linéaires de deux équations à une dimension d’espace, J. Math. Pure Appl. 65, 423–468 (1986)
M. Shearer, The Riemann problem for $2 \times 2$ systems of hyperbolic conservation laws with case I quadratic nonlinearities, J. Differential Equations 80, 343–363 (1989)
M. Shearer, D. Schaeffer, D. Marchesin, and P. Paes-Leme, Solution of the Riemann problem for a prototype $2 \times 2$ system of nonstrictly hyperbolic conservation laws, Arch. Rational Mech. Anal. 97, 299–320 (1987)
J. W. Shearer, Global existence and compactness in ${L^{p}}$ for the quasilinear wave equation, Comm. Partial Differential Equations 19 (11 & 12), 1829–1877 (1994)
J. A. Smoller, Shock waves and reaction-diffusion equations, Springer-Verlag, New York, 1983
J. A. Smoller and J. Johnson, Global solutions for an extended class of hyperbolic systems of conservation laws, Arch. Rat. Mech. Anal. 32, 169–189 (1969)
S. L. Sobolev, Partial Differential Equations of Mathematical Physics, Pergamon Press, 1964
L. Tartar, Compensated compactness and applications to partial differential equations, In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, IV, Research Notes in Math. 39, Pitman, Boston, MA, 1979, pp. 136 210
L. Tartar, The compensated compactness method applied to systems of conservation laws, In Systems of Nonlinear Partial Differential Equations (J. Ball, Ed.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci. 111, Reidel, Dordrecht, Boston, MA, 1983
I. Vecchi, Entropy compactification in Lagrangean gas dynamics, Mathematical Methods in the Applied Sciences 14, 207 216 (1991)
Z. X. Wang and D. R. Guo, Special Functions, Science Press, Beijing, 1965
K. Yosida, Functional Analysis, Springer-Verlag, New York, 1978
T. Zhang and Y. F. Guo, A class of initial-value problems for systems of areodynamic equations, Acta Math. Sinica 15, 386–396 (1965) (in Chinese)
H. J. Zhao, Solutions in the large for certain nonlinear parabolic systems in arbitrary spatial dimensions, Applicable Analysis 59, 349–376 (1995)
H. J. Zhao, A new compactness result of $L_{loc}^p\left ( {1 < p < \infty } \right )$ bounded approximate solutions to scalar conservation laws and its applications, Preprint, 1995
C. J. Zhu, C. Z. Li, and H. J. Zhao, Global continuous solutions to a class of nonstrictly hyperbolic conservation laws, Acta Mathematica Scientia 14 (1), 96 106 (1994) (in Chinese)
X. X. Ding and J. H. Wang, Global solutions for a semilinear parabolic system, Acta Mathematica Scientia 3, 397–412 (1983)
Y. G. Lu, Convergence of solutions to nonlinear dispersive equations without convexity conditions, Applicable Analysis 31, 239 246 (1989)
C. S. Morawetz, An alternative proof of DiPerna’s theorem, Comm. Pure Appl. Math. 44 (8 & 9), 1081–1090 (1991)
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