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Continuous dependence for \(2 \times 2\) conservation laws with boundary. (English) Zbl 0884.35091
The author deals with the strictly hyperbolic \(2\times 2\) system of conservation laws \(u_t=F(u)_x=0\) on the domain \(t>0\), \(x>g(t)\) with initial data \(u(0,x)=u_0(x)\) and boundary condition along \(x=g(t)\). Two different kinds of such a boundary condition are investigated: the characteristic \((u(t,g(t))=h(t))\) and the non-characteristic one. Sufficiently small data are considered. For both cases of boundary conditions a Lipschitzian flow is constructed whose trajectories are weak solutions of the problem. Consequently it is proved the continuous dependence of the solution upon the initial data, the boundary condition, and the boundary profile.
Reviewer: A.Doktor (Praha)

35L65 Hyperbolic conservation laws
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
Full Text: DOI
[1] Amadori, D., Initial-boundary value problems for nonlinear systems of conservation laws, Nodea, 4, 1-42, (1997) · Zbl 0868.35069
[2] D. Amadori, R. M. Colombo, Characterization of viscosity solutions for conservation laws with boundary, Rend. Sem. Mat. Univ. Padova · Zbl 0910.35078
[3] P. Baiti, A. Bressan, The semigroup generated by Temple class systems with large data, Differential Integral Equations · Zbl 0890.35083
[4] Bressan, A., Global solutions of systems of conservation laws by wave-front tracking, J. math. anal. appl., 170, 414-432, (1992) · Zbl 0779.35067
[5] A. Bressan, 1995, Lecture notes on systems of conservation laws, S.I.S.S.A. Trieste
[6] Bressan, A., The unique limit of the glimm scheme, Arch. rational mech. anal., 130, 205-230, (1995) · Zbl 0835.35088
[7] A. Bressan, The semigroup approach to systems of conservation laws, Math. Contemporanea · Zbl 0866.35064
[8] Bressan, A.; Colombo, R.M., The semigroup generated by 2×2 systems of conservation laws, Arch. rational mech. anal., 133, (1996)
[9] Bressan, A.; Colombo, R.M., Unique solutions of 2×2 conservation laws with large data, Indiana univ. math. J., 44, 677-725, (1995) · Zbl 0852.35092
[10] A. Bressan, G. Crasta, B. Piccoli, 1996, Well-posedness of the Cauchy problem fornn, S.I.S.S.A. · Zbl 0958.35001
[11] R. M. Colombo, 1995, Uniqueness and Continuous Dependence for 2×2 Conservation Laws, S.I.S.S.A.
[12] G. Crasta, B. Piccoli, 1995, Viscosity solutions and uniqueness for systems of inhomogeous balance laws, S.I.S.S.A. · Zbl 0949.35089
[13] Dubois, F.; Le Floch, P., Boundary conditions for nonlinear hyperbolic systems of conservation laws, J. differential equations, 71, 93-122, (1988) · Zbl 0649.35057
[14] Dubroca, B.; Gallice, G., Résultats d’existence et d’unicité du problème mixte pour des systèmes hyperboliques de lois de conservation monodimensionels, Comm. P.D.E., 15-1, 59-80, (1990) · Zbl 0735.35092
[15] Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 697-715, (1965) · Zbl 0141.28902
[16] J. Goodman, 1982, Initial Boundary Value Problems for Hyperbolic Systems of Conservation Laws, California University
[17] Lax, P., Hyperbolic systems of conservation laws II, Comm. pure appl. math., 10, 537-567, (1957) · Zbl 0081.08803
[18] Liu, T.P., Initial-boundary value problems for gas dynamics, Arch. rational mech. anal., 64, 137-168, (1977) · Zbl 0357.35016
[19] Nishida, T.; Smoller, J., Mixed problems for nonlinear conservation laws, J. differential equations, 23, 244-269, (1977) · Zbl 0303.35052
[20] Risebro, N.H., A front-tracking alternative to the random choice method, Proc. amer. math. soc., 117, 1125-1139, (1993) · Zbl 0799.35153
[21] Sablé-Tougeron, M., Méthode de glimm et problème mixte, Ann. inst. Henri Poincaré, 10, 423-443, (1993) · Zbl 0832.35093
[22] Smoller, J., Shock waves and reaction-diffusion equations, (1983), Springer-Verlag New York · Zbl 0508.35002
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