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Oleinik type estimates and uniqueness for \(n \times n\) conservation laws. (English) Zbl 0990.35095
It is investigated the strictly hyperbolic \(n\times n\) system of conservation laws \(u_t+f(u)_x=0\), \(t>0\), \(x\in\mathbb{R}\), with the initial condition \(u(0,x)= \overline u(x)\). The authors prove that the above Cauchy problem has a unique weak solution, which satisfies some assumptions (entropy condition, tame oscillation or decay estimate). So, they generalize a classical result proved by O. A. Oleinik in the scalar case [in Am. Math. Soc., Transl. II. Ser. 26, 95-172 (1963); translation from Usp. Mat. Nauk 12, No. 3, 3-73 (1957; Zbl 0080.07701)].
Reviewer: R.Luca (Iaşi)

35L65 Hyperbolic conservation laws
35B45 A priori estimates in context of PDEs
35L45 Initial value problems for first-order hyperbolic systems
Full Text: DOI
[1] Bressan, A., The unique limit of the glimm scheme, Arch. rational mech. anal., 130, 205-230, (1995) · Zbl 0835.35088
[2] Bressan, A., Analysis of solutions to hyperbolic systems by the front tracking method, (), 1-48 · Zbl 0945.35055
[3] Bressan, A., On the Cauchy problem for nonlinear hyperbolic systems, (), 23-36 · Zbl 0911.35070
[4] Bressan, A.; Colombo, R.M., The semigroup generated by 2×2 conservation laws, Arch. rational mech. anal., 133, 1-75, (1995) · Zbl 0849.35068
[5] Bressan, A.; Colombo, R.M., Decay of positive waves in nonlinear systems of conservation laws, Ann. scuola norm. sup. Pisa, 26, 133-160, (1998) · Zbl 0906.35059
[6] A. Bressan, G. Crasta, and, B. Piccoli, Well posedness of the Cauchy problem for n×n systems of conservation laws, Memoir Amer. Math. Soc, to appear. · Zbl 0958.35001
[7] Bressan, A.; LeFloch, P., Uniqueness of weak solutions to hyperbolic systems of conservation laws, Arch. rational mech. anal., 140, 301-317, (1997) · Zbl 0903.35039
[8] A. Bressan, T. P. Liu, and, T. Yang, L1 stability estimates for n×n conservation laws, Arch. Rational Mech. Anal, to appear.
[9] Dafermos, C.; Geng, X., Generalized characteristics, uniqueness and regularity of solutions in a hyperbolic system of conservation laws, Ann. inst. H. Poincaré nonlinear anal., 8, 231-269, (1991) · Zbl 0776.35033
[10] DiPerna, R.J., Uniqueness of solutions to hyperbolic conservation laws, Indiana univ. math. J., 28, 137-188, (1979) · Zbl 0409.35057
[11] Evans, L.C.; Gariepy, R.F., Measure theory and fine properties of functions, (1992), CRC Press Boca Raton · Zbl 0626.49007
[12] Glimm, J., Solutions in the large for nonlinear hyperbolic systems of equations, Comm. pure appl. math., 18, 697-715, (1965) · Zbl 0141.28902
[13] Heibig, A., Existence and uniqueness of solutions for some hyperbolic systems of conservation laws, Arch. rational mech. anal., 126, 79-101, (1994) · Zbl 0810.35058
[14] Kolmogorov, A.N.; Fomin, S.V., Introductory real analysis, (1975), Dover New York · Zbl 0213.07305
[15] Kruzhkov, S., First-order quasilinear equations with several space variables, Mat. sb., 123, 228-255, (1970) · Zbl 0202.11203
[16] Lax, P.D., Hyperbolic systems of conservation laws II, Comm. pure appl. math., 10, 537-566, (1957) · Zbl 0081.08803
[17] LeFloch, P.G.; Xin, Z.P., Uniqueness via the adjoint problems for systems of conservation laws, Comm. pure appl. math., 46, 1499-1533, (1993) · Zbl 0797.35116
[18] Liu, T.P., Uniqueness of weak solutions of the Cauchy problem for general 2×2 conservation laws, J. differential equations, 20, 369-388, (1976) · Zbl 0288.76031
[19] Liu, T.P., Admissible solutions of hyperbolic conservation laws, Amer. math. soc. memoir, 240, (1981) · Zbl 0446.76058
[20] Oleinik, O., Discontinuous solutions of nonlinear differential equations, Uspekhi mat. nauk, 12, 3-73, (1957)
[21] Oleinik, O., On the uniqueness of the generalized solution of the Cauchy problem for a nonlinear system of equations occurring in mechanics, Uspekhi mat. nauk, 12, 169-176, (1957) · Zbl 0080.07702
[22] Smoller, J., Shock waves and reaction – diffusion equations, (1983), Springer-Verlag New York · Zbl 0508.35002
[23] Volpert, A.I., The space BV and quasilinear equations, Math. USSR sb., 2, 257-267, (1967)
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