# zbMATH — the first resource for mathematics

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)

##### MSC:
 35L65 Hyperbolic conservation laws 35B45 A priori estimates in context of PDEs 35L45 Initial value problems for first-order hyperbolic systems
Full Text:
##### References:
 [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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.