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Existence of global bounded weak solutions to nonsymmetric systems of Keyfitz-Kranzer type. (English) Zbl 1235.35194

The author studies the Cauchy problem for nonsymmetric systems of Keyfitz-Kranzer type

${\rho }_{t}+{\left(\rho \varphi \left(\rho ,w\right)\right)}_{x}=0,\phantom{\rule{1.em}{0ex}}{\left(\rho w\right)}_{t}+{\left(\rho w\varphi \left(\rho ,w\right)\right)}_{x}=0,$

where the unknown vectors $\left(\rho ,w\right)\in ℝ×{ℝ}^{n}$ and $\varphi \left(\rho ,w\right)={\Phi }\left(w\right)-P\left(\rho \right)$. In the case $n=1$, ${\Phi }\left(w\right)=w$, this system coincides with the known Aw-Rascle traffic flow model. Using BV estimates on the Riemann invariants and the compensated compactness method applied to special approximate sequences, the author establishes the global existence of bounded entropy weak solutions.

##### MSC:
 35L65 Conservation laws 35B45 A priori estimates for solutions of PDE 35D30 Weak solutions of PDE 35A01 Existence problems for PDE: global existence, local existence, non-existence 90B20 Traffic problems
##### References:
 [1] Aw, A.; Rascle, M.: Resurrection of ”second order” models of traffic flow, SIAM J. Appl. math. 60, 916-938 (2000) · Zbl 0957.35086 · doi:10.1137/S0036139997332099 [2] Bouchut, F.; James, F.: Duality solutions for pressureless gases, monotone scalar conservation laws, and uniqueness, Comm. partial differential equations 24, 2173-2190 (1999) · Zbl 0937.35098 · doi:10.1080/03605309908821498 [3] Brenier, Y.; Grenier, E.: On the model of pressureless gases with sticky particles, SIAM J. Numer. anal. 35, 2317-2328 (1999) [4] Chen, G. -Q.: Hyperbolic system of conservation laws with a symmetry, Comm. partial differential equations 16, 1461-1487 (1991) · Zbl 0753.35051 · doi:10.1080/03605309108820806 [5] Diperna, R. J.: Convergence of the viscosity method for isentropic gas dynamics, Comm. math. Phys. 91, 1-30 (1983) · Zbl 0533.76071 · doi:10.1007/BF01206047 [6] Freistuhler, H.: On the Cauchy problem for a class of hyperbolic systems of conservation laws, J. differential equations 112, 170-178 (1994) · Zbl 0806.35113 · doi:10.1006/jdeq.1994.1099 [7] Garavello, M.; Piccoli, B.: Traffic flow on a road network using the aw-rascle model, Comm. partial differential equations 31, 243-275 (2006) · Zbl 1090.90032 · doi:10.1080/03605300500358053 [8] Godvik, M.; Hanche-Olsen, H.: Existence of solutions for the aw-rascle traffic flow model with vacuum, J. hyperbolic differ. Equ. 5, 45-64 (2008) · Zbl 1185.35144 · doi:10.1142/S0219891608001428 [9] Huang, F. M.: Weak solution to pressureless type system, Comm. partial differential equations 30, 283-304 (2005) · Zbl 1074.35021 · doi:10.1081/PDE-200050026 [10] Huang, F. M.; Wang, Z.: Well posedness for pressureless flow, Comm. math. Phys. 222, 117-146 (2001) · Zbl 0988.35112 · doi:10.1007/s002200100506 [11] James, F.; Peng, Y. -J.; Perthame, B.: Kinetic formulation for chromatography and some other hyperbolic systems, J. math. Pure appl. 74, 367-385 (1995) · Zbl 0847.35084 [12] Kearsley, A.; Reiff, A.: Existence of weak solutions to a class of nonstrictly hyperbolic conservation laws with non-interacting waves, Pacific J. Math. 205, 153-170 (2002) · Zbl 1065.35188 · doi:10.2140/pjm.2002.205.153 [13] Keyfitz, B.; Kranzer, H.: A system of nonstrictly hyperbolic conservation laws arising in elasticity, Arch. ration. Mech. anal. 72, 219-241 (1980) · Zbl 0434.73019 · doi:10.1007/BF00281590 [14] Liu, T. -P.; Wang, J. -H.: On a hyperbolic system of conservation laws which is not strictly hyperbolic, J. differential equations 57, 1-14 (1985) [15] Y.-G. Lu, An explicit lower bound for viscosity solutions to isentropic gas dynamics and to Euler equations, Preprint 95-18, SFB 359, Heidelberg University, Germany, 1995. [16] Lu, Y. -G.: Hyperbolic conservation laws and the compensated compactness method, vol. 128, (2002) · Zbl 1024.35002 [17] Lu, Y. -G.: Some results on general system of isentropic gas dynamics, Differ. equ. 43, 130-138 (2007) [18] Y.-G. Lu, Existence of global bounded weak solutions to a symmetric system of Keyfitz-Kranzer type, Nonlinear Anal. Real World Appl., doi:10.1016/j.nonrwa.2011.07.029, in press. [19] Murat, F.: Compacité par compensation, Ann. sc. Norm. super. Pisa 5, 489-507 (1978) · Zbl 0399.46022 · doi:numdam:ASNSP_1978_4_5_3_489_0 [20] Panov, E. Yu.: On the theory of generalized entropy solutions of the Cauchy problem for a class of non-strictly hyperbolic systems of conservation laws, Sb. math. 191, 121-150 (2000) · Zbl 0954.35107 · doi:10.1070/SM2000v191n01ABEH000450 [21] Perthame, B.: Kinetic formulations, (2002) [22] Serre, D.: Solutions à variations bornées pour certains systèmes hyperboliques de lois de conservation, J. differential equations 68, 137-168 (1987) · Zbl 0627.35062 · doi:10.1016/0022-0396(87)90189-6 [23] Serre, D.: Systems of conservation laws, (1999) [24] Tartar, T.: Compensated compactness and applications to partial differential equations, Research notes in mathematics (1979) [25] Temple, B.: Systems of conservation laws with invariant submanifolds, Trans. amer. Math. soc. 280, 781-795 (1983) · Zbl 0559.35046 · doi:10.2307/1999646 [26] Wang, Z.; Ding, X.: Uniqueness of Cauchy problem for transportation equation, Acta math. Sci. 17, 341-352 (1997)