zbMATH — the first resource for mathematics

Scalar nonlinear conservation laws with integrable boundary data. (English) Zbl 0919.35081
The authors study the initial-boundary value problem for a scalar nonlinear conservation law \[ u_t+ f(u)_x=0, \;u=u(t,x), \quad t,x\geq 0; \qquad u(0,x)=\overline u(x), \;u(t,0)=\widetilde u(x) \] with integrable (possibly unbounded) initial and boundary data \(\overline u\), \(\tilde u(x)\). The flux function \(f(u)\) is assumed to be a superlinear strictly convex function.
The authors generalize Le Floch’s explicit formula in the case \(\overline u \in L^1\), \(f(\widetilde u)\in L^1_{\text{loc}}\). A comparison principle for integrals of solutions is also derived. Then the problem under consideration is investigated for fixed \(\overline u=0\) as a control problem governed by the boundary data. The corresponding attainable profiles at fixed time and at a fixed space variable are described. Finally the authors apply their results to some optimization problem of traffic flow.

35L65 Hyperbolic conservation laws
93C20 Control/observation systems governed by partial differential equations
Full Text: DOI
[1] Ancona, F.; Marson, A., On the attainable set for scalar non-linear conservation laws with boundary control, SIAM J. control optim., 36, 290-312, (1998) · Zbl 0919.35082
[2] Aubin, J.; Cellina, A., Differential inclusion, (1984), Springer Berlin
[3] Bardi, M.; Evans, L.C., On hopf’s formulas for solutions of hamilton – jacobi equations, Nonlinear anal. theory meth. appl., 8, 1373-1381, (1984) · Zbl 0569.35011
[4] C. Bardos, A.Y. Leroux, J.C. Nedelec, First order quasilinear equations with boundary conditions, Comm. P.D.E. (4) 9 (1979) 1017-1034. · Zbl 0418.35024
[5] A. Bressan, Lectures Notes on Systems of Conservation Laws, S.I.S.S.A.-I.S.A.S., Trieste, Italy, 1995.
[6] Conway, E.D.; Hopf, E., Hamilton’s theory and generalized solutions of the hamilton – jacobi equations, J. math. mech., 13, 939-986, (1964) · Zbl 0178.11002
[7] M.G. Crandall, H. Hishii, P.L. Lions, Uniqueness of viscosity solutions of Hamilton-Jacobi equations revisited, J. Math. Soc. Japan 39 (4) (1987).
[8] Crandall, M.G.; Lions, P.L., Viscosity solutions of hamilton – jacobi equations, Trans. amer. math. soc., 277, 9, 1-42, (1983) · Zbl 0599.35024
[9] Dafermos, C., Generalized characteristic and the structure of solutions of hyperbolic conservation laws, Indiana math. J., 26, 1097-1119, (1977) · Zbl 0377.35051
[10] Douglis, A., An ordering principle and generalized certain quasi-linear partial differential equations, Comm. pure appl. math., 12, 87-112, (1959) · Zbl 0084.29503
[11] A.F. Filippov, Differential equations with discountinuous right-hand side, Math. Sb. 93 (1960) 99-128. English translation: Amer. Math. Soc. Transl., Ser. 2 42 (1960) 199-231. · Zbl 0138.32204
[12] A.D. Ioffe, On lower semicontinuity of integral functionals. I, SIAM J. Control Optim. 15 (4) (1977) 521-538. · Zbl 0361.46037
[13] Kruzkov, S.N., First order quasilinear equations in several independent variables, Math. USSR sbornik, 10, 2, 217-243, (1970) · Zbl 0215.16203
[14] Lax, P.D., Hyperbolic systems of conservation laws II, Comm. pure appl. math., 10, 537-566, (1957) · Zbl 0081.08803
[15] Lax, P.D., The formation and decay of shock waves, Amer. math. monthly, 79, 227-241, (1972) · Zbl 0228.35019
[16] P. Le Floch, Explicit formula for scalar non-linear conservation laws with boundary condition, Math. Meth. Appl. Sci. 10 (1988) 265-287. · Zbl 0679.35065
[17] J. Smoller, Shock Waves and Reaction-Diffusion Equations, 2nd ed., Springer, New York, 1994. · Zbl 0807.35002
[18] Terracina, A., Comparison properties for scalar conservation laws with boundary conditions, Nonlinear anal. theory meth. appl., 28, 633-653, (1997) · Zbl 0873.35049
[19] G.B. Whitham, Linear and Nonlinear Waves, Wiley, New York, 1974. · Zbl 0373.76001
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.