Hyperbolic equations and SBV functions

De Lellis, Camillo

Journées équations aux dérivées partielles (2010), article no. 6, 10 p.

In this article we survey some recent results in the regularity theory of admissible solutions to hyperbolic conservation laws and Hamilton-Jacobi equations.

[1] Alberti, G., and Ambrosio, L. A geometrical approach to monotone functions in ${\mathbf{R}}^n$. Math. Z. 230, 2 (1999), 259–316. | MR | Zbl

[2] Ambrosio, L., and De Lellis, C. A note on admissible solutions of 1D scalar conservation laws and 2D Hamilton-Jacobi equations. J. Hyperbolic Differ. Equ. 1, 4 (2004), 813–826. | MR | Zbl

[3] Ambrosio, L., De Lellis, C., and Malý, J. On the chain rule for the divergence of BV-like vector fields: applications, partial results, open problems. In Perspectives in nonlinear partial differential equations, vol. 446 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 31–67. | MR

[4] Ambrosio, L., Fusco, N., and Pallara, D. Functions of bounded variation and free discontinuity problems. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 2000. | MR | Zbl

[5] Ancona, F., and Coclite, G. M. On the attainable set for Temple class systems with boundary controls. SIAM J. Control Optim. 43, 6 (2005), 2166–2190 (electronic). | MR | Zbl

[6] Ancona, F., and Marson, A. On the attainable set for scalar nonlinear conservation laws with boundary control. SIAM J. Control Optim. 36, 1 (1998), 290–312 (electronic). | MR | Zbl

[7] Ancona, F., and Marson, A. Asymptotic stabilization of systems of conservation laws by controls acting at a single boundary point. In Control methods in PDE-dynamical systems, vol. 426 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 1–43. | MR | Zbl

[8] Ancona, F., and Nguyen, K. T. SBV regularity for solutions to genuinely nonlinear Temple systems of balance laws. In preparation.

[9] Bianchini, S., and Caravenna, L. SBV regularity for genuinely nonlinear, strictly hyperbolic systems of conservation laws. In preparation.

[10] Bianchini, S., De Lellis, C., and Robyr, R. SBV regularity for Hamilton-Jacobi equations in $R^n$. To appear in Arch. Rat. Mech. Anal. (2010).

[11] Bressan, A. Hyperbolic systems of conservation laws, vol. 20 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2000. The one-dimensional Cauchy problem. | MR | Zbl

[12] Bressan, A., and Marson, A. A maximum principle for optimally controlled systems of conservation laws. Rend. Sem. Mat. Univ. Padova 94 (1995), 79–94. | Numdam | Zbl

[13] Bressan, A., and Shen, W. Optimality conditions for solutions to hyperbolic balance laws. In Control methods in PDE-dynamical systems, vol. 426 of Contemp. Math. Amer. Math. Soc., Providence, RI, 2007, pp. 129–152. | MR

[14] Cannarsa, P., and Sinestrari, C. Semiconcave functions, Hamilton-Jacobi equations, and optimal control. Progress in Nonlinear Differential Equations and their Applications, 58. Birkhäuser Boston Inc., Boston, MA, 2004. | MR | Zbl

[15] Dafermos, C. M. Hyperbolic conservation laws in continuum physics, vol. 325 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin, 2000. | MR | Zbl

[16] Dafermos, C. M. Wave fans are special. Acta Math. Appl. Sin. Engl. Ser. 24, 3 (2008), 369–374. | MR | Zbl

[17] De Giorgi, E., and Ambrosio, L. New functionals in the calculus of variations. Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Natur. (8) 82, 2 (1988), 199–210 (1989). | MR | Zbl

[18] Evans, L. C. Partial differential equations, vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, 1998. | MR | Zbl

[19] Robyr, R. SBV regularity of entropy solutions for a class of genuinely nonlinear scalar balance laws with non-convex flux function. J. Hyperbolic Differ. Equ. 5, 2 (2008), 449–475. | MR | Zbl

[20] Tonon, D. Personal communication..