On the Dirichlet problem for first order quasilinear equations on a manifold

Panov, E.

doi:10.1090/S0002-9947-2010-05016-0

On the Dirichlet problem for first order quasilinear equations on a manifold
HTML articles powered by AMS MathViewer

by E. Yu. Panov

Trans. Amer. Math. Soc. 363 (2011), 2393-2446

DOI: https://doi.org/10.1090/S0002-9947-2010-05016-0

Published electronically: December 15, 2010

PDF | Request permission

Abstract:

We study the Dirichlet problem for a first order quasilinear equation on a smooth manifold with boundary. The existence and uniqueness of a generalized entropy solution are established. The uniqueness is proved under some additional requirement on the field of coefficients. It is shown that generally the uniqueness fails. The nonuniqueness occurs because of the presence of the characteristics not outgoing from the boundary (including closed ones). The existence is proved in a general case. Moreover, we establish that among generalized entropy solutions laying in the ball $\|u\|_\infty \le R$ there exist unique maximal and minimal solutions. To prove our results, we use the kinetic formulation similar to the one by C. Imbert and J. Vovelle.

References

Paulo Amorim, Matania Ben-Artzi, and Philippe G. LeFloch, Hyperbolic conservation laws on manifolds: total variation estimates and the finite volume method, Methods Appl. Anal. 12 (2005), no. 3, 291–323. MR 2254012, DOI 10.4310/MAA.2005.v12.n3.a6
C. Bardos, A. Y. le Roux, and J.-C. Nédélec, First order quasilinear equations with boundary conditions, Comm. Partial Differential Equations 4 (1979), no. 9, 1017–1034. MR 542510, DOI 10.1080/03605307908820117
Matania Ben-Artzi and Philippe G. LeFloch, Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré C Anal. Non Linéaire 24 (2007), no. 6, 989–1008. MR 2371116, DOI 10.1016/j.anihpc.2006.10.004
Gui-Qiang Chen and Hermano Frid, Divergence-measure fields and hyperbolic conservation laws, Arch. Ration. Mech. Anal. 147 (1999), no. 2, 89–118. MR 1702637, DOI 10.1007/s002050050146
Ronald J. DiPerna, Measure-valued solutions to conservation laws, Arch. Rational Mech. Anal. 88 (1985), no. 3, 223–270. MR 775191, DOI 10.1007/BF00752112
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, DOI 10.1007/BF01393835
R. Eymard, T. Gallouët, and R. Herbin, Existence and uniqueness of the entropy solution to a nonlinear hyperbolic equation, Chinese Ann. Math. Ser. B 16 (1995), no. 1, 1–14. A Chinese summary appears in Chinese Ann. Math. Ser. A 16 (1995), no. 1, 119. MR 1338923
Einar Hille and Ralph S. Phillips, Functional analysis and semi-groups, American Mathematical Society Colloquium Publications, Vol. 31, American Mathematical Society, Providence, R.I., 1957. rev. ed. MR 0089373
C. Imbert and J. Vovelle, A kinetic formulation for multidimensional scalar conservation laws with boundary conditions and applications, SIAM J. Math. Anal. 36 (2004), no. 1, 214–232. MR 2083859, DOI 10.1137/S003614100342468X
S. N. Kružkov, First order quasilinear equations with several independent variables. , Mat. Sb. (N.S.) 81 (123) (1970), 228–255 (Russian). MR 0267257
Philippe G. LeFloch and Baver Okutmustur, Hyperbolic conservation laws on manifolds with limited regularity, C. R. Math. Acad. Sci. Paris 346 (2008), no. 9-10, 539–543 (English, with English and French summaries). MR 2412792, DOI 10.1016/j.crma.2008.03.017
P.-L. Lions, B. Perthame, and E. Tadmor, A kinetic formulation of multidimensional scalar conservation laws and related equations, J. Amer. Math. Soc. 7 (1994), no. 1, 169–191. MR 1201239, DOI 10.1090/S0894-0347-1994-1201239-3
Felix Otto, Initial-boundary value problem for a scalar conservation law, C. R. Acad. Sci. Paris Sér. I Math. 322 (1996), no. 8, 729–734 (English, with English and French summaries). MR 1387428
E. Yu. Panov, Strong measure-valued solutions of the Cauchy problem for a first-order quasilinear equation with a bounded measure-valued initial function, Vestnik Moskov. Univ. Ser. I Mat. Mekh. 1 (1993), 20–23, 110 (Russian, with Russian summary); English transl., Moscow Univ. Math. Bull. 48 (1993), no. 1, 18–21. MR 1293931
E. Yu. Panov, On measure-valued solutions of the Cauchy problem for a first-order quasilinear equation, Izv. Ross. Akad. Nauk Ser. Mat. 60 (1996), no. 2, 107–148 (Russian, with Russian summary); English transl., Izv. Math. 60 (1996), no. 2, 335–377. MR 1399420, DOI 10.1070/IM1996v060n02ABEH000073
E. Yu. Panov, On the Cauchy problem for a first-order quasilinear equation on a manifold, Differ. Uravn. 33 (1997), no. 2, 257–266, 287 (Russian, with Russian summary); English transl., Differential Equations 33 (1997), no. 2, 257–266. MR 1609868
E. Yu. Panov, On the largest and smallest generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation, Mat. Sb. 193 (2002), no. 5, 95–112 (Russian, with Russian summary); English transl., Sb. Math. 193 (2002), no. 5-6, 727–743. MR 1918249, DOI 10.1070/SM2002v193n05ABEH000653
E. Yu. Panov, On the theory of generalized entropy solutions of the Cauchy problem for a first-order quasilinear equation in the class of locally integrable functions, Izv. Ross. Akad. Nauk Ser. Mat. 66 (2002), no. 6, 91–136 (Russian, with Russian summary); English transl., Izv. Math. 66 (2002), no. 6, 1171–1218. MR 1970354, DOI 10.1070/IM2002v066n06ABEH000411
Evgeniy Yu. Panov, On kinetic formulation of first-order hyperbolic quasilinear systems, Ukr. Mat. Visn. 1 (2004), no. 4, 548–563; English transl., Ukr. Math. Bull. 1 (2004), no. 4, 553–568. MR 2172652
E.Yu. Panov, On Kinetic Formulation of Measure Valued Solutions for Hyperbolic First-Order Quasilinear Equations with Several Space Variables, in the book “Analytical Approaches to Multidimensional Balance Laws”, O. Rozanova ed., Nova Science Publishers, New York, 2005.
E. Yu. Panov, Existence of strong traces for generalized solutions of multidimensional scalar conservation laws, J. Hyperbolic Differ. Equ. 2 (2005), no. 4, 885–908. MR 2195985, DOI 10.1142/S0219891605000658
E. Yu. Panov, Existence of strong traces for quasi-solutions of multidimensional conservation laws, J. Hyperbolic Differ. Equ. 4 (2007), no. 4, 729–770. MR 2374223, DOI 10.1142/S0219891607001343
Benoît Perthame, Kinetic formulation of conservation laws, Oxford Lecture Series in Mathematics and its Applications, vol. 21, Oxford University Press, Oxford, 2002. MR 2064166
L. Tartar, Compensated compactness and applications to partial differential equations, Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV, Res. Notes in Math., vol. 39, Pitman, Boston, Mass.-London, 1979, pp. 136–212. MR 584398
Alexis Vasseur, Strong traces for solutions of multidimensional scalar conservation laws, Arch. Ration. Mech. Anal. 160 (2001), no. 3, 181–193. MR 1869441, DOI 10.1007/s002050100157