A theory of $$L ^{1}$$-dissipative solvers for scalar conservation laws with discontinuous flux.(English)Zbl 1261.35088

The authors study $$L^1$$ contractive semigroups of solutions to conservation laws $$u_t+f(x,u)_x=0$$ with discontinuous flux $$f(x,u)=f^l(u)$$ for $$x<0$$, $$f(x,u)=f^r(u)$$ for $$x>0$$. It is known that such conservation laws usually admit many different $$L^1$$ contractive semigroups, and additional admissibility (entropy) criteria are necessary to extract unique solutions. The authors propose a general framework to these criteria, based on selection of a special family of piecewise constant weak solutions, which is called germ. For any given germ the authors formulate the corresponding “germ-based” admissibility criteria in the form of interface conditions on the discontinuity line $$x=0$$. They characterize the germs that lead to $$L^1$$ contractive semigroups. The suggested unified approach allows to provide new uniqueness results for conservation laws with discontinuous flux as well as to recover the known results under the weaker assumptions. The existence of admissible solutions is also discussed and is established for fluxes satisfying some additional conditions.

MSC:

 35L65 Hyperbolic conservation laws 47H20 Semigroups of nonlinear operators 35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
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 [1] Adimurthi , Ghoshal S.S., Dutta R., Veerappa Gowda G.D.: Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux. Comm. Pure Appl. Math. 64(1), 84–115 (2011) · Zbl 1223.35222 [2] Adimurthi , Veerappa Gowda G.D.: Conservation laws with discontinuous flux. J. Math. Kyoto University 43(1), 27–70 (2003) · Zbl 1063.35114 [3] Adimurthi , Jaffré J., Veerappa Gowda G.D.: Godunov-type methods for conservation laws with a flux function discontinuous in space. SIAM J. Numer. Anal. 42(1), 179–208 (2004) · Zbl 1081.65082 [4] Adimurthi , Mishra S., Veerappa Gowda G.D.: Optimal entropy solutions for conservation laws with discontinuous flux-functions. J. Hyperbolic Differ. Equ. 2(4), 783–837 (2005) · Zbl 1093.35045 [5] Alt H.W., Luckhaus S.: Quasilinear elliptic-parabolic differential equations. Mat. Z. 183, 311–341 (1983) · Zbl 0508.35046 [6] Ammar K., Wittbold P.: Existence of renormalized solutions of degenerate elliptic-parabolic problems. Proc. Roy. Soc. Edinburgh Sect. A 133(3), 477–496 (2003) · Zbl 1077.35103 [7] Ammar K., Wittbold P., Carrillo J.: Scalar conservation laws with general boundary condition and continuous flux function. J. Differ. Equ. 228(1), 111–139 (2006) · Zbl 1103.35069 [8] Andreianov B., Bénilan P., Kruzhkov S.N.: L 1 theory of scalar conservation law with continuous flux function. J. Funct. Anal. 171(1), 15–33 (2000) · Zbl 0944.35048 [9] Andreianov B., Goatin P., Seguin N.: Finite volume schemes for locally constrained conservation laws. Numer. Math. 115(4), 609–645 (2010) · Zbl 1196.65151 [10] Andreianov, B., Karlsen, K.H., Risebro, N.H.: A theory of L 1-dissipative solvers for scalar conservation laws with discontinuous flux. II (in preparation) · Zbl 1261.35088 [11] Andreianov B., Karlsen K.H., Risebro N.H.: On vanishing viscosity approximation of conservation laws with discontinuous flux. Netw. Heterog. Media 5(3), 617–633 (2010) · Zbl 1270.35305 [12] Andreianov B., Sbihi K.: Scalar conservation laws with nonlinear boundary conditions. C. R. Acad. Sci. Paris, Ser. I 345, 431–434 (2007) · Zbl 1157.35420 [13] Audusse E., Perthame B.: Uniqueness for scalar conservation laws with discontinuous flux via adapted entropies. Proc. Roy. Soc. Edinburgh A 135(2), 253–265 (2005) · Zbl 1071.35079 [14] Bachmann F., Vovelle J.: Existence and uniqueness of entropy solution of scalar conservation laws with a flux function involving discontinuous coefficients. Comm. Partial Differ. Equ. 31, 371–395 (2006) · Zbl 1102.35064 [15] Baiti P., Jenssen H.K.: Well-posedness for a class of 2 {$$\times$$} 2 conservation laws with L data. J. Differ. Equ. 140(1), 161–185 (1997) · Zbl 0892.35097 [16] Bardos C., LeRoux A.-Y., Nédélec J.-C.: First order quasilinear equations with boundary conditions. Comm. Partial Differ. Equ. 4(9), 1017–1034 (1979) · Zbl 0418.35024 [17] Bénilan, P.: Equations d’évolution dans un espace de Banach quelconques et applications. Thèse d’état, 1972 [18] Bénilan P., Carrillo J., Wittbold P.: Renormalized entropy solutions of scalar conservation laws. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 29(2), 313–327 (2000) · Zbl 0965.35021 [19] Bénilan P., Kruzhkov S.N.: Conservation laws with continuous flux functions. NoDEA Nonlinear Differ. Equ. Appl. 3(4), 395–419 (1996) · Zbl 0961.35088 [20] Bürger R., García A., Karlsen K.H., Towers J.D.: Difference schemes, entropy solutions, and speedup impulse for an inhomogeneous kinematic traffic flow model. Netw. Heterog. Media 3, 1–41 (2008) · Zbl 1171.35328 [21] Bürger, R., Karlsen, K.H., Mishra, S., Towers, J.D.: On conservation laws with discontinuous flux. Trends in Applications of Mathematics to Mechanics (Eds. Wang Y. and Hutter K.) Shaker Verlag, Aachen, 75–84, 2005 · Zbl 1076.76069 [22] Bürger R., Karlsen K.H., Towers J.: An Engquist-Osher type scheme for conservation laws with discontinuous flux adapted to flux connections. SIAM J. Numer. Anal. 47, 1684–1712 (2009) · Zbl 1201.35022 [23] Cancès C.: Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. I. Convergence to the optimal entropy solution. SIAM J. Math. Anal. 42(2), 946–971 (2010) · Zbl 1219.35136 [24] Cancès C.: Asymptotic behavior of two-phase flows in heterogeneous porous media for capillarity depending only on space. II. Nonclassical shocks to model oil-trapping. SIAM J. Math. Anal. 42(2), 972–995 (2010) · Zbl 1219.35139 [25] Cancès C.: On the effects of discontinuous capillarities for immiscible two-phase flows in porous media made of several rock-types. Netw. Heterog. Media 5(3), 635–647 (2010) · Zbl 1262.35163 [26] Carrillo J.: Entropy solutions for nonlinear degenerate problems. Arch. Ration. Mech. Anal. 147(4), 269–361 (1999) · Zbl 0935.35056 [27] Chechkin, G.A., Goritsky, A.Yu.: S. N. Kruzhkov’s lectures on first-order quasilinear PDEs. (Eds. Emmrich E. and Wittbold P.) Anal. Numer. Asp. Partial Differ. Equ., de Gruyter, 2009 · Zbl 1184.35114 [28] Chen G.-Q., Even N., Klingenberg C.: Hyperbolic conservation laws with discontinuous fluxes and hydrodynamic limit for particle systems. J. Differ. Equ. 245(11), 3095–3126 (2008) · Zbl 1195.35211 [29] Chen G.-Q., Frid H.: Divergence-measure fields and hyperbolic conservationlaws. Arch. Ration. Mech. Anal. 147, 89–118 (1999) · Zbl 0942.35111 [30] Chen G.-Q., Rascle M.: Initial layers and uniqueness of weak entropy solutions to hyperbolic conservation laws. Arch. Ration. Mech. Anal. 153, 205–220 (2000) · Zbl 0962.35122 [31] Colombo R.M., Goatin P.: A well posed conservation law with a variable unilateral constraint. J. Differ. Equ. 234(2), 654–675 (2007) · Zbl 1253.65122 [32] Crandall M.G.: The semigroup approach to first-order quasilinear equations in several space variables. Israel J. Math. 12, 108–132 (1972) · Zbl 0246.35018 [33] Crandall M.G., Tartar L.: Some relations between nonexpansive and order preserving mappings. Proc. Am. Math. Soc. 78(3), 385–390 (1980) · Zbl 0449.47059 [34] Diehl S.: On scalar conservation laws with point source and discontinuous flux function. SIAM J. Math. Anal. 26(6), 1425–1451 (1995) · Zbl 0852.35094 [35] Diehl, S.: Scalar conservation laws with discontinuous flux function. I. The viscous profile condition. II. On the stability of the viscous profiles. Commun. Math. Phys. 176(1), 23–44 and 45–71 (1996) · Zbl 0845.35067 [36] Diehl S.: A conservation law with point source and discontinuous flux function modelling continuous sedimentation. SIAM J. Appl. Math. 56(2), 388–419 (1996) · Zbl 0849.35142 [37] Diehl S.: A uniqueness condition for non-linear convection-diffusion equations with discontinuous coefficients. J. Hyperbolic Differ. Equ. 6(1), 127–159 (2009) · Zbl 1180.35305 [38] Eymard, R., Gallouët, T., Herbin, R.: Finite volume methods. Handbook of numerical analysis, Vol. VII, Handb. Numer. Anal., VII. North-Holland, Amsterdam, 713–1020, 2000 · Zbl 0981.65095 [39] Gallouët T., Hubert F.: On the convergence of the parabolic approximation of a conservation law in several space dimensions. Chin. Ann. Math. Ser. B 20(1), 7–10 (1999) · Zbl 0922.35091 [40] Garavello M., Natalini R., Piccoli B., Terracina A.: Conservation laws with discontinuous flux. Netw. Heterog. Media 2, 159–179 (2007) · Zbl 1142.35511 [41] Gelfand, I.M.: Some problems in the theory of quasilinear equations. (Russian) Uspekhi Mat. Nauk 14(2), 87–158, 1959; English tr. in Am. Math. Soc. Transl. Ser. 29(2), 295–381 (1963) [42] Gimse, T., Risebro, N.H.: Riemann problems with a discontinuous flux function. Proceedings of Third International Conference on Hyperbolic Problems, Vol. I, II Uppsala, 1990, 488–502, Studentlitteratur, Lund, 1991 · Zbl 0789.35102 [43] Gimse T., Risebro N.H.: Solution of the Cauchy problem for a conservation law with a discontinuous flux function. SIAM J. Math. Anal. 23(3), 635–648 (1992) · Zbl 0776.35034 [44] Holden, H., Risebro, N.H.: Front tracking for hyperbolic conservation laws. In: Applied Mathematical Sciences, vol. 152. Springer, New York, 2002 · Zbl 1006.35002 [45] Hopf E.: The partial differential equation u t + uu x = {$$\mu$$} u xx . Comm. Pure Appl. Math. 3(3), 201–230 (1950) · Zbl 0039.10403 [46] Jimenez J.: Some scalar conservation laws with discontinuous flux. Int. J. Evol. Equ. 2(3), 297–315 (2007) · Zbl 1133.35405 [47] Jimenez J., Lévi L.: Entropy formulations for a class of scalar conservations laws with space-discontinuous flux functions in a bounded domain. J. Engrg. Math. 60(3–4), 319–335 (2008) · Zbl 1133.74030 [48] Kaasschieter E.F.: Solving the Buckley-Leverett equation with gravity in a heterogeneous porous medium. Comput. Geosci. 3, 23–48 (1999) · Zbl 0952.76085 [49] Karlsen K.H., Towers J.D.: Convergence of the Lax-Friedrichs scheme and stability for conservation laws with a discontinuous space-time dependent flux. Chin. Ann. Math. Ser. B. 25(3), 287–318 (2004) · Zbl 1112.65085 [50] Karlsen, K.H., Risebro, N.H., Towers, J.D.: On a nonlinear degenerate parabolic transport-diffusion equation with a discontinuous coefficient. Electron J. Differ. Equ., pages No. 93, 23 pp. (electronic), (2002) · Zbl 1015.35049 [51] Karlsen K.H., Risebro N.H., Towers J.D.: Upwind difference approximations for degenerate parabolic convection-diffusion equations with a discontinuous coefficient. IMA J. Numer. Anal. 22, 623–664 (2002) · Zbl 1014.65073 [52] Karlsen K.H., Risebro N.H., Towers J.D.: L 1 stability for entropy solutions of nonlinear degenerate parabolic convection-diffusion equations with discontinuous coefficients. Skr. K. Nor. Vidensk. Selsk. 3, 1–49 (2003) · Zbl 1036.35104 [53] Klingenberg C., Risebro N.H.: Convex conservation laws with discontinuous coefficients: Existence, uniqueness and asymptotic behavior. Comm. Partial Differ. Equ. 20(11–12), 1959–1990 (1995) · Zbl 0836.35090 [54] Kruzhkov S.N.: First order quasi-linear equations in several independent variables. Math. USSR Sbornik 10(2), 217–243 (1970) · Zbl 0215.16203 [55] Kruzhkov S.N., Hildebrand F.: The Cauchy problem for first order quasilinear equations in the case when the domain of dependence on the initial data is infinite (Russian). Vestnik Moskov. Univ. Ser. I Mat. Meh. 29, 93–100 (1974) [56] Kruzhkov, S.N., Panov, E.Yu.: First-order quasilinear conservation laws with infinite initial data dependence area (Russian). Dokl. Akad. Nauk URSS 314(1), 79–84, 1990; English tr. in Sov. Math. Dokl. 42(2), 316–321 (1991) [57] Kwon Y.-S., Vasseur A.: Strong traces for solutions to scalar conservation laws with general flux. Arch. Ration. Mech. Anal. 185(3), 495–513 (2007) · Zbl 1121.35078 [58] LeFloch, P.G.: Hyperbolic systems of conservation laws. Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2002. The theory of classical and nonclassical shock waves · Zbl 1019.35001 [59] Lions, J.-L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. (French) Dunod, Paris, 1969 [60] Lions P.-L., Perthame B., Tadmor E.: A kinetic formulation of multidimensional scalar conservation laws and related equations. J. Am. Math. Soc. 7(1), 169–191 (1994) · Zbl 0820.35094 [61] Málek J., Nečas J., Rokyta M., Ružička M.: Weak and Measure-valued Solutions to Evolutionary PDEs. Chapman & Hall, London (1996) [62] Maliki M., Touré H.: Uniqueness of entropy solutions for nonlinear degenerate parabolic problems. J. Evol. Equ. 3(4), 603–622 (2003) · Zbl 1052.35106 [63] Mishra, S.: Analysis and Numerical Approximation of Conservation Laws with Discontinuous Coefficients. PhD thesis, Indian Institute of Science, Bangalore, India, 2005 [64] Mitrovič D.: Existence and stability of a multidimensional scalar conservation law with discontinuous flux. Networks Het. Media 5(1), 163–188 (2010) · Zbl 1262.35161 [65] Olenik O.A.: Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi–linear equation. Amer. Math. Soc Transl. Ser. 2 33, 285–290 (1963) [66] Ostrov D.N.: Solutions of Hamilton-Jacobi equations and scalar conservation laws with discontinuous space-time dependence. J. Differ. Equ. 182(1), 51–77 (2002) · Zbl 1009.35015 [67] Otto F.: Initial-boundary value problem for a scalar conservation law. C. R. Acad. Sci. Paris Sér. I Math. 322(8), 729–734 (1996) · Zbl 0852.35013 [68] Otto F.: L 1-contraction and uniqueness for quasilinear elliptic-parabolic equations. J. Differ. Equ. 131(1), 20–38 (1996) · Zbl 0862.35078 [69] Panov E.Yu.: Strong measure-valued solutions of the Cauchy problem for a first-order quasilinear equation with a bounded measure-valued initial function. Moscow Univ. Math. Bull. 48(1), 18–21 (1993) · Zbl 0830.35023 [70] Panov, E.Yu.: On sequences of measure valued solutions for a first order quasilinear equation (Russian). Mat. Sb. 185(2), 87–106 1994; Engl. tr. in Russian Acad. Sci. Sb. Math. 81(1), 211–227 (1995) [71] Panov E.Yu.: Existence of strong traces for generalized solutions of multidimensional scalar conservation laws. J. Hyperbolic Differ. Equ. 2(4), 885–908 (2005) · Zbl 1145.35429 [72] Panov E.Yu.: Existence of strong traces for quasi-solutions of multidimensional conservation laws. J. Hyperbolic Differ. Equ. 4(4), 729–770 (2007) · Zbl 1144.35037 [73] Panov E.Yu.: Existence and strong pre-compactness properties for entropy solutions of a first-order quasilinear equation with discontinuous flux. Arch. Ration. Mech. Anal. 195(2), 643–673 (2009) · Zbl 1191.35102 [74] Panov E.Yu.: On existence and uniqueness of entropy solutions to the Cauchy problem for a conservation law with discontinuous flux. J. Hyperbolic Differ. Equ. 6(3), 525–548 (2009) · Zbl 1181.35145 [75] Perthame, B.: Kinetic formulation of conservation laws, volume 21 of Oxford Lecture Series in Mathematics and its Applications. Oxford University Press, Oxford, 2002 [76] Quinn (Keyfitz), B.: Solutions with shocks, an example of an L 1-contractive semigroup. Comm. Pure Appl. Math. 24(1), 125–132 (1971) · Zbl 0206.10401 [77] Seguin N., Vovelle J.: Analysis and approximation of a scalar conservation law with a flux function with discontinuous coefficients. Math. Models Methods Appl. Sci. 13(2), 221–257 (2003) · Zbl 1078.35011 [78] Szepessy A.: Measure-valued solution of scalar conservation laws with boundary conditions. Arch. Ration. Mech. Anal. 107(2), 182–193 (1989) · Zbl 0702.35155 [79] Temple B.: Global solution of the Cauchy problem for a class of 2 {$$\times$$} 2 nonstrictly hyperbolic conservation laws. Adv. Appl. Math. 3(3), 335–375 (1982) · Zbl 0508.76107 [80] Towers J.D.: Convergence of a difference scheme for conservation laws with a discontinuous flux. SIAM J. Numer. Anal. 38(2), 681–698 (2000) · Zbl 0972.65060 [81] Towers J.D.: A difference scheme for conservation laws with a discontinuous flux: the nonconvex case. SIAM J. Numer. Anal. 39(4), 1197–1218 (2001) · Zbl 1055.65104 [82] Vallet G.: Dirichlet problem for a nonlinear conservation law. Rev. Math. Comput. 13(1), 231–250 (2000) · Zbl 0979.35099 [83] Vasseur A.: Strong traces of multidimensional scalar conservation laws. Arch. Ration. Mech. Anal. 160(3), 181–193 (2001) · Zbl 0999.35018 [84] Vol’pert A.I.: The spaces BV and quasi-linear equations. Math. USSR Sbornik 2(2), 225–267 (1967) · Zbl 0168.07402
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