BV regularity near the interface for nonuniform convex discontinuous flux.

*(English)*Zbl 1343.35050Summary: In this paper, we discuss the total variation bound for the solution of scalar conservation laws with discontinuous flux. We prove the smoothing effect of the equation forcing the \(BV_{\mathrm{loc}}\) solution near the interface for \(L^\infty\) initial data without the assumption on the uniform convexity of the fluxes made as in [Adimurthi et al., Commun. Pure Appl. Math. 64, No. 1, 84–115 (2011; Zbl 1223.35222); the author, J. Differ. Equations 258, No. 3, 980–1014 (2015; Zbl 1312.35032)]. The proof relies on the method of characteristics and the explicit formulas.

##### MSC:

35B65 | Smoothness and regularity of solutions to PDEs |

35L65 | Hyperbolic conservation laws |

35L67 | Shocks and singularities for hyperbolic equations |

35F21 | Hamilton-Jacobi equations |

##### References:

[1] | Adimurthi, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux,, Comm. Pure Appl. Math., 64, 84, (2011) · Zbl 1223.35222 |

[2] | Adimurthi, Finer regularity of an entropy solution for \(1\)-\(d\) scalar conservation laws with non uniform convex flux,, Rend. Sem. Mat. Univ. Padova, 132, 1, (2014) · Zbl 1307.35165 |

[3] | Adimurthi, Structure of an entropy solution of a scalar conservation law with strict convex flux,, J. Hyperbolic Differ. Equ., 9, 571, (2012) · Zbl 1272.35145 |

[4] | Adimurthi, Conservation laws with discontinuous flux,, J. Math. Kyoto Univ., 43, 27, (2003) · Zbl 0900.35140 |

[5] | Adimurthi, Godunov type methods for scalar conservation laws with flux function discontinuous in the space variable,, SIAM J. Numer. Anal., 42, 179, (2004) · Zbl 1081.65082 |

[6] | Adimurthi, Optimal entropy solutions for conservation laws with discontinuous flux-functions,, J. Hyperbolic Differ. Equ., 2, 783, (2005) · Zbl 1093.35045 |

[7] | Adimurthi, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients，, J. Differential Equations, 241, 1, (2007) · Zbl 1128.35067 |

[8] | Adimurthi, Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function,, Math. Comp., 76, 1219, (2007) · Zbl 1116.35084 |

[9] | B. Andreianov, A theory of \(L^1\) - dissipative solvers for scalar conservation laws with discontinuous flux,, Arch. Ration. Mech. Anal., 201, 27, (2011) · Zbl 1261.35088 |

[10] | B. Andreianov, The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions,, Appl. Math. Lett., 25, 1844, (2012) · Zbl 1253.65122 |

[11] | R. Bürger, A family of numerical schemes for kinematic flows with discontinuous flux,, J. Engrg. Math., 60, 387, (2008) · Zbl 1200.76126 |

[12] | R. Bürger, Well-posedness in \(BV_t\) and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units,, Numer. Math., 97, 25, (2004) · Zbl 1053.76047 |

[13] | R. Bürger, A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units,, Nonlin. Anal. Real World Appl., 4, 457, (2003) · Zbl 1013.35052 |

[14] | R. Bürger, A relaxation scheme for continuous sedimentation in ideal clarifier-thickner units,, Comput. Math. Applic., 50, 993, (2005) · Zbl 1122.76063 |

[15] | R. Bürger, Monotone difference approximations for the simulation of clarifier-thickener units,, Comput. Visual. Sci., 6, 83, (2004) · Zbl 1299.76283 |

[16] | R. Bürger, A mathematical model of continuous sedimentation of flocculated suspensions in clarifier-thickener units,, SIAM J. Appl. Math., 65, 882, (2005) · Zbl 1089.76061 |

[17] | S. Diehl, Dynamic and steady-state behaviour of continuous sedimentation,, SIAM J. Appl. Math., 57, 991, (1997) · Zbl 0889.35062 |

[18] | S. Diehl, Operating charts for continuous sedimentation II: Step responses,, J. Engrg. Math., 53, 139, (2005) · Zbl 1086.76069 |

[19] | T. Gimse, Solution of the Cauchy problem for a conservation law with a discontinuous flux function,, SIAM J. Math. Anal., 23, 635, (1992) · Zbl 0776.35034 |

[20] | T. Gimse, Riemann problems with discontinuous flux function,, Proc. 3rd Internat. Conf. Hyperbolic Problems, 488, (1991) · Zbl 0789.35102 |

[21] | S. S. Ghoshal, Optimal results on TV bounds for scalar conservation laws with discontinuous flux,, J. Differential Equations, 258, 980, (2015) · Zbl 1312.35032 |

[22] | S. S. Ghoshal, <em>Finer Analysis of Characteristic Curve and its Application to Exact, Optimal Controllability, Structure of the Entropy Solution of a Scalar Conservation Law with Convex Flux</em>,, Ph.D thesis, (2012) |

[23] | K. T. Joseph, Explicit formula for solution of convex conservation laws with boundary condition,, Duke Math. J., 62, 401, (1991) · Zbl 0739.35040 |

[24] | E. Kaasschieter, Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media,, Comput. Geosci., 3, 23, (1999) · Zbl 0952.76085 |

[25] | K. H. Karlsen, <em>On a Nonlinear Degenerate Parabolic Transport Diffusion Equation with Discontinuous Coefficient,</em>, Electron. J. Differential Equations, (2002) · Zbl 1015.35049 |

[26] | C. Klingenberg, Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymptotic behavior,, Comm. Partial Differential Equations, 20, 1959, (1995) · Zbl 0836.35090 |

[27] | S. N. Kružkov, First order quasilinear equations with several independent variables. (Russian),, Mat. Sb., 81, 228, (1970) |

[28] | P. D. Lax, Hyperbolic systems of conservation laws II,, Comm. Pure Appl. Math., 10, 537, (1957) · Zbl 0081.08803 |

[29] | S. Mochon, An analysis for the traffic on highways with changing surface conditions,, Math. Model., 9, 1, (1987) |

[30] | D. N. Ostrov, Solutions of Hamilton-Jacobi equations and conservation laws with discontinuous space-time dependence,, J. Differential Equations, 182, 51, (2002) · Zbl 1009.35015 |

[31] | D. Serre, Systémes De Lois De Conservation. I., Hyperbolicité, Entropies, Ondes de Choc,, Diderot Editeur, (1996) · Zbl 0930.35002 |

[32] | J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux,, SIAM J. Numer. Anal., 38, 681, (2000) · Zbl 0972.65060 |

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.