BV regularity near the interface for nonuniform convex discontinuous flux

Shyam Sundar  Ghoshal

doi:10.3934/nhm.2016.11.331

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2016, Volume 11, Issue 2: 331-348. Doi: 10.3934/nhm.2016.11.331

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BV regularity near the interface for nonuniform convex discontinuous flux

Shyam Sundar Ghoshal^1,

1.
Gran Sasso Science Institute, Viale Francesco Crispi, 7, 67100 LAquila

Received: March 31, 2015

Revised: August 31, 2015

Published: May 2016

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Abstract

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 $BV_{loc}$ solution near the interface for $L^\infty$ initial data without the assumption on the uniform convexity of the fluxes made as in [1,21]. The proof relies on the method of characteristics and the explicit formulas.

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Mathematics Subject Classification: Primary: 35B65, 35L65, 35L67.

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References

[1]	Adimurthi, R. Dutta, S. S. Ghoshal and G. D. Veerappa Gowda, Existence and nonexistence of TV bounds for scalar conservation laws with discontinuous flux, Comm. Pure Appl. Math., 64 (2011), 84-115.doi: 10.1002/cpa.20346.
[2]	Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Finer regularity of an entropy solution for $1$-$d$ scalar conservation laws with non uniform convex flux, Rend. Sem. Mat. Univ. Padova, 132 (2014), 1-24.doi: 10.4171/RSMUP/132-1.
[3]	Adimurthi, S. S. Ghoshal and G. D. Veerappa Gowda, Structure of an entropy solution of a scalar conservation law with strict convex flux, J. Hyperbolic Differ. Equ., 9 (2012), 571-611.doi: 10.1142/S0219891612500191.
[4]	Adimurthi and G. D. Veerappa Gowda, Conservation laws with discontinuous flux, J. Math. Kyoto Univ., 43 (2003), 27-70.
[5]	Adimurthi, J. Jaffre and G. D. Veerappa Gowda, Godunov type methods for scalar conservation laws with flux function discontinuous in the space variable, SIAM J. Numer. Anal., 42 (2004), 179-208.doi: 10.1137/S003614290139562X.
[6]	Adimurthi, S. Mishra and G. D. Veerappa Gowda, Optimal entropy solutions for conservation laws with discontinuous flux-functions, J. Hyperbolic Differ. Equ., 2 (2005), 783-837.doi: 10.1142/S0219891605000622.
[7]	Adimurthi, S. Mishra and G. D. Veerappa Gowda, Explicit Hopf-Lax type formulas for Hamilton-Jacobi equations and conservation laws with discontinuous coefficients， J. Differential Equations, 241 (2007), 1-31.doi: 10.1016/j.jde.2007.05.039.
[8]	Adimurthi, S. Mishra and G. D. Veerappa Gowda, Convergence of Godunov type methods for a conservation law with a spatially varying discontinuous flux function, Math. Comp., 76 (2007), 1219-1242.doi: 10.1090/S0025-5718-07-01960-6.
[9]	B. Andreianov, K. H. Karlsen and N. H. Risebro, A theory of $L^1$ - dissipative solvers for scalar conservation laws with discontinuous flux, Arch. Ration. Mech. Anal., 201 (2011), 27-86.doi: 10.1007/s00205-010-0389-4.
[10]	B. Andreianov and C. Cances, The Godunov scheme for scalar conservation laws with discontinuous bell-shaped flux functions, Appl. Math. Lett., 25 (2012), 1844-1848.doi: 10.1016/j.aml.2012.02.044.
[11]	R. Bürger, A. García, K. H. Karlsen and J. D. Towers, A family of numerical schemes for kinematic flows with discontinuous flux, J. Engrg. Math., 60 (2008), 387-425.doi: 10.1007/s10665-007-9148-4.
[12]	R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Well-posedness in $BV_t$ and convergence of a difference scheme for continuous sedimentation in ideal clarifier thickener units, Numer. Math., 97 (2004), 25-65.doi: 10.1007/s00211-003-0503-8.
[13]	R. Bürger, K. H. Karlsen, C. Klingenberg and N. H. Risebro, A front tracking approach to a model of continuous sedimentation in ideal clarifier-thickener units, Nonlin. Anal. Real World Appl., 4 (2003), 457-481.doi: 10.1016/S1468-1218(02)00071-8.
[14]	R. Bürger, K. H. Karlsen and N. H. Risebro, A relaxation scheme for continuous sedimentation in ideal clarifier-thickner units, Comput. Math. Applic., 50 (2005), 993-1009.doi: 10.1016/j.camwa.2005.08.019.
[15]	R. Bürger, K. H. Karlsen, N. H. Risebro and J. D. Towers, Monotone difference approximations for the simulation of clarifier-thickener units, Comput. Visual. Sci., 6 (2004), 83-91.doi: 10.1007/s00791-003-0112-1.
[16]	R. Bürger, K. H. Karlsen and J. D. Towers, A mathematical model of continuous sedimentation of flocculated suspensions in clarifier-thickener units, SIAM J. Appl. Math., 65 (2005), 882-940.doi: 10.1137/04060620X.
[17]	S. Diehl, Dynamic and steady-state behaviour of continuous sedimentation, SIAM J. Appl. Math., 57 (1997), 991-1018.doi: 10.1137/S0036139995290101.
[18]	S. Diehl, Operating charts for continuous sedimentation II: Step responses, J. Engrg. Math., 53 (2005), 139-185.doi: 10.1007/s10665-005-6430-1.
[19]	T. Gimse and N. H. Risebro, Solution of the Cauchy problem for a conservation law with a discontinuous flux function, SIAM J. Math. Anal., 23 (1992), 635-648.doi: 10.1137/0523032.
[20]	T. Gimse and N. H. Risebro, Riemann problems with discontinuous flux function, Proc. 3rd Internat. Conf. Hyperbolic Problems, Studentlitteratur, Uppsala, (1991), 488-502.
[21]	S. S. Ghoshal, Optimal results on TV bounds for scalar conservation laws with discontinuous flux, J. Differential Equations, 258 (2015), 980-1014.doi: 10.1016/j.jde.2014.10.014.
[22]	S. S. Ghoshal, 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, Ph.D thesis, TIFR, Centre for Applicable Mathematics, 2012.
[23]	K. T. Joseph and G. D. Veerappa Gowda, Explicit formula for solution of convex conservation laws with boundary condition, Duke Math. J., 62 (1991), 401-416.doi: 10.1215/S0012-7094-91-06216-2.
[24]	E. Kaasschieter, Solving the Buckley-Leverret equation with gravity in a heterogeneous porous media, Comput. Geosci., 3 (1999), 23-48.doi: 10.1023/A:1011574824970.
[25]	K. H. Karlsen, N. H. Risebro and J. D. Towers, On a Nonlinear Degenerate Parabolic Transport Diffusion Equation with Discontinuous Coefficient, Electron. J. Differential Equations, 2002.
[26]	C. Klingenberg and N. H. Risebro, Convex conservation laws with discontinuous coefficients, existence, uniqueness and asymptotic behavior, Comm. Partial Differential Equations, 20 (1995), 1959-1990.doi: 10.1080/03605309508821159.
[27]	S. N. Kružkov, First order quasilinear equations with several independent variables. (Russian), Mat. Sb., 81 (1970), 228-255. English transl. in Math. USSR Sb., 10 (1970), 217-243.
[28]	P. D. Lax, Hyperbolic systems of conservation laws II, Comm. Pure Appl. Math., 10 (1957), 537-566.doi: 10.1002/cpa.3160100406.
[29]	S. Mochon, An analysis for the traffic on highways with changing surface conditions, Math. Model., 9 (1987), 1-11.doi: 10.1016/0270-0255(87)90068-6.
[30]	D. N. Ostrov, Solutions of Hamilton-Jacobi equations and conservation laws with discontinuous space-time dependence, J. Differential Equations, 182 (2002), 51-77.doi: 10.1006/jdeq.2001.4088.
[31]	D. Serre, Systémes De Lois De Conservation. I., Hyperbolicité, Entropies, Ondes de Choc, Diderot Editeur, Paris, 1996.
[32]	J. D. Towers, Convergence of a difference scheme for conservation laws with a discontinuous flux, SIAM J. Numer. Anal., 38 (2000), 681-698.doi: 10.1137/S0036142999363668.