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Vanishing viscosity limit of a conservation law regularised by a Riesz-Feller operator. (English) Zbl 1441.35165

Summary: We study a nonlocal regularisation of a scalar conservation law given by a fractional derivative of order between one and two. The nonlocal operator is of Riesz-Feller type with skewness two minus its order. This equation describes the internal structure of hydraulic jumps in a shallow water model. The main purpose of the paper is the study of the vanishing viscosity limit of the Cauchy problem for this equation. First, we study the properties of the solution of the regularised problem and then we show that the difference between the regularised solution and the entropy solution of the scalar conservation law converges to zero in this limit in \(C([0, T]; L^1_{loc}(\mathbb{R}))\) for initial data in \(L^\infty (\mathbb{R})\), and in \(C([0, T]; L^1(\mathbb{R}))\) for initial data in \(L^\infty (\mathbb{R}) \cap BV(\mathbb{R})\). In order to prove these results we use weak entropy inequalities and the double scale technique of Kruzhkov. Such techniques also allow to show the \(L^1(\mathbb{R})\) contraction of the regularised problem. For completeness, we study the behaviour in the tail of travelling wave solutions for genuinely nonlinear fluxes. These waves converge to shock waves in the vanishing viscosity limit, but decay algebraically as \(x - ct \rightarrow \infty\), rather than exponentially, the latter being a behaviour that they exhibit as \(x - ct \rightarrow - \infty\), however. Finally, we generalise the results concerning the vanishing viscosity limit to Riesz-Feller operators.

MSC:

35L65 Hyperbolic conservation laws
35L60 First-order nonlinear hyperbolic equations
35S30 Fourier integral operators applied to PDEs
35L45 Initial value problems for first-order hyperbolic systems
35B25 Singular perturbations in context of PDEs
35R11 Fractional partial differential equations
35C07 Traveling wave solutions
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[1] Achleitner, F.; Cuesta, CM; Hittmeir, S., Travelling waves for a non-local Korteweg-de Vries-Burgers equation, J. Differ. Equ., 257, 3, 720-758 (2014) · Zbl 1293.35273
[2] Achleitner, F.; Hittmeir, S.; Schmeiser, C., On nonlinear conservation laws with a nonlocal diffusion term, J. Differ. Equ., 250, 4, 2177-2196 (2011) · Zbl 1213.47066
[3] Achleitner, F., Hittmeir, S., Schmeiser, C.: On nonlinear conservation laws regularized by a Riesz-Feller operator. In: Hyperbolic Problems: Theory, Numerics, Applications. AIMS Series on Applied Mathematics, vol. 8, pp. 241-248. Am. Inst. Math. Sci. (AIMS), Springfield (2014)
[4] Achleitner, F.; Kuehn, C., Traveling waves for a bistable equation with nonlocal diffusion, Adv. Differ. Equ., 20, 9-10, 887-936 (2015) · Zbl 1327.35053
[5] Alibaud, N., Entropy formulation for fractal conservation laws, J. Evol. Equ., 7, 1, 145-175 (2007) · Zbl 1116.35013
[6] Alvarez-Samaniego, B.; Azerad, P., Existence of travelling-wave solutions and local well-posedness of the Fowler equation, Discrete Contin. Dyn. Syst. Ser. B, 12, 4, 671-692 (2009) · Zbl 1193.35239
[7] Cifani, S.; Jakobsen, ER, Entropy solution theory for fractional degenerate convection-diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 28, 3, 413-441 (2011) · Zbl 1217.35204
[8] Cuesta, C.M., Achleitner, F.: Addendum to “Travelling waves for a non-local Korteweg-de Vries-Burgers equation”. J. Differ. Equ. 257(3), 720-758 (2014). J. Differ. Equ. 262(2), 1155-1160 (2017) · Zbl 06652621
[9] de la Hoz, F.; Cuesta, CM, A pseudo-spectral method for a non-local KdV-Burgers equation posed on \({\mathbb{R}} \), J. Comput. Phys., 311, 45-61 (2016) · Zbl 1349.65532
[10] Droniou, J.; Gallouet, T.; Vovelle, J., Global solution and smoothing effect for a non-local regularization of a hyperbolic equation, J. Evol. Equ., 3, 3, 499-521 (2003) · Zbl 1036.35123
[11] Droniou, J., Vanishing non-local regularization of a scalar conservation law, Electron. J. Differ. Equ., 117, 20 (2003) · Zbl 1039.35061
[12] Droniou, J.; Imbert, C., Fractal first-order partial differential equations, Arch. Ration. Mech. Anal., 182, 2, 299-331 (2006) · Zbl 1111.35144
[13] Endal, J.; Jakobsen, ER, \(L^1\) contraction for bounded (nonintegrable) solutions of degenerate parabolic equations, SIAM J. Math. Anal., 46, 6, 3957-3982 (2014) · Zbl 1323.35096
[14] Gorenflo, R., Mainardi, F.: Fractional calculus: integral and differential equations of fractional order. In: Fractals and Fractional Calculus in Continuum Mechanics (Udine. 1996). CISM Courses and Lectures, vol. 378, pp. 223-276. Springer, Vienna (1997)
[15] Karlsen, KH; Risebro, NH, On the uniqueness and stability of entropy solutions of nonlinear degenerate parabolic equations with rough coefficients, Discrete Contin. Dyn. Syst., 9, 5, 1081-1104 (2003) · Zbl 1027.35057
[16] Kluwick, A.; Cox, EA; Exner, A.; Grinschgl, C., On the internal structure of weakly nonlinear bores in laminar high reynolds number flow, Acta Mech., 210, 135-157 (2010) · Zbl 1308.76055
[17] Kružkov, SN, First order quasilinear equations with several independent variables, Mat. Sb. (N.S.), 81, 123, 228-255 (1970)
[18] Mainardi, F.; Luchko, Y.; Pagnini, G., The fundamental solution of the space-time fractional diffusion equation, Fract. Calc. Appl. Anal., 4, 2, 153-192 (2001) · Zbl 1054.35156
[19] Marchaud, A.: Sur les dérivées et sur les différences des fonctions de variables réelles. Ph.D. thesis, (1927). Numdam, Thèses de l’ entre-deux-guerres, Tome 78 · JFM 53.0232.02
[20] Sato, K., Lévy Processes and Infinitely Divisible Distributions. Cambridge Studies in Advanced Mathematics (1999), Cambridge: Cambridge University Press, Cambridge
[21] Serre, D.: Systems of Conservation Laws. 1. Cambridge University Press, Cambridge (1999). Hyperbolicity, entropies, shock waves. Translated from the 1996 French original by I. N. Sneddon · Zbl 0930.35001
[22] Sugimoto, N.; Kakutani, T., “Generalized Burgers equation” for nonlinear viscoelastic waves, Wave Motion, 7, 5, 447-458 (1985) · Zbl 0588.73046
[23] Tartar, L., An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana (2007), Berlin: Springer, Berlin · Zbl 1126.46001
[24] Weyl, H., Bemerkungen zum Begriff de Differentialquotienten gebrochener Ordnung, Vierteljahr. Naturforsch. Ges. Zürich, 62, 296-302 (1917) · JFM 46.0437.01
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