Article
| Title: | Large time behaviour of a conservation law regularised by a Riesz-Feller operator: the sub-critical case (English) |
| Author: | Cuesta, Carlota Maria |
| Author: | Diez-Izagirre, Xuban |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 73 |
| Issue: | 4 |
| Year: | 2023 |
| Pages: | 1057-1080 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We study the large time behaviour of the solutions of a nonlocal regularisation of a scalar conservation law. This regularisation is given by a fractional derivative of order $1+\alpha $, with $\alpha \in (0,1)$, which is a Riesz-Feller operator. The nonlinear flux is given by the locally Lipschitz function $|u|^{q-1}u/q$ for $q>1$. We show that in the sub-critical case, $1<q < 1 +\alpha $, the large time behaviour is governed by the unique entropy solution of the scalar conservation law. Our proof adapts the proofs of the analogous results for the local case (where the regularisation is the Laplacian) and, more closely, the ones for the regularisation given by the fractional Laplacian with order larger than one, see L. I. Ignat and D. Stan (2018). The main difference is that our operator is not symmetric and its Fourier symbol is not real. We can also adapt the proof and obtain similar results for general Riesz-Feller operators. (English) |
| Keyword: | large time asymptotic |
| Keyword: | regularisation of conservation law |
| Keyword: | Riesz-Feller \hbox {operator} |
| MSC: | 26A33 |
| MSC: | 35B40 |
| MSC: | 47J35 |
| DOI: | 10.21136/CMJ.2023.0235-22 |
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| Date available: | 2023-11-23T12:21:13Z |
| Last updated: | 2023-11-27 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/151947 |
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| Reference: | [1] Achleitner, F., Hittmeir, S., Schmeiser, C.: On nonlinear conservation laws with a nonlocal diffusion term.J. Differ. Equations 250 (2011), 2177-2196. Zbl 1213.47066, MR 2763569, 10.1016/j.jde.2010.11.015 |
| Reference: | [2] Achleitner, F., Kuehn, C.: Traveling waves for a bistable equation with nonlocal diffusion.Adv. Differ. Equ. 20 (2015), 887-936. Zbl 1327.35053, MR 3360395, 10.57262/ade/1435064517 |
| Reference: | [3] Achleitner, F., Ueda, Y.: Asymptotic stability of traveling wave solutions for nonlocal viscous conservation laws with explicit decay rates.J. Evol. Equ. 18 (2018), 923-946. Zbl 06932128, MR 3820428, 10.1007/s00028-018-0426-6 |
| Reference: | [4] Adams, R. A.: Sobolev Spaces.Pure and Applied Mathematics 65. Academic Press, New York (1975). Zbl 0314.46030, MR 0450957 |
| Reference: | [5] Bertoin, J.: Lévy Processes.Cambridge Tracts in Mathematics 121. Cambridge University Press, Cambridge (1996). Zbl 0861.60003, MR 1406564 |
| Reference: | [6] Biler, P., Karch, G., Woyczyński, W. A.: Asymptotics for conservation laws involving Lévy diffusion generators.Stud. Math. 148 (2001), 171-192. Zbl 0990.35023, MR 1881259, 10.4064/sm148-2-5 |
| Reference: | [7] Biler, P., Karch, G., Woyczyński, W. A.: Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws.Ann. Inst. H. Poincaré, Anal. Non Linéaire 18 (2001), 613-637. Zbl 0991.35009, MR 1849690, 10.1016/S0294-1449(01)00080-4 |
| Reference: | [8] Bouharguane, A., Carles, R.: Splitting methods for the nonlocal Fowler equation.Math. Comput. 83 (2014), 1121-1141. Zbl 1286.65109, MR 3167452, 10.1090/S0025-5718-2013-02757-3 |
| Reference: | [9] Cazacu, C. M., Ignat, L. I., Pazoto, A. F.: On the asymptotic behavior of a subcritical convection-diffusion equation with nonlocal diffusion.Nonlinearity 30 (2017), 3126-3150. Zbl 1372.35040, MR 3685664, 10.1088/1361-6544/aa773a |
| Reference: | [10] Christ, F. M., Weinstein, M. I.: Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation.J. Funct. Anal. 100 (1991), 87-109. Zbl 0743.35067, MR 1124294, 10.1016/0022-1236(91)90103-C |
| Reference: | [11] Cifani, S., Jakobsen, E. R.: Entropy solution theory for fractional degenerate convection-diffusion equations.Ann. Inst. H. Poincaré, Anal. Non Linéaire 28 (2011), 413-441. Zbl 1217.35204, MR 2795714, 10.1016/j.anihpc.2011.02.006 |
| Reference: | [12] Cuesta, C. M., Achleitner, F.: Addendum to: "Travelling waves for a non-local Korteweg de Vries-Burgers equation".J. Differ. Equations 262 (2017), 1155-1160. Zbl 06652621, MR 3569418, 10.1016/j.jde.2016.09.029 |
| Reference: | [13] J. Diestel, J. J. Uhl, Jr.: Vector Measures.Mathematical Surveys 15. AMS, Providence (1977). Zbl 0369.46039, MR 0453964, 10.1090/surv/015 |
| Reference: | [14] Diez-Izagirre, X.: Non-Local Regularisations of Scalar Conservation Laws: Doctoral Thesis.University of the Basque Country, Azpeitia (2021), Spanish. |
| Reference: | [15] Diez-Izagirre, X., Cuesta, C. M.: Vanishing viscosity limit of a conservation law regularised by a Riesz-Feller operator.Monatsh. Math. 192 (2020), 513-550. Zbl 1441.35165, MR 4109515, 10.1007/s00605-020-01413-8 |
| Reference: | [16] Droniou, J., Imbert, C.: Fractal first-order partial differential equations.Arch. Ration. Mech. Anal. 182 (2006), 299-331. Zbl 1111.35144, MR 2259335, 10.1007/s00205-006-0429-2 |
| Reference: | [17] Escobedo, M., Vázquez, J. L., Zuazua, E.: A diffusion-convection equation in several space dimensions.Indiana Univ. Math. J. 42 (1993), 1413-1440. Zbl 0791.35059, MR 1266100, 10.1512/iumj.1993.42.42065 |
| Reference: | [18] Escobedo, M., Vázquez, J. L., Zuazua, E.: Asymptotic behaviour and source-type solutions for a diffusion-convection equation.Arch. Ration. Mech. Anal. 124 (1993), 43-65. Zbl 0807.35059, MR 1233647, 10.1007/BF00392203 |
| Reference: | [19] Escobedo, M., Zuazua, E.: Large time behavior for convection-diffusion equations in $\Bbb R^N$.J. Funct. Anal. 100 (1991), 119-161. Zbl 0762.35011, MR 1124296, 10.1016/0022-1236(91)90105-E |
| Reference: | [20] Fowler, A. C.: Evolution equations for dunes and drumlins.RACSAM, Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 96 (2002), 377-387. Zbl 1229.86014, MR 1985743 |
| Reference: | [21] Gatto, A. E.: Product rule and chain rule estimates for fractional derivatives on spaces that satisfy the doubling condition.J. Funct. Anal. 188 (2002), 27-37. Zbl 1031.43005, MR 1878630, 10.1006/jfan.2001.3836 |
| Reference: | [22] Ignat, L. I., Stan, D.: Asymptotic behavior of solutions to fractional diffusion-convection equations.J. Lond. Math. Soc., II. Ser. 97 (2018), 258-281. Zbl 1387.35051, MR 3789847, 10.1112/jlms.12110 |
| Reference: | [23] Kamin, S., Vázquez, J. L.: Fundamental solutions and asymptotic behaviour for the $p$-Laplacian equation.Rev. Mat. Iberoam. 4 (1988), 339-354. Zbl 0699.35158, MR 1028745, 10.4171/RMI/77 |
| Reference: | [24] Kluwick, A., Cox, E. A., Exner, A., Grinschgl, C.: On the internal structure of weakly nonlinear bores in laminar high Reynolds number flow.Acta Mech. 210 (2010), 135-157. Zbl 1308.76055, 10.1007/s00707-009-0188-x |
| Reference: | [25] Kružkov, S. N.: First order quasilinear equations in several independent variables.Math. USSR, Sb. 10 (1970), 217-243. Zbl 0215.16203, MR 0267257, 10.1070/SM1970v010n02ABEH002156 |
| Reference: | [26] Kwaśnicki, M.: Ten equivalent definitions of the fractional Laplace operator.Fract. Calc. Appl. Anal. 20 (2017), 7-51. Zbl 1375.47038, MR 3613319, 10.1515/fca-2017-0002 |
| Reference: | [27] Liu, T.-P., Pierre, M.: Source-solutions and asymptotic behavior in conservation laws.J. Differ. Equations 51 (1984), 419-441. Zbl 0545.35057, MR 0735207, 10.1016/0022-0396(84)90096-2 |
| Reference: | [28] Mainardi, F., Luchko, Y., Pagnini, G.: The fundamental solution of the space-time fractional diffusion equation.Fract. Calc. Appl. Anal. 4 (2001), 153-192. Zbl 1054.35156, MR 1829592 |
| Reference: | [29] Marchaud, A.: Sur les dérivées et sur les différences des fonctions de variables réelles.J. Math. Pures Appl. (9) 6 (1927), 337-425 French \99999JFM99999 53.0232.02. MR 3532941 |
| Reference: | [30] Pruitt, W. E., Taylor, S. J.: The potential kernel and hitting probabilities for the general stable process in $R^N$.Trans. Am. Math. Soc. 146 (1969), 299-321. Zbl 0229.60052, MR 0250372, 10.1090/S0002-9947-1969-0250372-3 |
| Reference: | [31] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications.Gordon and Breach, New York (1993). Zbl 0818.26003, MR 1347689 |
| Reference: | [32] Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions.Cambridge Studies in Advanced Mathematics 68. Cambridge University Press, Cambridge (1999). Zbl 0973.60001, MR 1739520 |
| Reference: | [33] Simon, J.: Compact sets in the space $L^p(0,T;B)$.Ann. Mat. Pura Appl., IV. Ser. 146 (1987), 65-96. Zbl 0629.46031, MR 0916688, 10.1007/BF01762360 |
| Reference: | [34] Sugimoto, N., Kakutani, T.: "Generalized Burgers' equation" for nonlinear viscoelastic waves.Wave Motion 7 (1985), 447-458. Zbl 0588.73046, MR 0802984, 10.1016/0165-2125(85)90019-8 |
| Reference: | [35] Viertl, N.: Viscous Regularisation of Weak Laminar Hydraulic Jumps and Bores in Two Layer Shallow Water Flow: Ph.D. Thesis.Technische Universität Wien, Wien (2005). |
| Reference: | [36] Visan, M.: The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions.Duke Math. J. 138 (2007), 281-374. Zbl 1131.35081, MR 2318286, 10.1215/S0012-7094-07-13825-0 |
| Reference: | [37] Weyl, H.: Bemerkungen zum Begriff des Differentialquotienten gebrochener Ordnung.Vierteljschr. Naturforsch. Ges. Zürich 62 (1917), 296-302 German \99999JFM99999 46.0437.01. MR 3618577 |
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