Viscous conservation laws in 1D with measure initial data
Authors:
Miriam Bank, Matania Ben-Artzi and Maria E. Schonbek
Journal:
Quart. Appl. Math. 79 (2021), 103-124
MSC (2010):
Primary 35K15; Secondary 35K59
DOI:
https://doi.org/10.1090/qam/1572
Published electronically:
May 12, 2020
MathSciNet review:
4188625
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Abstract:
The one-dimensional viscous conservation law is considered on the whole line \begin{equation*} u_t + f(u)_x=\varepsilon u_{xx},\quad (x,t)\in \mathbb {R}\times \overline {\mathbb {R}_{+}},\quad \varepsilon >0, \end{equation*} subject to positive measure initial data.
The flux $f\in C^1(\mathbb {R})$ is assumed to satisfy a $p-$condition, a weak form of convexity. In particular, any flux of the form $f(u)=\sum _{i=1}^Ja_iu^{m_i}$ is admissible if $a_i>0, m_i>1, i=1,2,\ldots ,J.$
The only case treated hitherto in the literature is $f(u)=u^m$ [Arch. Rat. Mech. Anal. 124 (1993), pp. 43–65] and the initial data is a “single source”, namely, a multiple of the delta function. The corresponding solutions have been labeled as “source-type” and the treatment made substantial use of the special form of both the flux and the initial data.
In this paper existence and uniqueness of solutions is established. The method of proof relies on sharp decay estimates for the viscous Hamilton-Jacobi equation. Some estimates are independent of the viscosity coefficient, thus leading to new estimates for the (inviscid) hyperbolic conservation law.
References
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- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Maria Elena Schonbek, Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations 5 (1980), no. 5, 449–473. MR 571048, DOI https://doi.org/10.1080/0360530800882145
- Maria Elena Schonbek and Endre Süli, Decay of the total variation and Hardy norms of solutions to parabolic conservation laws, Nonlinear Anal. 45 (2001), no. 5, Ser. A: Theory Methods, 515–528. MR 1838944, DOI https://doi.org/10.1016/S0362-546X%2899%2900404-6
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- Denis Serre, Stabilité $L^1$ d’ondes progressives de lois de conservation scalaires, Seminaire: Équations aux Dérivées Partielles, 1998–1999, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 1999, pp. Exp. No. VIII, 13 (French, with English summary). MR 1721326
- C. M. Dafermos and E. Feireisl (eds.), Evolutionary equations. Vol. II, Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam, 2005. MR 2194153
- Philippe Souplet and Fred B. Weissler, Poincaré’s inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), no. 3, 335–371 (English, with English and French summaries). MR 1687278, DOI https://doi.org/10.1016/S0294-1449%2899%2980017-1
References
- William Beckner, Geometric proof of Nash’s inequality, Internat. Math. Res. Notices 2 (1998), 67–71. MR 1604804, DOI https://doi.org/10.1155/S1073792898000063
- Saïd Benachour, Matania Ben-Artzi, and Philippe Laurençot, Sharp decay estimates and vanishing viscosity for diffusive Hamilton-Jacobi equations, Adv. Differential Equations 14 (2009), no. 1-2, 1–25. MR 2478927
- Saïd Benachour, Grzegorz Karch, and Philippe Laurençot, Asymptotic profiles of solutions to convection-diffusion equations, C. R. Math. Acad. Sci. Paris 338 (2004), no. 5, 369–374 (English, with English and French summaries). MR 2057166, DOI https://doi.org/10.1016/j.crma.2004.01.001
- Matania Ben-Artzi, Planar Navier-Stokes equations: vorticity approach, Handbook of mathematical fluid dynamics, Vol. II, North-Holland, Amsterdam, 2003, pp. 143–167. MR 1984152, DOI https://doi.org/10.1016/S1874-5792%2803%2980007-1
- Eric A. Carlen and Michael Loss, Sharp constant in Nash’s inequality, Internat. Math. Res. Notices 7 (1993), 213–215. MR 1230297, DOI https://doi.org/10.1155/S1073792893000224
- Eric A. Carlen and Michael Loss, Optimal smoothing and decay estimates for viscously damped conservation laws, with applications to the $2$-D Navier-Stokes equation, Duke Math. J. 81 (1995), no. 1, 135–157 (1996). A celebration of John F. Nash, Jr. MR 1381974, DOI https://doi.org/10.1215/S0012-7094-95-08110-1
- Jaywan Chung, Yong-Jung Kim, and Marshall Slemrod, An explicit solution of Burgers equation with stationary point source, J. Differential Equations 257 (2014), no. 7, 2520–2542. MR 3228975, DOI https://doi.org/10.1016/j.jde.2014.05.046
- Michael G. Crandall and Luc Tartar, Some relations between nonexpansive and order preserving mappings, Proc. Amer. Math. Soc. 78 (1980), no. 3, 385–390. MR 553381, DOI https://doi.org/10.2307/2042330
- Miguel Escobedo and Enrike Zuazua, Large time behavior for convection-diffusion equations in $\mathbf {R}^N$, J. Funct. Anal. 100 (1991), no. 1, 119–161. MR 1124296, DOI https://doi.org/10.1016/0022-1236%2891%2990105-E
- Miguel Escobedo, Juan Luis Vázquez, and Enrike Zuazua, Asymptotic behaviour and source-type solutions for a diffusion-convection equation, Arch. Rational Mech. Anal. 124 (1993), no. 1, 43–65. MR 1233647, DOI https://doi.org/10.1007/BF00392203
- Lawrence C. Evans, Partial differential equations, Graduate Studies in Mathematics, vol. 19, American Mathematical Society, Providence, RI, 1998. MR 1625845
- Eduard Feireisl and Philippe Laurençot, The $L^1$-stability of constant states of degenerate convection-diffusion equations, Asymptot. Anal. 19 (1999), no. 3-4, 267–288. MR 1696218
- Heinrich Freistühler and Denis Serre, $\mathcal {L}^1$ stability of shock waves in scalar viscous conservation laws, Comm. Pure Appl. Math. 51 (1998), no. 3, 291–301. MR 1488516, DOI https://doi.org/10.1002/%28SICI%291097-0312%28199803%2951%3A3%24%5Clangle%24291%3A%3AAID-CPA4%24%5Crangle%243.3.CO%3B2-S
- Avner Friedman, Partial differential equations, Holt, Rinehart and Winston, Inc., New York-Montreal, Que.-London, 1969. MR 0445088
- Edwige Godlewski and Pierre-Arnaud Raviart, Hyperbolic systems of conservation laws, Mathématiques & Applications (Paris) [Mathematics and Applications], vol. 3/4, Ellipses, Paris, 1991. MR 1304494
- Grzegorz Karch and Maria Elena Schonbek, On zero mass solutions of viscous conservation laws, Comm. Partial Differential Equations 27 (2002), no. 9-10, 2071–2100. MR 1941667, DOI https://doi.org/10.1081/PDE-120016137
- Yong Jung Kim, An Oleinik-type estimate for a convection-diffusion equation and convergence to $N$-waves, J. Differential Equations 199 (2004), no. 2, 269–289. MR 2047911, DOI https://doi.org/10.1016/j.jde.2003.10.014
- Yong-Jung Kim and Young-Ran Lee, Dynamics in the fundamental solution of a non-convex conservation law, Proc. Roy. Soc. Edinburgh Sect. A 146 (2016), no. 1, 169–193. MR 3457635, DOI https://doi.org/10.1017/S0308210515000359
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- Gary M. Lieberman, Second order parabolic differential equations, World Scientific Publishing Co., Inc., River Edge, NJ, 1996. MR 1465184
- Tai-Ping Liu and Michel Pierre, Source-solutions and asymptotic behavior in conservation laws, J. Differential Equations 51 (1984), no. 3, 419–441. MR 735207, DOI https://doi.org/10.1016/0022-0396%2884%2990096-2
- J. Nash, Continuity of solutions of parabolic and elliptic equations, Amer. J. Math. 80 (1958), 931–954. MR 100158, DOI https://doi.org/10.2307/2372841
- I. P. Natanson, Theory of functions of a real variable. Vol. II, Translated from the Russian by Leo F. Boron, Frederick Ungar Publishing Co., New York, 1961. MR 0148805
- Murray H. Protter and Hans F. Weinberger, Maximum principles in differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1967. MR 0219861
- Maria Elena Schonbek, Decay of solutions to parabolic conservation laws, Comm. Partial Differential Equations 5 (1980), no. 5, 449–473. MR 571048, DOI https://doi.org/10.1080/0360530800882145
- Maria Elena Schonbek and Endre Süli, Decay of the total variation and Hardy norms of solutions to parabolic conservation laws, Nonlinear Anal. 45 (2001), no. 5, Ser. A: Theory Methods, 515–528. MR 1838944, DOI https://doi.org/10.1016/S0362-546X%2899%2900404-6
- D. Serre, $L^1$-decay and the stability of shock profiles, Partial differential equations (Praha, 1998) Chapman & Hall/CRC Res. Notes Math., vol. 406, Chapman & Hall/CRC, Boca Raton, FL, 2000, pp. 312–321. MR 1713897
- Denis Serre, Stabilité $L^1$ d’ondes progressives de lois de conservation scalaires, Seminaire: Équations aux Dérivées Partielles, 1998–1999, Sémin. Équ. Dériv. Partielles, École Polytech., Palaiseau, 1999, pp. Exp. No. VIII, 13 (French, with English summary). MR 1721326
- C. M. Dafermos and E. Feireisl (eds.), Evolutionary equations. Vol. II, Handbook of Differential Equations, Elsevier/North-Holland, Amsterdam, 2005. MR 2194153
- Philippe Souplet and Fred B. Weissler, Poincaré’s inequality and global solutions of a nonlinear parabolic equation, Ann. Inst. H. Poincaré Anal. Non Linéaire 16 (1999), no. 3, 335–371 (English, with English and French summaries). MR 1687278, DOI https://doi.org/10.1016/S0294-1449%2899%2980017-1
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Additional Information
Miriam Bank
Affiliation:
Azrieli College of Engineering, Jerusalem 91035, Israel
MR Author ID:
899939
Email:
miriam.bank@mail.huji.ac.il
Matania Ben-Artzi
Affiliation:
Institute of Mathematics, Hebrew University, Jerusalem 91904, Israel
MR Author ID:
34290
ORCID:
0000-0002-6782-4085
Email:
mbartzi@math.huji.ac.il
Maria E. Schonbek
Affiliation:
Department of Mathematics, University of California Santa Cruz, Santa Cruz, California 95064
MR Author ID:
156790
ORCID:
0000-0002-9917-8495
Email:
schonbek@math.ucsc.edu
Keywords:
Scalar conservation law,
viscosity,
measure initial data,
p-condition,
sup-norm estimates,
decay estimates
Received by editor(s):
December 19, 2019
Received by editor(s) in revised form:
March 30, 2020
Published electronically:
May 12, 2020
Article copyright:
© Copyright 2020
Brown University