March  2018, 38(3): 1405-1425. doi: 10.3934/dcds.2018057

Improved energy methods for nonlocal diffusion problems

Departamento de Matemática Aplicada, Universidad de Granada, 18071 Granada, Spain

Departamento de Análisis Matemático, Universidad de Granada, 18071 Granada, Spain

Received  May 2017 Revised  September 2017 Published  December 2017

We prove an energy inequality for nonlocal diffusion operators of the following type, and some of its generalisations:
\begin{equation*} Lu (x) := \int_{\mathbb{R}^N} K(x, y) (u(y) -u(x)) \,\mathrm{d} y, \end{equation*}
where
$L$
acts on a real function
$u$
defined on
$\mathbb{R}^N$
, and we assume that
$K(x, y)$
is uniformly strictly positive in a neighbourhood of
$x=y$
. The inequality is a nonlocal analogue of the Nash inequality, and plays a similar role in the study of the asymptotic decay of solutions to the nonlocal diffusion equation
$\partial_t u = L u$
as the Nash inequality does for the heat equation. The inequality allows us to give a precise decay rate of the
$L^p$
norms of
$u$
and its derivatives. As compared to existing decay results in the literature, our proof is perhaps simpler and gives new results in some cases.
Citation: José A. Cañizo, Alexis Molino. Improved energy methods for nonlocal diffusion problems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (3) : 1405-1425. doi: 10.3934/dcds.2018057
References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems American Mathematical Society; Real Sociedad Matemática Española, 2010, http://www.worldcat.org/isbn/9780821852309.  Google Scholar

[2]

A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of Many-Particle systems: An essay on recent research, Monatshefte für Mathematik, 142 (2004), 35–43, URL http://dx.doi.org/10.1007/s00605-004-0239-2. doi: 10.1007/s00605-004-0239-2.  Google Scholar

[3]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités XIX 1983/84 (eds. J. Azéma and M. Yor), vol. 1123 of Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 1985, chapter 13, 177-206, URL http://dx.doi.org/10.1007/bfb0075847. Google Scholar

[4]

M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proceedings of the National Academy of Sciences, 107 (2010), 16459–16464, URL http://dx.doi.org/10.1073/pnas.1003972107. doi: 10.1073/pnas.1003972107.  Google Scholar

[5]

C. Brändle and A. de Pablo, Nonlocal heat equations: decay estimates and Nash inequalities, 2015, http://arxiv.org/abs/1312.4661. Google Scholar

[6]

E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Annales de l'Institute Henri Poincaré. Probabilités et statistiques, 23 (1987), 245–287, URL http://eudml.org/doc/77309.  Google Scholar

[7]

J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte für Mathematik, 133 (2001), 1–82, URL http://dx.doi.org/10.1007/s006050170032. doi: 10.1007/s006050170032.  Google Scholar

[8]

D. Chafaï, Entropies, convexity, and functional inequalities, Journal of Mathematics of Kyoto University, 44 (2004), 325–363, URL http://projecteuclid.org/euclid.kjm/1250283556. doi: 10.1215/kjm/1250283556.  Google Scholar

[9]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86 (2006), 271–291, URL http://dx.doi.org/10.1016/j.matpur.2006.04.005. doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[10]

C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process, Journal of Differential Equations, 241 (2007), 332–358, URL http://www.sciencedirect.com/science/article/pii/S0022039607002082. doi: 10.1016/j.jde.2007.06.002.  Google Scholar

[11]

C. CortázarM. ElguetaJ. García-Melián and S. Martínez, Stationary sign changing solutions for an inhomogeneous nonlocal problem, Indiana Univ. Math. J., 60 (2011), 209-232.  doi: 10.1512/iumj.2011.60.4385.  Google Scholar

[12]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, Finite mass solutions for a nonlocal inhomogeneous dispersal equation, Discrete and Continuous Dynamical Systems, 35 (2015), 1409–1419, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10560.  Google Scholar

[13]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, An inhomogeneous nonlocal diffusion problem with unbounded steps, Journal of Evolution Equations, 16 (2016), 209–232, URL http://dx.doi.org/10.1007/s00028-015-0299-x. doi: 10.1007/s00028-015-0299-x.  Google Scholar

[14]

C. Cortázar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics, 170 (2009), 53–60, URL http://dx.doi.org/10.1007/s11856-009-0019-8. doi: 10.1007/s11856-009-0019-8.  Google Scholar

[15]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Archive for Rational Mechanics and Analysis, 187 (2008), 137–156, URL http://dx.doi.org/10.1007/s00205-007-0062-8.  Google Scholar

[16]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Inventiones mathematicae, 159 (2004), 245–316, URL http://dx.doi.org/10.1007/s00222-004-0389-9. doi: 10.1007/s00222-004-0389-9.  Google Scholar

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence Wiley Series in Probability and Statistics, Wiley, 1986, URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0471081868.  Google Scholar

[18]

M. -H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations vol. 79 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. , Boston, MA, 2010, URL http://dx.doi.org/10.1007/978-0-8176-4651-6, Asymptotic behavior of solutions and self-similar solutions.  Google Scholar

[19]

L. Gross, Logarithmic sobolev inequalities, American Journal of Mathematics, 97 (1975), 1061–1083, URL http://www.jstor.org/stable/2373688. doi: 10.2307/2373688.  Google Scholar

[20]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, Journal of Functional Analysis, 251 (2007), 399–437, URL http://dx.doi.org/10.1016/j.jfa.2007.07.013. doi: 10.1016/j.jfa.2007.07.013.  Google Scholar

[21]

L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations, Journal of Evolution Equations, 8 (2008), 617–629, URL http://dx.doi.org/10.1007/s00028-008-0372-9. doi: 10.1007/s00028-008-0372-9.  Google Scholar

[22]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, Journal de Mathématiques Pures et Appliquées, 92 (2009), 163–187, URL http://dx.doi.org/10.1016/j.matpur.2009.04.009. doi: 10.1016/j.matpur.2009.04.009.  Google Scholar

[23]

P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, Comptes Rendus Mathematique, 338 (2004), 697–702, URL http://dx.doi.org/10.1016/j.crma.2004.03.006. doi: 10.1016/j.crma.2004.03.006.  Google Scholar

[24]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1235–1260, URL http://www.sciencedirect.com/science/article/pii/S0021782405000528. doi: 10.1016/j.matpur.2005.04.001.  Google Scholar

[25]

S. Mischler and I. Tristani, Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations, J. Éc. Polytech. Math., 4 (2017), 389–433, URL http://arxiv.org/abs/1507.04861. doi: 10.5802/jep.46.  Google Scholar

[26]

A. Molino and J. D. Rossi, Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence, Zeitschrift für angewandte Mathematik und Physik 67 (2016), Art. 41, 14 pp, URL http://dx.doi.org/10.1007/s00033-016-0649-8.  Google Scholar

[27]

J. Nash, Continuity of solutions of parabolic and elliptic equations, URL http://dx.doi.org/10.2307/2372841. Google Scholar

[28]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361–400, URL http://dx.doi.org/10.1006/jfan.1999.3557. doi: 10.1006/jfan.1999.3557.  Google Scholar

[29]

T. Rey and G. Toscani, Large-time behavior of the solutions to Rosenau-type approximations to the heat equation, SIAM Journal on Applied Mathematics, 73 (2013), 1416–1438, URL http://dx.doi.org/10.1137/120876290. doi: 10.1137/120876290.  Google Scholar

[30]

M. E. Schonbek, Decay of solution to parabolic conservation laws, Communications in Partial Differential Equations, 5 (1980), 449–473, URL http://dx.doi.org/10.1080/0360530800882145. doi: 10.1080/0360530800882145.  Google Scholar

[31]

J.-W. SunW.-T. Li and F.-Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3501-3509.  doi: 10.1016/j.na.2011.02.034.  Google Scholar

[32]

G. Toscani, A Rosenau-type approach to the approximation of the linear Fokker-Planck equation, 2017, http://arxiv.org/abs/1703.10909. Google Scholar

[33]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), Elsevier, Amsterdam, Netherlands; Boston, U.S.A., 1 (2002), 71–305, URL http://www.umpa.ens-lyon.fr/~{}cvillani/GZPDF/B01.Handbook.pdf.gz.  Google Scholar

show all references

References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. J. Toledo-Melero, Nonlocal Diffusion Problems American Mathematical Society; Real Sociedad Matemática Española, 2010, http://www.worldcat.org/isbn/9780821852309.  Google Scholar

[2]

A. Arnold, J. A. Carrillo, L. Desvillettes, J. Dolbeault, A. Jüngel, C. Lederman, P. A. Markowich, G. Toscani and C. Villani, Entropies and equilibria of Many-Particle systems: An essay on recent research, Monatshefte für Mathematik, 142 (2004), 35–43, URL http://dx.doi.org/10.1007/s00605-004-0239-2. doi: 10.1007/s00605-004-0239-2.  Google Scholar

[3]

D. Bakry and M. Émery, Diffusions hypercontractives, in Séminaire de Probabilités XIX 1983/84 (eds. J. Azéma and M. Yor), vol. 1123 of Lecture Notes in Mathematics, Springer Berlin / Heidelberg, 1985, chapter 13, 177-206, URL http://dx.doi.org/10.1007/bfb0075847. Google Scholar

[4]

M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proceedings of the National Academy of Sciences, 107 (2010), 16459–16464, URL http://dx.doi.org/10.1073/pnas.1003972107. doi: 10.1073/pnas.1003972107.  Google Scholar

[5]

C. Brändle and A. de Pablo, Nonlocal heat equations: decay estimates and Nash inequalities, 2015, http://arxiv.org/abs/1312.4661. Google Scholar

[6]

E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Annales de l'Institute Henri Poincaré. Probabilités et statistiques, 23 (1987), 245–287, URL http://eudml.org/doc/77309.  Google Scholar

[7]

J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatshefte für Mathematik, 133 (2001), 1–82, URL http://dx.doi.org/10.1007/s006050170032. doi: 10.1007/s006050170032.  Google Scholar

[8]

D. Chafaï, Entropies, convexity, and functional inequalities, Journal of Mathematics of Kyoto University, 44 (2004), 325–363, URL http://projecteuclid.org/euclid.kjm/1250283556. doi: 10.1215/kjm/1250283556.  Google Scholar

[9]

E. Chasseigne, M. Chaves and J. D. Rossi, Asymptotic behavior for nonlocal diffusion equations, Journal de Mathématiques Pures et Appliquées, 86 (2006), 271–291, URL http://dx.doi.org/10.1016/j.matpur.2006.04.005. doi: 10.1016/j.matpur.2006.04.005.  Google Scholar

[10]

C. Cortázar, J. Coville, M. Elgueta and S. Martínez, A nonlocal inhomogeneous dispersal process, Journal of Differential Equations, 241 (2007), 332–358, URL http://www.sciencedirect.com/science/article/pii/S0022039607002082. doi: 10.1016/j.jde.2007.06.002.  Google Scholar

[11]

C. CortázarM. ElguetaJ. García-Melián and S. Martínez, Stationary sign changing solutions for an inhomogeneous nonlocal problem, Indiana Univ. Math. J., 60 (2011), 209-232.  doi: 10.1512/iumj.2011.60.4385.  Google Scholar

[12]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, Finite mass solutions for a nonlocal inhomogeneous dispersal equation, Discrete and Continuous Dynamical Systems, 35 (2015), 1409–1419, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=10560.  Google Scholar

[13]

C. Cortázar, M. Elgueta, J. García-Melián and S. Martínez, An inhomogeneous nonlocal diffusion problem with unbounded steps, Journal of Evolution Equations, 16 (2016), 209–232, URL http://dx.doi.org/10.1007/s00028-015-0299-x. doi: 10.1007/s00028-015-0299-x.  Google Scholar

[14]

C. Cortázar, M. Elgueta and J. D. Rossi, Nonlocal diffusion problems that approximate the heat equation with Dirichlet boundary conditions, Israel Journal of Mathematics, 170 (2009), 53–60, URL http://dx.doi.org/10.1007/s11856-009-0019-8. doi: 10.1007/s11856-009-0019-8.  Google Scholar

[15]

C. Cortázar, M. Elgueta, J. D. Rossi and N. Wolanski, How to approximate the heat equation with Neumann boundary conditions by nonlocal diffusion problems, Archive for Rational Mechanics and Analysis, 187 (2008), 137–156, URL http://dx.doi.org/10.1007/s00205-007-0062-8.  Google Scholar

[16]

L. Desvillettes and C. Villani, On the trend to global equilibrium for spatially inhomogeneous kinetic systems: the Boltzmann equation, Inventiones mathematicae, 159 (2004), 245–316, URL http://dx.doi.org/10.1007/s00222-004-0389-9. doi: 10.1007/s00222-004-0389-9.  Google Scholar

[17]

S. N. Ethier and T. G. Kurtz, Markov Processes: Characterization and Convergence Wiley Series in Probability and Statistics, Wiley, 1986, URL http://www.amazon.com/exec/obidos/redirect?tag=citeulike07-20&path=ASIN/0471081868.  Google Scholar

[18]

M. -H. Giga, Y. Giga and J. Saal, Nonlinear Partial Differential Equations vol. 79 of Progress in Nonlinear Differential Equations and their Applications, Birkhäuser Boston, Inc. , Boston, MA, 2010, URL http://dx.doi.org/10.1007/978-0-8176-4651-6, Asymptotic behavior of solutions and self-similar solutions.  Google Scholar

[19]

L. Gross, Logarithmic sobolev inequalities, American Journal of Mathematics, 97 (1975), 1061–1083, URL http://www.jstor.org/stable/2373688. doi: 10.2307/2373688.  Google Scholar

[20]

L. I. Ignat and J. D. Rossi, A nonlocal convection-diffusion equation, Journal of Functional Analysis, 251 (2007), 399–437, URL http://dx.doi.org/10.1016/j.jfa.2007.07.013. doi: 10.1016/j.jfa.2007.07.013.  Google Scholar

[21]

L. I. Ignat and J. D. Rossi, Refined asymptotic expansions for nonlocal diffusion equations, Journal of Evolution Equations, 8 (2008), 617–629, URL http://dx.doi.org/10.1007/s00028-008-0372-9. doi: 10.1007/s00028-008-0372-9.  Google Scholar

[22]

L. I. Ignat and J. D. Rossi, Decay estimates for nonlocal problems via energy methods, Journal de Mathématiques Pures et Appliquées, 92 (2009), 163–187, URL http://dx.doi.org/10.1016/j.matpur.2009.04.009. doi: 10.1016/j.matpur.2009.04.009.  Google Scholar

[23]

P. Michel, S. Mischler and B. Perthame, General entropy equations for structured population models and scattering, Comptes Rendus Mathematique, 338 (2004), 697–702, URL http://dx.doi.org/10.1016/j.crma.2004.03.006. doi: 10.1016/j.crma.2004.03.006.  Google Scholar

[24]

P. Michel, S. Mischler and B. Perthame, General relative entropy inequality: An illustration on growth models, Journal de Mathématiques Pures et Appliquées, 84 (2005), 1235–1260, URL http://www.sciencedirect.com/science/article/pii/S0021782405000528. doi: 10.1016/j.matpur.2005.04.001.  Google Scholar

[25]

S. Mischler and I. Tristani, Uniform semigroup spectral analysis of the discrete, fractional and classical Fokker-Planck equations, J. Éc. Polytech. Math., 4 (2017), 389–433, URL http://arxiv.org/abs/1507.04861. doi: 10.5802/jep.46.  Google Scholar

[26]

A. Molino and J. D. Rossi, Nonlocal diffusion problems that approximate a parabolic equation with spatial dependence, Zeitschrift für angewandte Mathematik und Physik 67 (2016), Art. 41, 14 pp, URL http://dx.doi.org/10.1007/s00033-016-0649-8.  Google Scholar

[27]

J. Nash, Continuity of solutions of parabolic and elliptic equations, URL http://dx.doi.org/10.2307/2372841. Google Scholar

[28]

F. Otto and C. Villani, Generalization of an inequality by Talagrand and links with the logarithmic Sobolev inequality, J. Funct. Anal., 173 (2000), 361–400, URL http://dx.doi.org/10.1006/jfan.1999.3557. doi: 10.1006/jfan.1999.3557.  Google Scholar

[29]

T. Rey and G. Toscani, Large-time behavior of the solutions to Rosenau-type approximations to the heat equation, SIAM Journal on Applied Mathematics, 73 (2013), 1416–1438, URL http://dx.doi.org/10.1137/120876290. doi: 10.1137/120876290.  Google Scholar

[30]

M. E. Schonbek, Decay of solution to parabolic conservation laws, Communications in Partial Differential Equations, 5 (1980), 449–473, URL http://dx.doi.org/10.1080/0360530800882145. doi: 10.1080/0360530800882145.  Google Scholar

[31]

J.-W. SunW.-T. Li and F.-Y. Yang, Approximate the Fokker-Planck equation by a class of nonlocal dispersal problems, Nonlinear Analysis: Theory, Methods & Applications, 74 (2011), 3501-3509.  doi: 10.1016/j.na.2011.02.034.  Google Scholar

[32]

G. Toscani, A Rosenau-type approach to the approximation of the linear Fokker-Planck equation, 2017, http://arxiv.org/abs/1703.10909. Google Scholar

[33]

C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, (eds. S. Friedlander and D. Serre), Elsevier, Amsterdam, Netherlands; Boston, U.S.A., 1 (2002), 71–305, URL http://www.umpa.ens-lyon.fr/~{}cvillani/GZPDF/B01.Handbook.pdf.gz.  Google Scholar

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