Improved interpolation inequalities, relative entropy and fast diffusion equations
Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 5, pp. 917-934.

We consider a family of Gagliardo–Nirenberg–Sobolev interpolation inequalities which interpolate between Sobolevʼs inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy–entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.

DOI : 10.1016/j.anihpc.2012.12.004
Classification : 26D10, 46E35, 35K55
Mots-clés : Gagliardo–Nirenberg–Sobolev inequalities, Improved inequalities, Manifold of optimal functions, Entropy–entropy production method, Fast diffusion equation, Barenblatt solutions, Second moment, Intermediate asymptotics, Sharp rates, Optimal constants 
@article{AIHPC_2013__30_5_917_0,
     author = {Dolbeault, Jean and Toscani, Giuseppe},
     title = {Improved interpolation inequalities, relative entropy and fast diffusion equations},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {917--934},
     publisher = {Elsevier},
     volume = {30},
     number = {5},
     year = {2013},
     doi = {10.1016/j.anihpc.2012.12.004},
     zbl = {06295446},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.anihpc.2012.12.004/}
}
TY  - JOUR
AU  - Dolbeault, Jean
AU  - Toscani, Giuseppe
TI  - Improved interpolation inequalities, relative entropy and fast diffusion equations
JO  - Annales de l'I.H.P. Analyse non linéaire
PY  - 2013
SP  - 917
EP  - 934
VL  - 30
IS  - 5
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.anihpc.2012.12.004/
DO  - 10.1016/j.anihpc.2012.12.004
LA  - en
ID  - AIHPC_2013__30_5_917_0
ER  - 
%0 Journal Article
%A Dolbeault, Jean
%A Toscani, Giuseppe
%T Improved interpolation inequalities, relative entropy and fast diffusion equations
%J Annales de l'I.H.P. Analyse non linéaire
%D 2013
%P 917-934
%V 30
%N 5
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.anihpc.2012.12.004/
%R 10.1016/j.anihpc.2012.12.004
%G en
%F AIHPC_2013__30_5_917_0
Dolbeault, Jean; Toscani, Giuseppe. Improved interpolation inequalities, relative entropy and fast diffusion equations. Annales de l'I.H.P. Analyse non linéaire, Tome 30 (2013) no. 5, pp. 917-934. doi : 10.1016/j.anihpc.2012.12.004. http://www.numdam.org/articles/10.1016/j.anihpc.2012.12.004/

[1] M. Abramowitz, I.A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, National Bureau of Standards Appl. Math. Ser. vol. 55 (1964) | Zbl

[2] A. Arnold, P. Markowich, G. Toscani, A. Unterreiter, On convex Sobolev inequalities and the rate of convergence to equilibrium for Fokker–Planck type equations, Comm. Partial Differential Equations 26 (2001), 43-100 | Zbl

[3] T. Aubin, Problèmes isopérimétriques et espaces de Sobolev, J. Differential Geom. 11 (1976), 573-598 | Zbl

[4] D. Bakry, M. Émery, Hypercontractivité de semi-groupes de diffusion, C. R. Math. Acad. Sci. Paris 299 (1984), 775-778 | Zbl

[5] G. Bianchi, H. Egnell, A note on the Sobolev inequality, J. Funct. Anal. 100 (1991), 18-24 | Zbl

[6] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, J.-L. Vázquez, Hardy–Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. 344 (2007), 431-436 | Zbl

[7] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo, J.-L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Arch. Ration. Mech. Anal. 191 (2009), 347-385 | Zbl

[8] M. Bonforte, J. Dolbeault, G. Grillo, J.L. Vázquez, Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities, Proc. Natl. Acad. Sci. 107 (2010), 16459-16464 | Zbl

[9] H. Brezis, E.H. Lieb, Sobolev inequalities with remainder terms, J. Funct. Anal. 62 (1985), 73-86 | Zbl

[10] M.J. Cáceres, J.A. Carrillo, J. Dolbeault, Nonlinear stability in L p for a confined system of charged particles, SIAM J. Math. Anal. 34 (2002), 478-494 | Zbl

[11] J.A. Carrillo, A. Jüngel, P.A. Markowich, G. Toscani, A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math. 133 (2001), 1-82 | Zbl

[12] J.A. Carrillo, G. Toscani, Asymptotic L 1 -decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J. 49 (2000), 113-142 | Zbl

[13] J.A. Carrillo, J.L. Vázquez, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations 28 (2003), 1023-1056 | Zbl

[14] A. Cianchi, Quantitative Sobolev and Hardy inequalities, and related symmetrization principles, Sobolev Spaces in Mathematics. I, Int. Math. Ser. (N. Y.) vol. 8, Springer, New York (2009), 87-116 | Zbl

[15] A. Cianchi, N. Fusco, F. Maggi, A. Pratelli, The sharp Sobolev inequality in quantitative form, J. Eur. Math. Soc. (JEMS) 11 (2009), 1105-1139 | EuDML | Zbl

[16] M. Del Pino, J. Dolbeault, Best constants for Gagliardo–Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. 9 no. 81 (2002), 847-875 | Zbl

[17] J. Dolbeault, Sobolev and Hardy–Littlewood–Sobolev inequalities: duality and fast diffusion, Math. Res. Lett. 18 (2011), 1037-1050 | Zbl

[18] J. Dolbeault, G. Karch, Large time behaviour of solutions to nonhomogeneous diffusion equations, Banach Center Publ. 74 (2006), 133-147 | EuDML | Zbl

[19] J. Dolbeault, G. Toscani, Fast diffusion equations: Matching large time asymptotics by relative entropy methods, Kinet. Relat. Models 4 (2011), 701-716 | Zbl

[20] L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061-1083 | Zbl

[21] W.I. Newman, A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. I, J. Math. Phys. 25 (1984), 3120-3123 | Zbl

[22] J. Ralston, A Lyapunov functional for the evolution of solutions to the porous medium equation to self-similarity. II, J. Math. Phys. 25 (1984), 3124-3127 | Zbl

[23] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. (4) 110 (1976), 353-372 | Zbl

[24] A. Unterreiter, A. Arnold, P. Markowich, G. Toscani, On generalized Csiszár–Kullback inequalities, Monatsh. Math. 131 (2000), 235-253 | Zbl

[25] F.B. Weissler, Logarithmic Sobolev inequalities for the heat-diffusion semigroup, Trans. Amer. Math. Soc. 237 (1978), 255-269 | Zbl

Cité par Sources :