Article Contents
Article Contents

# Weighted fast diffusion equations (Part Ⅰ): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities

• * Corresponding author.
• In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities (CKN), with two radial power law weights and exponents in a subcritical range. We address the question of symmetry breaking: are the optimal functions radially symmetric, or not? Our intuition comes from a weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy -entropy production inequality which governs the intermediate asymptotics is indeed equivalent to (CKN), and the self-similar profiles are optimal for (CKN).

We establish an explicit symmetry breaking condition by proving the linear instability of the radial optimal functions for (CKN). Symmetry breaking in (CKN) also has consequences on entropy -entropy production inequalities and on the intermediate asymptotics for (WFD). Even when no symmetry holds in (CKN), asymptotic rates of convergence of the solutions to (WFD) are determined by a weighted Hardy-Poincaré inequality which is interpretedas a linearized entropy -entropy production inequality. All our results rely on the study of the bottom of the spectrum of the linearized diffusion operator around the self-similar profiles, which is equivalent to the linearization of (CKN) around the radial optimal functions, and on variational methods.Consequences for the (WFD) flow will be studied in Part Ⅱ of this work.

Mathematics Subject Classification: Primary: 35K55, 46E35, 49K30; Secondary: 26D10, 35B06, 49K20, 35J20.

 Citation:

• Figure 1.  We consider the admissible range for the $(\beta,\gamma)$ parameters. The grey area is the area of validity of (2) and it is given by $\gamma-2 < \beta < \tfrac{d-2}d\,\gamma$ if $\gamma < d$ and $\tfrac{d-2}d\,\gamma < \beta < \gamma-2$ if $\gamma>d$: the cones corresponding to $\gamma < d$ and $\gamma>d$ are in one-to-one correspondance by an inversion symmetry: see details in Section 2.1 Notice that the case $\gamma>d$ has been excluded in (3) in order to simplify the statements. The hyperbola defined by the Felli & Schneider curve determines a region (dark grey area) of symmetry breaking which is valid for any $p\in(1,p_\star)$, and independent of $p$. However, since $p_\star$ depends on $\beta$ and $\gamma$, this induces an additional restriction on the admissible range of $(\beta,\gamma)$, which depends on $p$: see Fig.2 and Fig.3. Here we consider the special case $d=5$.

Figure 2.  With $d=3$, the left figure is essentially an enlargement of Fig.1 and represents the symmetry breaking region, while on the right figure, we choose $p=2$, so that the admissible range of parameters $(\beta,\gamma)$ is restricted by the condition $p\le p_\star(\beta,\gamma)$, i.e., $\beta\ge d-2-(d-\gamma)/p$. This lower bound corresponds to the line determined by the points $(\beta,\gamma)=(d-2,d)$ and $(\beta,\gamma)$ given by the condition $\Lambda_\star=\Lambda_{0,1}=\Lambda_{1,0}$. The curve $\beta=\sigma(\gamma,p)$ in Theorem 3 is represented by a dotted curve. To $\beta\ge\sigma(\gamma,p)$ corresponds the case $\Lambda=\Lambda_{0,1}\le\Lambda_{1,0}$, while $\beta\le\sigma(\gamma,p)$ corresponds to the case $\Lambda_{0,1}\ge\Lambda_{1,0}=\Lambda$, when $\gamma\in(-\infty,d)$.

Figure 3.  Enlargement of Fig.2 in a neighborhood of $(\beta,\gamma)=(0,0)$. On the right, the equality case $\Lambda_{0,1}=\Lambda_{1,0}$ determines the dotted curve $\beta=\sigma(\gamma,p)$. Notice that the symmetry breaking region is contained in the region in which the spectral gap is $\Lambda=\Lambda_{0,1}$.

Figure 4.  In the dark grey region, symmetry breaking occurs. The plot is done for $p=2$ and $d=3$. See Appendix B for a more detailed description of the properties of the lowest eigenvalues.

Figure 5.  The spectral gap and the lowest eigenvalues of $\mathcal L$ for $n=3$. The parabola represents $\Lambda_{\rm ess}$ as a function of $\delta$, $\Lambda_{1,0}$ is tangent to the parabola and $\Lambda_{0,1}$ is shown for $\eta=3.5$ (left), $\eta=1.4$ (center) and $\eta=0.35$ (right). The eigenvalues $\Lambda_{1,0}$ and $\Lambda_{0,1}$ are represented by a plain line only if the corresponding eigenvalues are in the space $\mathrm L^2(\mathbb{R}^d,\mathcal B^{2-m}\,|x|^{n-d}\,dx)$.

•  [1] S. Angenent, Large time asymptotics for the porous media equation, in Nonlinear diffusion equations and their equilibrium states, I (Berkeley, CA, 1986), vol. 12 of Math. Sci. Res. Inst. Publ., Springer, New York, 1988, 21–34, URL http://dx.doi.org/10.1007/978-1-4613-9605-5_2. doi: 10.1007/978-1-4613-9605-5_2. [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, Monatsh. Math., 142 (2004), 35–43, URL http://dx.doi.org/10.1007/978-3-7091-0609-9_5. doi: 10.1007/s00605-004-0239-2. [3] A. Arnold, P. Markowich, G. Toscani and 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, URL https://dx.doi.org/10.1081/PDE-100002246. doi: 10.1081/PDE-100002246. [4] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. -L. Vázquez, Hardy-Poincaré inequalities and applications to nonlinear diffusions, C. R. Math. Acad. Sci. Paris, 344 (2007), 431–436, URL http://dx.doi.org/10.1016/j.crma.2007.01.011. doi: 10.1016/j.crma.2007.01.011. [5] A. Blanchet, M. Bonforte, J. Dolbeault, G. Grillo and J. L. Vázquez, Asymptotics of the fast diffusion equation via entropy estimates, Archive for Rational Mechanics and Analysis, 191 (2009), 347–385, URL http://dx.doi.org/10.1007/s00205-008-0155-z. doi: 10.1007/s00205-008-0155-z. [6] S. G. Bobkov and M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures, Ann. Probab., 37 (2009), 403–427, URL https://dx.doi.org/10.1214/08-AOP407. doi: 10.1214/08-AOP407. [7] 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, Proc. Natl. Acad. Sci. USA, 107 (2010), 16459–16464, URL http://dx.doi.org/10.1073/pnas.1003972107. doi: 10.1073/pnas.1003972107. [8] M. Bonforte, J. Dolbeault, M. Muratori and B. Nazaret, Weighted fast diffusion equations (Part Ⅱ): Sharp asymptotic rates of convergence in relative error by entropy methods, to appear in Kinet. Relat. Models, Preprint hal-01279327 & arXiv: 1602.08315, (2016). [9] M. Bonforte, G. Grillo and J. L. Vázquez, Special fast diffusion with slow asymptotics: Entropy method and flow on a Riemannian manifold, Archive for Rational Mechanics and Analysis, 196 (2010), 631–680, URL http://dx.doi.org/10.1007/s00205-009-0252-7. doi: 10.1007/s00205-009-0252-7. [10] M. Bonforte and J. L. Vázquez, Global positivity estimates and Harnack inequalities for the fast diffusion equation, J. Funct. Anal., 240 (2006), 399–428, URL http://dx.doi.org/10.1016/j.jfa.2006.07.009. doi: 10.1016/j.jfa.2006.07.009. [11] M. J. Cáceres and G. Toscani, Kinetic approach to long time behavior of linearized fast diffusion equations, J. Stat. Phys., 128 (2007), 883–925, URL http://dx.doi.org/10.1007/s10955-007-9329-6. doi: 10.1007/s10955-007-9329-6. [12] L. Caffarelli, R. Kohn and L. Nirenberg, First order interpolation inequalities with weights, Compositio Math., 53 (1984), 259–275, URL http://eudml.org/doc/89687. [13] J. A. Carrillo, A. Jüngel, P. A. Markowich, G. Toscani and A. Unterreiter, Entropy dissipation methods for degenerate parabolic problems and generalized Sobolev inequalities, Monatsh. Math., 133 (2001), 1–82, URL http://dx.doi.org/10.1007/s006050170032. doi: 10.1007/s006050170032. [14] J. A. Carrillo, C. Lederman, P. A. Markowich and G. Toscani, Poincaré inequalities for linearizations of very fast diffusion equations, Nonlinearity, 15 (2002), 565–580, URL http://dx.doi.org/10.1088/0951-7715/15/3/303. doi: 10.1088/0951-7715/15/3/303. [15] J. A. Carrillo, P. A. Markowich and A. Unterreiter, Large-time asymptotics of porous-medium type equations, in Free boundary problems: theory and applications, I (Chiba, 1999), vol. 13 of GAKUTO Internat. Ser. Math. Sci. Appl., Gakk¯otosho, Tokyo, 2000, 24–36. [16] J. A. Carrillo and G. Toscani, Asymptotic L1-decay of solutions of the porous medium equation to self-similarity, Indiana Univ. Math. J., 49 (2000), 113–142. doi: 10.1512/iumj.2000.49.1756. [17] J. A. Carrillo and G. Toscani, Contractive probability metrics and asymptotic behavior of dissipative kinetic equations, Riv. Mat. Univ. Parma (7), 6 (2007), 75–198. [18] J. A. Carrillo and J. L. Vázquez, Fine asymptotics for fast diffusion equations, Comm. Partial Differential Equations, 28 (2003), 1023–1056, URL http://dx.doi.org/10.1081/PDE-120021185. doi: 10.1081/PDE-120021185. [19] F. Catrina and Z. -Q. Wang, On the Caffarelli-Kohn-Nirenberg inequalities: Sharp constants, existence (and nonexistence), and symmetry of extremal functions, Comm. Pure Appl. Math., 54 (2001), 229–258, URL http://dx.doi.org/10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. doi: 10.1002/1097-0312(200102)54:2<229::AID-CPA4>3.0.CO;2-I. [20] M. Del Pino and J. Dolbeault, Best constants for Gagliardo-Nirenberg inequalities and applications to nonlinear diffusions, J. Math. Pures Appl. (9), 81 (2002), 847–875, URL http://dx.doi.org/10.1016/S0021-7824(02)01266-7. doi: 10.1016/S0021-7824(02)01266-7. [21] M. Del Pino, J. Dolbeault, S. Filippas and A. Tertikas, A logarithmic Hardy inequality, Journal of Functional Analysis, 259 (2010), 2045–2072, URL http://dx.doi.org/10.1016/j.jfa.2010.06.005. doi: 10.1016/j.jfa.2010.06.005. [22] J. Denzler, H. Koch and R. J. McCann, Higher-order time asymptotics of fast diffusion in Euclidean space: A dynamical systems approach, Mem. Amer. Math. Soc., 234 (2015), vi+81pp, URL https://dx.doi.org/10.1090/memo/1101. doi: 10.1090/memo/1101. [23] J. Denzler and R. J. McCann, Phase transitions and symmetry breaking in singular diffusion, Proc. Natl. Acad. Sci. USA, 100 (2003), 6922–6925 (electronic), URL http://dx.doi.org/10.1073/pnas.1231896100. doi: 10.1073/pnas.1231896100. [24] J. Denzler and R. J. McCann, Fast diffusion to self-similarity: Complete spectrum, longtime asymptotics, and numerology, Arch. Ration. Mech. Anal., 175 (2005), 301–342, URL http://dx.doi.org/10.1007/s00205-004-0336-3. doi: 10.1007/s00205-004-0336-3. [25] J. Dolbeault and M. J. Esteban, Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities, Proc. Roy. Soc. Edinburgh Sect. A, 142 (2012), 745–767, URL http://dx.doi.org/10.1017/S0308210510001101. doi: 10.1017/S0308210510001101. [26] J. Dolbeault, M. J. Esteban, S. Filippas and A. Tertikas, Rigidity results with applications to best constants and symmetry of Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities, Calc. Var. Partial Differential Equations, 54 (2015), 2465–2481, URL http://dx.doi.org/10.1007/s00526-015-0871-9. doi: 10.1007/s00526-015-0871-9. [27] J. Dolbeault, M. J. Esteban and M. Loss, Rigidity versus symmetry breaking via nonlinear flows on cylinders and Euclidean spaces, to appear in Inventiones Mathematicae, URL http://link.springer.com/article/10.1007/s00222-016-0656-6, (2016). [28] J. Dolbeault, M. J. Esteban, M. Loss and M. Muratori, Symmetry for extremal functions in subcritical Caffarelli-Kohn-Nirenberg inequalities, to appear in C. R. Mathématique, Preprint hal-01318727 & arXiv: 1605.06373, (2016). doi: 10.1016/j.crma.2017.01.004. [29] J. Dolbeault, M. J. Esteban, G. Tarantello and A. Tertikas, Radial symmetry and symmetry breaking for some interpolation inequalities, Calc. Var. Partial Differential Equations, 42 (2011), 461–485, URL https://dx.doi.org/10.1007/s00526-011-0394-y. doi: 10.1007/s00526-011-0394-y. [30] J. Dolbeault, M. Muratori and B. Nazaret, Weighted interpolation inequalities: a perturbation approach, to appear in Mathematische Annalen, Preprint hal-01207009 & arXiv: 1509.09127, (2016). [31] J. Dolbeault and G. Toscani, Fast diffusion equations: matching large time asymptotics by relative entropy methods, Kinetic and Related Models, 4 (2011), 701–716, URL http://dx.doi.org/10.3934/krm.2011.4.701. doi: 10.3934/krm.2011.4.701. [32] J. Dolbeault and G. Toscani, Improved interpolation inequalities, relative entropy and fast diffusion equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 30 (2013), 917–934, URL http://dx.doi.org/10.1016/j.anihpc.2012.12.004. doi: 10.1016/j.anihpc.2012.12.004. [33] J. Dolbeault and G. Toscani, Best matching Barenblatt profiles are delayed, Journal of Physics A: Mathematical and Theoretical, 48 (2015), 065206, 14pp, URL http://dx.doi.org/10.1088/1751-8113/48/6/065206. doi: 10.1088/1751-8113/48/6/065206. [34] J. Dolbeault and G. Toscani, Nonlinear diffusions: Extremal properties of Barenblatt profiles, best matching and delays, Nonlinear Analysis: Theory, Methods & Applications, URL www.sciencedirect.com/science/article/pii/S0362546X15003880. [35] J. Dolbeault and G. Toscani, Stability results for logarithmic Sobolev and Gagliardo– Nirenberg inequalities, International Mathematics Research Notices, URL http://imrn.oxfordjournals.org/content/early/2015/05/15/imrn.rnv131.abstract. [36] V. Felli and M. Schneider, Perturbation results of critical elliptic equations of Caffarelli-KohnNirenberg type, J. Differential Equations, 191 (2003), 121–142, URL http://dx.doi.org/10.1016/S0022-0396(02)00085-2. doi: 10.1016/S0022-0396(02)00085-2. [37] A. Friedman and S. Kamin, The asymptotic behavior of gas in an n-dimensional porous medium, Trans. Amer. Math. Soc., 262 (1980), 551–563, URL http://dx.doi.org/10.2307/1999846. doi: 10.2307/1999846. [38] Y. J. Kim and R. J. McCann, Potential theory and optimal convergence rates in fast nonlinear diffusion, J. Math. Pures Appl. (9), 86 (2006), 42–67, URL http://dx.doi.org/10.1016/j.matpur.2006.01.002. doi: 10.1016/j.matpur.2006.01.002. [39] C. Lederman and P. A. Markowich, On fast-diffusion equations with infinite equilibrium entropy and finite equilibrium mass, Comm. Partial Differential Equations, 28 (2003), 301– 332, URL http://dx.doi.org/10.1081/PDE-120019384. doi: 10.1081/PDE-120019384. [40] A. Persson, Bounds for the discrete part of the spectrum of a semi-bounded Schrödinger operator, Math. Scand., 8 (1960), 143–153, URL http://gdz.sub.uni-goettingen.de/dms/load/?PID=GDZPPN002346214. doi: 10.7146/math.scand.a-10602. [41] T. P. Witelski and A. J. Bernoff, Self-similar asymptotics for linear and nonlinear diffusion equations, Stud. Appl. Math., 100 (1998), 153–193, URL http://dx.doi.org/10.1111/1467-9590.00074. doi: 10.1111/1467-9590.00074. [42] Y. B. Zel'dovič and G. I. Barenblatt, Asymptotic properties of self-preserving solutions of equations of unsteady motion of gas through porous media, Dokl. Akad. Nauk SSSR (N. S. ), 118 (1958), 671–674.

Figures(5)