# American Institute of Mathematical Sciences

September  2019, 39(9): 4929-4943. doi: 10.3934/dcds.2019201

## Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem

 ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland

Received  February 2016 Published  May 2019

In this paper we study the asymptotic behavior of a very fast diffusion PDE in 1D with periodic boundary conditions. This equation is motivated by the gradient flow approach to the problem of quantization of measures introduced in [3]. We prove exponential convergence to equilibrium under minimal assumptions on the data, and we also provide sufficient conditions for $W_2$-stability of solutions.

Citation: Mikaela Iacobelli. Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem. Discrete & Continuous Dynamical Systems - A, 2019, 39 (9) : 4929-4943. doi: 10.3934/dcds.2019201
##### References:
 [1] L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in Calculus of variations and nonlinear partial differential equations, 1–41, Lecture Notes in Math., 1927, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_1.  Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.  Google Scholar [3] E. Caglioti, F. Golse and M. Iacobelli, A gradient flow approach to quantization of measures, Math. Models Methods Appl. Sci., 25 (2015), 1845-1885.  doi: 10.1142/S0218202515500475.  Google Scholar [4] E. Caglioti, F. Golse and M. Iacobelli, Quantization of measures and gradient flows: A perturbative approach in the $2$ dimensional case, Ann. Inst. N. Poincaré Anal. Non Linéaire, 35 (2018), 1531-1555.  doi: 10.1016/j.anihpc.2017.12.003.  Google Scholar [5] J. A. Carrillo and D. Slepcev, Example of a displacement convex functional of first order, Calc. Var. Partial Differential Equations, 36 (2009), 547-564.  doi: 10.1007/s00526-009-0243-4.  Google Scholar [6] J. R. Esteban, A. Rodríguez and J. L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations, 13 (1988), 985-1039.  doi: 10.1080/03605308808820566.  Google Scholar [7] S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Math. 1730, Springer-Verlag, Berlin Heidelberg, 2000. doi: 10.1007/BFb0103945.  Google Scholar [8] M. Iacobelli, F. S. Patacchini and F. Santambrogio, Weighted Ultrafast Diffusion Equations: From Well-Posedness to Long-Time Behaviour, Arch. Ration. Mech. Anal., 232 (2019), 1165-1206.  doi: 10.1007/s00205-018-01341-w.  Google Scholar [9] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.  Google Scholar [10] F. Otto and M. Westdickenberg, Eulerian calculus for the contraction in the Wasserstein distance,, SIAM J. Math. Anal., 37 (2005), 1227-1255.  doi: 10.1137/050622420.  Google Scholar [11] A. Rodríguez and J. L. Vázquez, A well-posed problem in singular Fickian diffusion, Arch. Rational Mech. Anal., 110 (1990), 141-163.  doi: 10.1007/BF00873496.  Google Scholar [12] F. Santambrogio and X.-J. Wang, Convexity of the support of the displacement interpolation: Counterexamples, Appl. Math. Lett., 58 (2016), 152-158.  doi: 10.1016/j.aml.2016.02.016.  Google Scholar [13] J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures. Appl., 71 (1992), 503-526.   Google Scholar [14] J. L. Vázquez, Failure of the strong maximum principle in nonlinear diffusion. Existence of needles, Comm. Partial Differential Equations, 30 (2005), 1263-1303.  doi: 10.1080/10623320500258759.  Google Scholar [15] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and its Applications, 33. Oxford University Press, Oxford, 2006.  doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar [16] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.   Google Scholar [17] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Math. Soc., Providence RI, 2003. doi: 10.1007/b12016.  Google Scholar

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##### References:
 [1] L. Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, in Calculus of variations and nonlinear partial differential equations, 1–41, Lecture Notes in Math., 1927, Springer, Berlin, 2008. doi: 10.1007/978-3-540-75914-0_1.  Google Scholar [2] L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, 2$^{nd}$ edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008.  Google Scholar [3] E. Caglioti, F. Golse and M. Iacobelli, A gradient flow approach to quantization of measures, Math. Models Methods Appl. Sci., 25 (2015), 1845-1885.  doi: 10.1142/S0218202515500475.  Google Scholar [4] E. Caglioti, F. Golse and M. Iacobelli, Quantization of measures and gradient flows: A perturbative approach in the $2$ dimensional case, Ann. Inst. N. Poincaré Anal. Non Linéaire, 35 (2018), 1531-1555.  doi: 10.1016/j.anihpc.2017.12.003.  Google Scholar [5] J. A. Carrillo and D. Slepcev, Example of a displacement convex functional of first order, Calc. Var. Partial Differential Equations, 36 (2009), 547-564.  doi: 10.1007/s00526-009-0243-4.  Google Scholar [6] J. R. Esteban, A. Rodríguez and J. L. Vázquez, A nonlinear heat equation with singular diffusivity, Comm. Partial Differential Equations, 13 (1988), 985-1039.  doi: 10.1080/03605308808820566.  Google Scholar [7] S. Graf and H. Luschgy, Foundations of Quantization for Probability Distributions, Lecture Notes in Math. 1730, Springer-Verlag, Berlin Heidelberg, 2000. doi: 10.1007/BFb0103945.  Google Scholar [8] M. Iacobelli, F. S. Patacchini and F. Santambrogio, Weighted Ultrafast Diffusion Equations: From Well-Posedness to Long-Time Behaviour, Arch. Ration. Mech. Anal., 232 (2019), 1165-1206.  doi: 10.1007/s00205-018-01341-w.  Google Scholar [9] F. Otto, The geometry of dissipative evolution equations: The porous medium equation, Comm. Partial Differential Equations, 26 (2001), 101-174.  doi: 10.1081/PDE-100002243.  Google Scholar [10] F. Otto and M. Westdickenberg, Eulerian calculus for the contraction in the Wasserstein distance,, SIAM J. Math. Anal., 37 (2005), 1227-1255.  doi: 10.1137/050622420.  Google Scholar [11] A. Rodríguez and J. L. Vázquez, A well-posed problem in singular Fickian diffusion, Arch. Rational Mech. Anal., 110 (1990), 141-163.  doi: 10.1007/BF00873496.  Google Scholar [12] F. Santambrogio and X.-J. Wang, Convexity of the support of the displacement interpolation: Counterexamples, Appl. Math. Lett., 58 (2016), 152-158.  doi: 10.1016/j.aml.2016.02.016.  Google Scholar [13] J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures. Appl., 71 (1992), 503-526.   Google Scholar [14] J. L. Vázquez, Failure of the strong maximum principle in nonlinear diffusion. Existence of needles, Comm. Partial Differential Equations, 30 (2005), 1263-1303.  doi: 10.1080/10623320500258759.  Google Scholar [15] J. L. Vázquez, Smoothing and Decay Estimates for Nonlinear Diffusion Equations, Oxford Lecture Series in Mathematics and its Applications, 33. Oxford University Press, Oxford, 2006.  doi: 10.1093/acprof:oso/9780199202973.001.0001.  Google Scholar [16] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007.   Google Scholar [17] C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Math. Soc., Providence RI, 2003. doi: 10.1007/b12016.  Google Scholar
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