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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

Mikaela Iacobelli

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.

Keywords: Very fast diffusion, quantization of measures, gradient flows, asymptotic.
Mathematics Subject Classification: Primary: 35K15, 35K65; Secondary: 70F45.
Citation: Mikaela Iacobelli. Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem. Discrete and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures. Appl., 71 (1992), 503-526. 

[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.

[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.
[16] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. 
[17]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Math. Soc., Providence RI, 2003. doi: 10.1007/b12016.

show all references

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[13]

J. L. Vázquez, Nonexistence of solutions for nonlinear heat equations of fast-diffusion type, J. Math. Pures. Appl., 71 (1992), 503-526. 

[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.

[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.
[16] J. L. Vázquez, The Porous Medium Equation. Mathematical Theory,, Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, Oxford, 2007. 
[17]

C. Villani, Topics in Optimal Transportation, Graduate Studies in Mathematics 58, American Math. Soc., Providence RI, 2003. doi: 10.1007/b12016.

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