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Asymptotic analysis for a very fast diffusion equation arising from the 1D quantization problem
ETH Zürich, Rämistrasse 101, 8092, Zürich, Switzerland |
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 [
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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