Nonlinear parabolic equations for measures

Manita, O.; Shaposhnikov, S.

doi:10.1090/S1061-0022-2013-01279-9

Nonlinear parabolic equations for measures
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by O. A. Manita and S. V. Shaposhnikov
Translated by: O. A. Manita

St. Petersburg Math. J. 25 (2014), 43-62

DOI: https://doi.org/10.1090/S1061-0022-2013-01279-9

Published electronically: November 20, 2013

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Abstract:

A new existence result is established for weak parabolic equations for probability measures. Sufficient conditions are given for the existence of local and global-in-time probability solutions of the Cauchy problem for such equations. Some conditions under which global-in-time solutions do not exist are indicated.

References

Ya. I. Belopol′skaya, On a probabilistic approach to the solution of systems of nonlinear parabolic equations, Teor. Veroyatn. Primen. 49 (2004), no. 4, 625–652 (Russian, with Russian summary); English transl., Theory Probab. Appl. 49 (2005), no. 4, 589–611. MR 2142559, DOI 10.1137/S0040585X97981287
V. I. Bogachev, N. V. Krylov, and M. Rëkner, Elliptic and parabolic equations for measures, Uspekhi Mat. Nauk 64 (2009), no. 6(390), 5–116 (Russian, with Russian summary); English transl., Russian Math. Surveys 64 (2009), no. 6, 973–1078. MR 2640966, DOI 10.1070/RM2009v064n06ABEH004652
V. I. Bogachev, M. Rëkner, and S. V. Shaposhnikov, Nonlinear evolution and transport equations for measures, Dokl. Akad. Nauk 429 (2009), no. 1, 7–11 (Russian); English transl., Dokl. Math. 80 (2009), no. 3, 785–789. MR 2606196, DOI 10.1134/S1064562409060015
V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. MR 2267655, DOI 10.1007/978-3-540-34514-5
Yu. L. Daletskiĭ and S. V. Fomin, Mery i differentsial′nye uravneniya v beskonechnomernykh prostranstvakh, “Nauka”, Moscow, 1983 (Russian). MR 720545
Ju. L. Daleckiĭ, Infinite-dimensional elliptic operators and the corresponding parabolic equations, Uspehi Mat. Nauk 22 (1967), no. 4 (136), 3–54 (Russian). MR 0223773
Nelson Dunford and Jacob T. Schwartz, Linear operators. Part I, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. General theory; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1958 original; A Wiley-Interscience Publication. MR 1009162
R. L. Dobrušin, Vlasov equations, Funktsional. Anal. i Prilozhen. 13 (1979), no. 2, 48–58, 96 (Russian). MR 541637
V. V. Kozlov, The generalized Vlasov kinetic equation, Uspekhi Mat. Nauk 63 (2008), no. 4(382), 93–130 (Russian, with Russian summary); English transl., Russian Math. Surveys 63 (2008), no. 4, 691–726. MR 2483200, DOI 10.1070/RM2008v063n04ABEH004549
—, Kinetic Vlasov equation, dynamics of continua and turbulence, Nonlinear Dynamics 6 (2010), no. 3, 489–512. (Russian)
È. Mitidieri and S. I. Pokhozhaev, A priori estimates and the absence of solutions of nonlinear partial differential equations and inequalities, Tr. Mat. Inst. Steklova 234 (2001), 1–384 (Russian, with English and Russian summaries); English transl., Proc. Steklov Inst. Math. 3(234) (2001), 1–362. MR 1879326
È. Mitidieri and S. I. Pokhozhaev, Liouville theorems for some classes of nonlinear nonlocal problems, Tr. Mat. Inst. Steklova 248 (2005), no. Issled. po Teor. Funkts. i Differ. Uravn., 164–184 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 1(248) (2005), 158–178. MR 2165926
O. A. Oleĭnik and E. V. Radkevich, Second order equations with non-negative characteristic form, Moskov. Gos. Univ., Moscow, 2010. (Russian)
L. G. Tonoyan, Nonlinear elliptic equations for measures, Dokl. Akad. Nauk 439 (2011), no. 2, 174–177 (Russian); English transl., Dokl. Math. 84 (2011), no. 1, 558–561. MR 2884629, DOI 10.1134/S1064562411040132
S. V. Shaposhnikov, On the uniqueness of the probabilistic solution of the Cauchy problem for the Fokker-Planck-Kolmogorov equation, Teor. Veroyatn. Primen. 56 (2011), no. 1, 77–99 (Russian, with Russian summary); English transl., Theory Probab. Appl. 56 (2012), no. 1, 96–115. MR 2848417, DOI 10.1137/S0040585X97985212
Luigi Ambrosio, Transport equation and Cauchy problem for $BV$ vector fields, Invent. Math. 158 (2004), no. 2, 227–260. MR 2096794, DOI 10.1007/s00222-004-0367-2
Luigi Ambrosio, Transport equation and Cauchy problem for non-smooth vector fields, Calculus of variations and nonlinear partial differential equations, Lecture Notes in Math., vol. 1927, Springer, Berlin, 2008, pp. 1–41. MR 2408257, DOI 10.1007/978-3-540-75914-0_{1}
Andrea L. Bertozzi, José A. Carrillo, and Thomas Laurent, Blow-up in multidimensional aggregation equations with mildly singular interaction kernels, Nonlinearity 22 (2009), no. 3, 683–710. MR 2480108, DOI 10.1088/0951-7715/22/3/009
V. I. Bogachev, G. Da Prato, and M. Röckner, On parabolic equations for measures, Comm. Partial Differential Equations 33 (2008), no. 1-3, 397–418. MR 2398235, DOI 10.1080/03605300701382415
V. I. Bogachev, G. Da Prato, M. Röckner, and S. V. Shaposhnikov, Nonlinear evolution equations for measures on infinite dimensional spaces, Stochastic partial differential equations and applications, Quad. Mat., vol. 25, Dept. Math., Seconda Univ. Napoli, Caserta, 2010, pp. 51–64. MR 2985081
V. I. Bogachev, M. Röckner, and S. V. Shaposhnikov, On uniqueness problems related to the Fokker-Planck-Kolmogorov equation for measures, J. Math. Sci. (N.Y.) 179 (2011), no. 1, 7–47. Problems in mathematical analysis. No. 61. MR 3014097, DOI 10.1007/s10958-011-0581-6
José A. Carrillo, Robert J. McCann, and Cédric Villani, Kinetic equilibration rates for granular media and related equations: entropy dissipation and mass transportation estimates, Rev. Mat. Iberoamericana 19 (2003), no. 3, 971–1018. MR 2053570, DOI 10.4171/RMI/376
J. A. Carrillo, M. DiFrancesco, A. Figalli, T. Laurent, and D. Slepčev, Global-in-time weak measure solutions and finite-time aggregation for nonlocal interaction equations, Duke Math. J. 156 (2011), no. 2, 229–271. MR 2769217, DOI 10.1215/00127094-2010-211
R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511–547. MR 1022305, DOI 10.1007/BF01393835
Tadahisa Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations, Z. Wahrsch. Verw. Gebiete 67 (1984), no. 3, 331–348. MR 762085, DOI 10.1007/BF00535008
Hailiang Li and Giuseppe Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal. 172 (2004), no. 3, 407–428. MR 2062430, DOI 10.1007/s00205-004-0307-8
Thomas Lorenz, Mutational analysis, Lecture Notes in Mathematics, vol. 1996, Springer-Verlag, Berlin, 2010. A joint framework for Cauchy problems in and beyond vector spaces. MR 2656173, DOI 10.1007/978-3-642-12471-6
Stefania Maniglia, Probabilistic representation and uniqueness results for measure-valued solutions of transport equations, J. Math. Pures Appl. (9) 87 (2007), no. 6, 601–626 (English, with English and French summaries). MR 2335089, DOI 10.1016/j.matpur.2007.04.001
Daniel W. Stroock and S. R. Srinivasa Varadhan, Multidimensional diffusion processes, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 233, Springer-Verlag, Berlin-New York, 1979. MR 532498