×

Nonlinear parabolic equations for measures. (English) Zbl 1286.35137

St. Petersbg. Math. J. 25, No. 1, 43-62 (2014); translation from Algebra Anal. 25, No. 1, 64-93 (2013).
Summary: 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.

MSC:

35K59 Quasilinear parabolic equations
35K15 Initial value problems for second-order parabolic equations
35R60 PDEs with randomness, stochastic partial differential equations
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ya. I. Belopol\(^{\prime}\)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. · Zbl 1092.60039
[2] 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. · Zbl 1194.35481
[3] 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. · Zbl 1200.35047
[4] V. I. Bogachev, Measure theory. Vol. I, II, Springer-Verlag, Berlin, 2007. · Zbl 1120.28001
[5] Меры и дифференциал\(^{\приме}\)ные уравнения в бесконечномерных пространствах, ”Наука”, Мосцощ, 1983 (Руссиан). Ю. Л. Далецкы анд С. В. Фомин, Меасурес анд дифферентиал ечуатионс ин инфините-дименсионал спаце, Матхематицс анд иц Апплицатионс (Совиет Сериес), вол. 76, Клущер Ацадемиц Публишерс Гроуп, Дордречт, 1991. Транслатед фром тхе Руссиан; Щитх аддитионал материал бы В. Р. Стебловская, Ю. В. Богданскы [Ю. В. Богданский] анд Н. Ю. Гончарук.
[6] Ju. L. Daleckiĭ, Infinite-dimensional elliptic operators and the corresponding parabolic equations, Uspehi Mat. Nauk 22 (1967), no. 4 (136), 3 – 54 (Russian).
[7] 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. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part II, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral theory. Selfadjoint operators in Hilbert space; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1963 original; A Wiley-Interscience Publication. Nelson Dunford and Jacob T. Schwartz, Linear operators. Part III, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1988. Spectral operators; With the assistance of William G. Bade and Robert G. Bartle; Reprint of the 1971 original; A Wiley-Interscience Publication. · Zbl 0635.47002
[8] R. L. Dobrušin, Vlasov equations, Funktsional. Anal. i Prilozhen. 13 (1979), no. 2, 48 – 58, 96 (Russian).
[9] 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. · Zbl 1181.37006
[10] -, Kinetic Vlasov equation, dynamics of continua and turbulence, Nonlinear Dynamics 6 (2010), no. 3, 489-512. (Russian)
[11] 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. · Zbl 0987.35002
[12] 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. · Zbl 1123.35087
[13] O. A. Oleĭnik and E. V. Radkevich, Second order equations with non-negative characteristic form, Moskov. Gos. Univ., Moscow, 2010. (Russian)
[14] 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. · Zbl 1241.35074
[15] 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. · Zbl 1238.35168
[16] Luigi Ambrosio, Transport equation and Cauchy problem for \?\? vector fields, Invent. Math. 158 (2004), no. 2, 227 – 260. · Zbl 1075.35087
[17] 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. · Zbl 1159.35041
[18] 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. · Zbl 1194.35053
[19] 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. · Zbl 1323.35058
[20] 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. · Zbl 1275.60051
[21] 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. · Zbl 1291.35425
[22] 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. · Zbl 1073.35127
[23] 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. · Zbl 1215.35045
[24] R. J. DiPerna and P.-L. Lions, Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math. 98 (1989), no. 3, 511 – 547. · Zbl 0696.34049
[25] Tadahisa Funaki, A certain class of diffusion processes associated with nonlinear parabolic equations, Z. Wahrsch. Verw. Gebiete 67 (1984), no. 3, 331 – 348. · Zbl 0546.60081
[26] Hailiang Li and Giuseppe Toscani, Long-time asymptotics of kinetic models of granular flows, Arch. Ration. Mech. Anal. 172 (2004), no. 3, 407 – 428. · Zbl 1116.82025
[27] 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. · Zbl 1198.37003
[28] 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). · Zbl 1123.60048
[29] 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. · Zbl 0426.60069
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.