Probabilistic weak solutions for nonlinear stochastic evolution problems involving pseudomonotone operators
Анотація
УДК 519.21
Імовiрнiснi слабкi розв’язки нелiнiйних стохастичних еволюцiйних задач, що мiстять псевдомонотоннi оператори
Вивчається важливий клас стохастичних нелiнiйних еволюцiйних задач iз псевдомонотонними елiптичними частинами. Встановлено iснування ймовiрнiсних слабких (або мартингальних) розв’язкiв. На сьогоднi не iснує теорiї розв’язностi, розробленої для таких рiвнянь, незважаючи на те, що є багато робiт, в яких вивчаються рiзнi узагальнення умов монотонностi. Ключем до нашої роботи є знаковий результат для диференцiала Iто наближеного розв’язку, який ми встановлюємо, а також кiлька результатiв щодо компактностi аналiтичної та ймовiрнiсної природи i характеризацiя псевдомонотонних операторiв по Ф. E. Браудеру
Посилання
A. Bensoussan, Some existence results for stochastic partial differential equations, Stoch. Partial Different. Equat. and Appl. (Trento, 1990), 268, 37 – 53 (1992).
A. Bensoussan, Stochastic Navier – Stokes equations, Acta Appl. Math., 38, № 3, 267 – 304 (1995), https://doi.org/10.1007/BF00996149 DOI: https://doi.org/10.1007/BF00996149
A. Bensoussan, R. Temam, Équations stochastiques du type Navier-Stokes. (French), J. Funct. Anal., 13, № 1, 195 – 222 (1973), https://doi.org/10.1016/0022-1236(73)90045-1 DOI: https://doi.org/10.1016/0022-1236(73)90045-1
A. Bensoussan, R. Temam, Équations aux dérivées partielles stochastiques non linéaires. I (French), Israel J. Math., 11, № 1, 95 – 129 (1972), https://doi.org/10.1007/BF02761449 DOI: https://doi.org/10.1007/BF02761449
H. Brezis, Équations et inéquations non linéaires dans les espaces vectoriels en dualité (French), Ann. Inst. Fourier (Grenoble), 18, 115 – 175 (1968). DOI: https://doi.org/10.5802/aif.280
H. Brézis, F. E. Browder, Strongly nonlinear elliptic boundary value problems, Ann. Scuola Norm. Super. Pisa, Cl. Sci. (4e ), 5, № 3, 587 – 603 (1978).
H. Brézis, F. E. Browder, Strongly nonlinear parabolic initial-boundary value problems, Proc. Natl. Acad. Sci. USA, 76, № 1, 38 – 40 (1979), https://doi.org/10.1073/pnas.76.1.38 DOI: https://doi.org/10.1073/pnas.76.1.38
H. Brézis, F. Browder, Some properties of higher order Sobolev spaces, J. Math. Pures et Appl., 61, № 3, 245 – 259 (1982).
F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Proc. Sympos. Pure Math., 18, pt 2 (1976). DOI: https://doi.org/10.1090/pspum/018.2
F. E. Browder, Pseudo-monotone operators and nonlinear elliptic boundary value problems on unbounded domains, Proc. Natl. Acad. Sci. USA, 74, № 7, 2659 – 2661 (1977), https://doi.org/10.1073/pnas.74.7.2659 DOI: https://doi.org/10.1073/pnas.74.7.2659
F. E. Browder, Strongly nonlinear parabolic equations of higher order, Convegno celebrativo del centenario della nascita di Mauro Picone e di Leonida Tonell, Accad. Naz. Lincei (1986).
F. E. Browder, Strongly nonlinear parabolic boundary value problems, Amer. J. Math., 86, № 2, 339 – 357 (1964), https://doi.org/10.2307/2373169 DOI: https://doi.org/10.2307/2373169
F. E. Browder, H. Brézis, Strongly nonlinear parabolic variational inequalities, Proc. Natl. Acad. Sci. USA, 77, № 2, 713 – 715 (1980), https://doi.org/10.1073/pnas.77.2.713 DOI: https://doi.org/10.1073/pnas.77.2.713
Z. Brzeźniak, W. Liu, J. Zhu, Strong solutions for SPDE with locally monotone coefficients driven by Lévy noise, Nonlinear Anal. Real World Appl., 17, 283 – 310 (2014), https://doi.org/10.1016/j.nonrwa.2013.12.005 DOI: https://doi.org/10.1016/j.nonrwa.2013.12.005
G. Da Prato, J. Zabczyk, Stochastic equations in infinite dimensions, Cambridge Univ. Press (1992), https://doi.org/10.1017/CBO9780511666223 DOI: https://doi.org/10.1017/CBO9780511666223
Yu. A. Dubinskii, Itogi Nauki i Tekhniki, Sovr. Probl. Mat., 9, 5 – 130 (1976).
Yu. A. Dubinskii, Itogi Nauki i Tekhniki, Sovr. Probl. Mat., 37, 89 – 166 (1990).
I. Gyöngy, A. Millet, On discretization schemes for stochastic evolution equations, Potential Anal., 23, № 2, 99 – 134 (2005), https://doi.org/10.1007/s11118-004-5393-6 DOI: https://doi.org/10.1007/s11118-004-5393-6
I. Gyöngy, N. V. Krylov, On stochastic equations with respect to semimartingales I, Stochastics, 4, № 1, 1 – 21 (1980/1981), https://doi.org/10.1080/03610918008833154 DOI: https://doi.org/10.1080/03610918008833154
N. V. Krylov, B. L. Rozovskii, Stochastic evolution equations, J. Soviet Math., 14, 1233 – 1277 (1981). DOI: https://doi.org/10.1007/BF01084893
R. Landes, A Note on strongly nonlinear parabolic equations of higher order, Different. and Integral Equat., 3, № 5, 851 – 862 (1990).
W. Liu, M. Röckner, Local and global well-posedness of SPDE with generalized coercivity conditions, J. Different. quat., 254, № 2, 725 – 755 (2013), https://doi.org/10.1016/j.jde.2012.09.014 DOI: https://doi.org/10.1016/j.jde.2012.09.014
W. Liu, Existence and uniqueness of solutions to nonlinear evolution equations with locally monotone operators, Nonlinear Anal., 74, № 18, 7543 – 7561 (2011), https://doi.org/10.1016/j.na.2011.08.018 DOI: https://doi.org/10.1016/j.na.2011.08.018
W. Liu, M. Röckner, SPDE in Hilbert space with locally monotone coefficients, J. Funct. Anal., 259, № 11, 2902 – 2922 (2010), https://doi.org/10.1016/j.jfa.2010.05.012 DOI: https://doi.org/10.1016/j.jfa.2010.05.012
J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Dunod Gauthier-Villar, Paris (1969).
G. I. Minty, On a monotonicity method for the solution of non-linear equations in Banach spaces, Proc. Natl. Acad. Sci. USA, 50, № 6, 1038 – 1041 (1963), https://doi.org/10.1073/pnas.50.6.1038 DOI: https://doi.org/10.1073/pnas.50.6.1038
E. Pardoux, Équations aux dérivées partielles stochastiques de type monotone. (French), Etude de solutions fortes de type Ito, These Doct. Sci. Math., Univ. Paris Sud (1975). ´
C. Prévôt, M. Röckner, A concise course on stochastic partial differential equations, Lect. Notes Math., 1905 (2007)
Y. Prokhorov, Convergence of random processes and limit theorems in probability theory, Theory Probab. and Appl., 1, № 2, 157 – 214 (1956). DOI: https://doi.org/10.1137/1101016
T. Roubicek, Nonlinear partial differential equations with applications, Birkhauser-Verlag, Basel etc. (2005).
B. L. Rozovsky, S. V. Lototsky, Stochastic evolution systems. Linear theory and applications to non-linear filtering, Second ed., Probab. Theory and Stoch. Model., 89, Springer, Cham (2018), https://doi.org/10.1007/978-3-319-94893-5 DOI: https://doi.org/10.1007/978-3-319-94893-5
M. Sango, Density dependent stochastic Navier – Stokes equations with non-Lipschitz random forcing, Rev. Math. Phys., 22, № 6, 669 – 697 (2010), https://doi.org/10.1142/S0129055X10004041 DOI: https://doi.org/10.1142/S0129055X10004041
M. Sango, Magnetohydrodynamic turbulent flows: existence results, Physica D, 239, № 12, 912 – 923 (2010), https://doi.org/10.1016/j.physd.2010.01.009 DOI: https://doi.org/10.1016/j.physd.2010.01.009
J. Simon, Compact sets in the space L p (O, T; B), Ann. Mat. Pura ed Appl., 146, № 1, 65 – 96 (1986), https://doi.org/10.1007/BF01762360 DOI: https://doi.org/10.1007/BF01762360
A. V. Skorokhod, Limit theorems for stochastic processes, Theory Probability and Appl., 1, № 3, 261 – 290 (1956). DOI: https://doi.org/10.1137/1101022
I. V. Skrypnik, Higher-order nonlinear elliptic equations (in Russian), J. Soviet Math., 56, № 4, 2505 – 2557 (1991). DOI: https://doi.org/10.1007/BF01097352
I. V. Skrypnik, Methods for analysis of nonlinear elliptic boundary value problems, Transl. Math. Monogr., 139, Amer. Math. Soc., Providence, RI (1994), https://doi.org/10.1090/mmono/139 DOI: https://doi.org/10.1090/mmono/139
R. Temam, Navier – Stokes equations, Theory and numerical analysis, Stud. Math. and Appl., 2, North-Holland Publ. Co., Amsterdam etc. (1977).
M. Viot, Solutions faibles d’equations aux dérivées partielles stochastiques non lineaires, These, Univ. Pierre et Marie Curie, Paris (1976).
M. I. Višik, Solvability of boundary-value problems for quasilinear parabolic equations of higher orders (in Russian), Mat. Sb. (N. S.), 59 (101), 289 – 325 (1962) (Translared in Amer. Math. Soc. Transl. Ser. 2, 65, Amer. Math. Soc., Providence, R.I. (1967)).
J. R. L. Webb, Boundary value problems for strongly nonlinear elliptic equations, J. London Math. Soc. (2), 21, № 1, 123 – 132 (1980), https://doi.org/10.1112/jlms/s2-21.1.123 DOI: https://doi.org/10.1112/jlms/s2-21.1.123
Авторські права (c) 2022 M SANGO
Для цієї роботи діють умови ліцензії Creative Commons Attribution 4.0 International License.
