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DENSITY DEPENDENT STOCHASTIC NAVIER–STOKES EQUATIONS WITH NON-LIPSCHITZ RANDOM FORCING

MAMADOU SANGO

School of Mathematics, Institute for Advanced Study, 1, Einstein Drive, Princeton, NJ 08540, USA

Department of Mathematics and Applied Mathematics, University of Pretoria, Pretoria 0002, South Africa

https://doi.org/10.1142/S0129055X10004041Cited by:24 (Source: Crossref)

Abstract

In this work, we investigate the question of existence of weak solutions to the density dependent stochastic Navier–Stokes equations. The noise considered contains functions which depend nonlinearly on the velocity and which do not satisfy the Lipschitz condition. Furthermore, the initial density is allowed to vanish. We introduce a suitable notion of probabilistic weak solution for the problem and prove its existence.

Keywords:

AMSC: 35R60, 35D05, 35Q35

References

S. N. Antontsev , A. V. Kazhikhov and V. N. Monakhov , Boundary Value Problems in Mechanics of Nonhomogeneous Fluids , Studies in Mathematics and Its Applications 22 ( North-Holland Publishing Co. , Amsterdam , 1990 ) . Google Scholar
A. Bensoussan, Stochastic Partial Differential Equations and Applications (Trento, 1990), Pitman Res. Notes Math. Ser. 268 (Longman Scientific and Technical, Harlow, UK, 1992) pp. 37–53. Google Scholar
A. Bensoussan, Control of Partial Differential Equations (Trento, 1993), Lecture Notes in Pure and Appl. Math. 165 (Dekker, New York, 1994) pp. 11–21. Google Scholar
A. Bensoussan, Acta Appl. Math. 38, 267 (1995), DOI: 10.1007/BF00996149. Crossref, Web of Science, Google Scholar
A. Bensoussan and R. Temam, J. Funct. Anal. 13, 195 (1973), DOI: 10.1016/0022-1236(73)90045-1. Crossref, Web of Science, Google Scholar
J. Bergh and J. Löfström , Interpolation Spaces. An Introduction , Grundlehren der Mathematischen Wissenschaften 223 ( Springer-Verlag , Berlin-New York , 1976 ) . Crossref, Google Scholar
J. Bricmont, A. Kupiainen and R. Lefevere, Comm. Math. Phys. 230(1), 87 (2002), DOI: 10.1007/s00220-002-0708-1. Crossref, Web of Science, ADS, Google Scholar
J. Bricmont, A. Kupiainen and R. Lefevere, Comm. Math. Phys. 224(1), 65 (2001). Crossref, Web of Science, ADS, Google Scholar
J. Bricmont, A. Kupiainen and R. Lefevere, J. Statist. Phys. 100(3–4), 743 (2000), DOI: 10.1023/A:1018627609718. Crossref, Web of Science, Google Scholar
Z. Brzezniak, M. Capinski and F. Flandoli, Stochastic Anal. Appl. 10(5), 523 (1992), DOI: 10.1080/07362999208809288. Crossref, Web of Science, Google Scholar
Z. Brzeźniak and Y. Li, Trans. Amer. Math. Soc. 358(12), 5587 (2006), DOI: 10.1090/S0002-9947-06-03923-7. Crossref, Web of Science, Google Scholar
N. J. Cutland and B. Enright, J. Differential Equations 228(1), 140 (2006), DOI: 10.1016/j.jde.2006.04.009. Crossref, Web of Science, ADS, Google Scholar
G. Da Prato and J. Zabczyk , Stochastic Equations in Infinite Dimensions , Encyclopedia of Mathematics and Its Applications 44 ( Cambridge University Press , Cambridge , 1992 ) . Crossref, Google Scholar
G. Da Prato and J. Zabczyk, Ergodicity for Infinite-Dimensional Systems 229, (1996), DOI: 10.1017/CBO9780511662829. Google Scholar
G. Deugoue and M. Sango, Abstr. Appl. Anal. 2009, 27 (2009), DOI: 10.1155/2009/723236. Google Scholar
G. Deugoue and M. Sango, Boundary Value Problems 2010, 31 (2010), DOI: 10.1155/2010/723018. Google Scholar
E. Weinan and Ya. G. Sinai, Russian Math. Surveys 55(4), 635 (2000). Crossref, Web of Science, Google Scholar
E. Feireisl , Dynamics of Viscous Compressible Fluids , Oxford Lecture Series in Mathematics and Its Applications 26 ( Oxford University Press , Oxford , 2004 ) . Google Scholar
F. Flandoli and D. Gatarek, Probab. Theory Relat. Fields 102, 367 (1995), DOI: 10.1007/BF01192467. Crossref, Web of Science, Google Scholar
J.-F. Gerbeau and C. Le Bris, Adv. Differential Equations 2(3), 427 (1997). Google Scholar
I. I. Gikhman and A. V. Skorohod , Stochastic Differential Equations , Ergebnisse der Mathematik und ihrer Grenzgebiete 72 ( Springer-Verlag , New York-Heidelberg , 1972 ) . Crossref, Google Scholar
Y. Cho and H. Kim, Nonlinear Anal. 59(4), 465 (2004). Crossref, Web of Science, Google Scholar
H. J. Choe and H. Kim, Comm. Partial Differential Equations 28(5–6), 1183 (2003), DOI: 10.1081/PDE-120021191. Crossref, Web of Science, Google Scholar
A. V. Kazhikhov, Dokl. Akad. Nauk SSSR 216, 1008 (1974). Google Scholar
S. B. Kuksin, Randomly Forced Nonlinear PDEs and Statistical Hydrodynamics in 2 Space Dimensions, Zurich Lectures in Advanced Mathematics (European Mathematical Society (EMS), Zürich, 2006) pp. x–93. Crossref, Google Scholar
S. Kuksin and A. Shirikyan, Math. Phys. Anal. Geom. 4(2), 147 (2001), DOI: 10.1023/A:1011989910997. Crossref, Web of Science, Google Scholar
O. A. Ladyzhenskaya , The Mathematical Theory of Viscous Incompressible Flow , 2nd edn. ( Gordon and Breach, Science Publishers , New York-London-Paris , 1969 ) . Google Scholar
O. A. Ladyzhenskaja and V. A. Solonnikov, Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst. Steklov. LOMI 52, 52 (1975). Google Scholar
J. Leray, C. R. Acad. Sci. Paris 192, 1180 (1931). Google Scholar
J. Leray, Acta Math. 63(1), 193 (1934), DOI: 10.1007/BF02547354. Crossref, Google Scholar
J. Leray, J. Math. Pure Appl. (9) 12, 1 (1933). Web of Science, Google Scholar
J. L. Lions , Quelques méthodes de résolution des problèmes aux limites non linéaires ( Dunod, Gauthiers-Villars , Paris , 1969 ) . Google Scholar
J.-L. Lions, On some problems connected with Navier–Stokes equations, Nonlinear Evolution Equations (Proc. Sympos.), Publ. Math. Res. Center Univ. Wisconsin 40 (Academic Press, New York-London, 1978) pp. 59–84. Google Scholar
J.-L. Lions and E. Magenes , Non-Homogeneous Boundary Value Problems and Applications, Vol. I , Die Grundlehren der Mathematischen Wissenschaften 181 ( Springer-Verlag , New York-Heidelberg , 1972 ) . Google Scholar
P.-L. Lions , Mathematical Topics in Fluid Mechanics , Incompressible Models 1 ( The Clarendon Press, Oxford University Press , New York , 1996 ) . Google Scholar
P.-L. Lions , Mathematical Topics in Fluid Mechanics , Compressible Models 2 ( The Clarendon Press, Oxford University Press , New York , 1998 ) . Google Scholar
P.-L. Lions, C. R. Acad. Sci. Paris Sér. I Math. 317(12), 1197 (1993). Google Scholar
P.-L. Lions, C. R. Acad. Sci. Paris Sér. I Math. 317(1), 115 (1993). Google Scholar
P.-L. Lions, C. R. Acad. Sci. Paris Sér. I Math. 316(12), 1335 (1993). Google Scholar
R. Mikulevicius and B. L. Rozovskii, Ann. Probab. 33(1), 137 (2005), DOI: 10.1214/009117904000000630. Crossref, Web of Science, Google Scholar
R. Mikulevicius and B. L. Rozovskii, SIAM J. Math. Anal. 35(5), 1250 (2004), DOI: 10.1137/S0036141002409167. Crossref, Web of Science, Google Scholar
S.-E. Mohammed and T.S. Zhang, Dynamics of Stochastic 2D Navier–Stokes, to appear in J. Funct. Anal . Google Scholar
A. S. Monin and A. M. Yaglom , Statistical Fluid Mechanics: Mechanics of Turbulence I, II ( Dover Publications , Dover Ed Edition , 2007 ) . Google Scholar
A. Novotny and I. Stravskraba, Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Series in Mathematics and Its Applications 27 (Oxford University Press, Oxford, 2004) pp. xx–506. Crossref, Google Scholar
Yu. V. Prohorov, Teor. Veroyatnost. i Primenen. 1, 177 (1956). Google Scholar
P. A. Razafimandimby and M. Sango, Boundary Value Problems 2010, 47 (2010), DOI: 10.1155/2010/636140. Google Scholar
P. A. Razafimandimby and M. Sango, Asymptotic behavior of solutions of stochastic evolution equations for second grade fluids, to appear in C. R. Math. Acad. Sci. Paris . Google Scholar
M. Sango, Proc. Japan Acad. Ser. A Math. Sci. 81(5), 89 (2005), DOI: 10.3792/pjaa.81.89. Crossref, Google Scholar
M. Sango, Discrete Contin. Dyn. Syst. Ser. B 7(4), 885 (2007). Crossref, Web of Science, Google Scholar
M. Sango, Phys. D 239, 912 (2010), DOI: 10.1016/j.physd.2010.01.009. Crossref, Web of Science, Google Scholar
J. Simon, Ann. Mat. Pura Appl. 146(4), 65 (1987), DOI: 10.1007/BF01762360. Crossref, Web of Science, Google Scholar
J. Simon, C. R. Acad. Sci. Paris Sér. I Math. 309(7), 447 (1989). Google Scholar
J. Simon, C. R. Acad. Sci. Paris Sér. A-B 287(15), A1009 (1978). Google Scholar
J. Simon, SIAM J. Math. Anal. 21(5), 1093 (1990), DOI: 10.1137/0521061. Crossref, Web of Science, Google Scholar
A. V. Skorokhod, Teor. Veroyatnost. i Primenen 1, 289 (1956). Google Scholar
A. V. Skorokhod , Studies in the Theory of Random Processes ( Scripta Technica, Inc. Addison-Wesley Publishing Co., Inc. , Reading, Mass , 1965 ) . Google Scholar
R. Temam , Navier–Stokes Equations. Theory and Numerical Analysis , Studies in Mathematics and Its Applications 2 ( North-Holland Publishing Co. , Amsterdam-New York-Oxford , 1977 ) . Google Scholar
E. Tornatore and H. Fujita Yashima, Ricerche Mat. 46(2), 255 (1997). Google Scholar
E. Tornatore, NoDEA Nonlinear Differential Equations Appl. 7(4), 343 (2000), DOI: 10.1007/PL00001429. Crossref, Web of Science, Google Scholar
M. I. Vishik, A. I. Komech and A. V. Fursikov, Uspekhi Mat. Nauk 34(5), 135 (1979). Google Scholar
M. I. Vishik and A. V. Fursikov , Mathematical Problems of Statistical Hydromechanics , Mathematics and Its Applications ( Kluwer , Drodrecht , 1988 ) . Crossref, Google Scholar
M. Viot, Solutions faibles d'equations aux derivees partielles stochastiques non lineaires, Doctor of Sciences thesis, Parix 6 (1973) . Google Scholar
H. F. Yashima, Acta Math. Vietnam 26(2), 147 (2001). Google Scholar
H. F. Yashima, Equations de Navier–Stokes stochastiques non homogènes et applications, Tesi di Perfezionamento, Scuola Normale Superiore, Pisa (1992), 169 pp . Google Scholar