A $W^1_2$-Theory of Stochastic Partial Differential Systems of Divergence Type on $C^1$ Domains

Lee Kijung (Ajou University)
Kim Kyeong-Hun (Korea University)


In this paper we study the stochastic partial differential systems of divergence type with $\mathcal{C}^1$ space domains in $\mathbb{R}^d$. Existence and uniqueness results are obtained in terms of Sobolev spaces with weights so that we allow the derivatives of the solution to blow up near the boundary. The coefficients of the systems are only measurable and are allowed to blow up near the boundary.

Full Text: Download PDF | View PDF online (requires PDF plugin)

Pages: 1296-1317

Publication Date: July 7, 2011

DOI: 10.1214/EJP.v16-913


  1. Flandoli, Franco. Dirichlet boundary value problem for stochastic parabolic equations: compatibility relations and regularity of solutions. Stochastics Stochastics Rep. 29 (1990), no. 3, 331--357. MR1042066 (91c:35169)
  2. Flandoli, Franco. An introduction to 3D stochastic fluid dynamics. SPDE in hydrodynamic: recent progress and prospects, 51--150, Lecture Notes in Math., 1942, Springer, Berlin, 2008. MR2459085 (2009j:76191)
  3. T. Funaki, Random motion of strings and related stochastic evolution equations, Nagoya Math. J. 89 (1983), 129--193.
  4. Gilbarg, David; Hörmander, Lars. Intermediate Schauder estimates. Arch. Rational Mech. Anal. 74 (1980), no. 4, 297--318. MR0588031 (82a:35038)
  5. Kim, Kyeong-Hun. On $Lsb p$-theory of stochastic partial differential equations of divergence form in $Csp 1$ domains. Probab. Theory Related Fields 130 (2004), no. 4, 473--492. MR2102888 (2005h:60185)
  6. Kim, Kyeong-Hun. On stochastic partial differential equations with variable coefficients in $Csp 1$ domains. Stochastic Process. Appl. 112 (2004), no. 2, 261--283. MR2073414 (2005e:60128)
  7. K. Kim, $L_p$ estimates for SPDE with discontinuous coefficients in domains/}, Electronic Journal of Probability, 10 (2005), no. 1, 1-20.
  8. K. Lee and K. Kim, A $W^n_2$-Theory of Stochastic Partial Differential Systems of Non-divergent type on $C^1$ domains, preprint.
  9. Kim, Kyeong-Hun; Krylov, N. V. On SPDEs with variable coefficients in one space dimension. Potential Anal. 21 (2004), no. 3, 209--239. MR2075669 (2005i:60120)
  10. Kim, Kyeong-Hun; Krylov, N. V. On the Sobolev space theory of parabolic and elliptic equations in $Csp 1$ domains. SIAM J. Math. Anal. 36 (2004), no. 2, 618--642. MR2111792 (2005k:35168)
  11. N.V. Krylov, Ito's formula for the $L_p$-norm of stochastic $W^1_p$-valued processes. Probab. Theory Related Fields 147 (2010), no. 3-4, 583-605.
  12. Krylov, N. V. Some properties of traces for stochastic and deterministic parabolic weighted Sobolev spaces. J. Funct. Anal. 183 (2001), no. 1, 1--41. MR1837532 (2002c:46069)
  13. Krylov, N. V. An analytic approach to SPDEs. Stochastic partial differential equations: six perspectives, 185--242, Math. Surveys Monogr., 64, Amer. Math. Soc., Providence, RI, 1999. MR1661766 (99j:60093)
  14. Krylov, N. V. Weighted Sobolev spaces and Laplace's equation and the heat equations in a half space. Comm. Partial Differential Equations 24 (1999), no. 9-10, 1611--1653. MR1708104 (2000j:46065)
  15. Krylov, N. V.; Lototsky, S. V. A Sobolev space theory of SPDEs with constant coefficients in a half space. SIAM J. Math. Anal. 31 (1999), no. 1, 19--33 (electronic). MR1720129 (2001a:60072)
  16. Lototsky, S. V. Sobolev spaces with weights in domains and boundary value problems for degenerate elliptic equations. Methods Appl. Anal. 7 (2000), no. 1, 195--204. MR1796011 (2001m:46077)
  17. Mikulevicius, R. On strong $Hsb 2sp 1$-solutions of stochastic Navier-Stokes equation in a bounded domain. SIAM J. Math. Anal. 41 (2009), no. 3, 1206--1230. MR2529962 (2011a:60232)
  18. Mikulevicius, R. On the Cauchy problem for stochastic Stokes equations. SIAM J. Math. Anal. 34 (2002), no. 1, 121--141 (electronic). MR1950829 (2003k:60154)
  19. Mikulevicius, R.; Rozovskii, B. L. Global $Lsb 2$-solutions of stochastic Navier-Stokes equations. Ann. Probab. 33 (2005), no. 1, 137--176. MR2118862 (2005k:60198)
  20. Mikulevicius, R.; Rozovskii, B. A note on Krylov's $Lsb p$-theory for systems of SPDEs. Electron. J. Probab. 6 (2001), no. 12, 35 pp. (electronic). MR1831807 (2003a:60097)
  21. Mueller, C.; Tribe, R. Hitting properties of a random string. Electron. J. Probab. 7 (2002), no. 10, 29 pp. (electronic). MR1902843 (2003g:60111)
  22. Pardoux, E. Stochastic partial differential equations and filtering of diffusion processes. Stochastics 3, no. 2, 127--167. (1979), MR0553909 (81b:60059)
  23. Rozovskiĭ, B. L. Stochastic evolution systems.Linear theory and applications to nonlinear filtering.Translated from the Russian by A. Yarkho.Mathematics and its Applications (Soviet Series), 35. Kluwer Academic Publishers Group, Dordrecht, 1990. xviii+315 pp. ISBN: 0-7923-0037-8 MR1135324 (92k:60136)
  24. Yoo, Hyek. $Lsb p$-estimates for stochastic PDEs with discontinuous coefficients. Stochastic Anal. Appl. 17 (1999), no. 4, 687--711. MR1693563 (2000f:60093)

Creative Commons License
This work is licensed under a Creative Commons Attribution 3.0 License.