On Lp-theory of stochastic partial differential equations in the whole spaces

N. V. Krylov

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73 Scopus citations

Abstract

It is shown that equations like du = (aijuxixj + biuxi + cu + f) dt + (σikuxi + νku + gk) dwkt, t > 0, with variable random coefficients and with zero initial condition have unique solutions in the Sobolev spaces W2p, p ∈ [2, ∞), under natural ellipticity condition and under conditions that (i) a is uniformly continuous with respect to x, (ii) σ, ν have bounded first derivatives in x and all other coefficients are bounded, (iii) f ∈ Lp, g ∈ W1p. A corresponding result in the spaces of Bessel potentials Hnp is proved, which implies that better differentiability properties of the coefficients and free terms of the equations lead to the better regularity of solutions. Applications to equations with space-time white noise are given.

Original languageEnglish (US)
Pages (from-to)313-340
Number of pages28
JournalSIAM Journal on Mathematical Analysis
Volume27
Issue number2
DOIs
StatePublished - Mar 1996

Keywords

  • Bessel potentials
  • Cylindrical white noise
  • Stochastic equations

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