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The critical variational setting for stochastic evolution equations

Antonio Agresti, Mark Veraar

Abstract

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn-Hilliard equation, tamed Navier-Stokes equations, and Allen-Cahn equation.

The critical variational setting for stochastic evolution equations

Abstract

In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. In addition, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn-Hilliard equation, tamed Navier-Stokes equations, and Allen-Cahn equation.
Paper Structure (25 sections, 17 theorems, 244 equations, 1 table)

This paper contains 25 sections, 17 theorems, 244 equations, 1 table.

Table of Contents

  1. Introduction
  2. The classical variational setting
  3. The critical variational setting
  4. Main results
  5. Embedding into $L^p(L^q)$-theory
  6. Preliminaries
  7. Variational setting
  8. Stochastic calculus
  9. Main results
  10. Setting
  11. Global existence and uniqueness
  12. Continuous dependence on initial data
  13. Proofs of the main results
  14. Proof of Theorem \ref{['thm:local']}: local existence, uniqueness and blow-up criterion
  15. Proof of Theorem \ref{['thm:globalclas']}: global existence and uniqueness
  16. ...and 10 more sections

Key Result

Theorem 3.3

Suppose that Assumption ass:condFG holds. Then for every $u_0\in L^0_{{\mathscr F}_0}(\Omega;H)$, there exists a (unique) maximal solution $(u,\sigma)$ to eq:SEE such that a.s. $u\in C([0,\sigma);H)\cap L^2_{\rm loc}([0,\sigma);V)$. Moreover, the following blow-up criteria holds

Theorems & Definitions (53)

  • Example 2.1: Weak setting
  • Example 2.2: Strong setting
  • Definition 3.2: Solution
  • Theorem 3.3: Local existence, uniqueness and blow-up criterion
  • Theorem 3.4: Global existence and uniqueness I
  • Theorem 3.5: Global existence and uniqueness II
  • Remark 3.6: Variants of the coercivity condition
  • Remark 3.7: $\Omega$-localization of $(u_0,\phi)$
  • Theorem 3.8: Continuous dependence on initial data
  • Remark 3.9: Feller property
  • ...and 43 more