Well-Posedness of the Multidimensional Fractional Stochastic Navier–Stokes Equations on the Torus and on Bounded Domains

Abstract

In this work, we introduce and study the well-posedness of the multidimensional fractional stochastic Navier-Stokes equations on bounded domains and on the torus (briefly dD-FSNSE). For the subcritical regime, we establish thresholds for which a maximal local mild solution exists and satisfies required space and time regularities. We prove that under conditions of Beale-Kato-Majda type, these solutions are global and unique. These conditions are automatically satisfied for the 2D-FSNSE on the torus if the initial data has H (1)-regularity and the diffusion term satisfies growth and Lipschitz conditions corresponding to H (1)-spaces. The case of 2D-FSNSE on the torus is studied separately. In particular, we established thresholds for the global existence, uniqueness, space and time regularities of the weak (strong in probability) solutions in the subcritical regime. For the general regime, we prove the existence of a martingale solution and we establish the uniqueness under a condition of Serrin's type on the fractional Sobolev spaces.

Optimal control problems constrained by the stochastic Navier–Stokes equations with multiplicative Lévy noise

“…and the Lévy process is given by a Q-Wiener process, then the solution of system (4.1) is a global mild solution in the sense that ℙ( = ) = 1, see [13].…”

“…Another approach is given by using semigroup theory leading to the definition of mild solutions. The existence and uniqueness of a mild solution over an arbitrary time interval can be obtained under certain additional assumptions, see [11,13]. In general, a unique mild solution of the stochastic Navier-Stokes equations does not exist.…”

“…In [25], an existence and uniqueness result for strong pathwise solutions is given. For further definitions of solutions to the fractional stochastic Navier-Stokes equations, we refer to [13].…”

“…(cf. Definition 3.2,[13]). Let be a stopping time taking values in (0, ] and let ( ) ∈ℕ be an increasing sequence of stopping times taking values in (0, ] satisfying lim →∞ = .A predictable process ( ( )) ∈[0, ) with values in ( ) is called a local mild solution of system (4.1) if for fixed ∈ ℕsup ∈[0, ) ‖ ( )‖ 2 ( ) < ∞ and we have for each ∈ ℕ, all ∈ [0, ] and ℙ-a.s.…”

Martingale and weak solutions for a stochastic nonlocal Burgers equation on finite intervals

“…This is similar to the textbook definition of a fractional power for a positive-definite symmetric matrix in linear algebra. For example, Debbi [11] considered the fractional stochastic Navier-Stokes equations on bounded domains with this fractional Laplacian operator. We remark that this fractional Laplacian operator is different from the nonlocal Laplacian operator (1.2) which we use here in this paper.…”