Markov selections for the 3D stochastic Navier–Stokes equations

By Franco Flandoli, Marco Romito. 2006

158 citations recorded by the source corpus.

We investigate the Markov property and the continuity with respect to the initial conditions (strong Feller property) for the solutions to the Navier–Stokes equations forced by an additive noise. First, we prove, by means of an abstract selection principle, that there are Markov solutions to the Navier–Stokes equations. Due to the lack of continuity of solutions in the space of finite energy, the Markov property holds almost everywhere in time. Then, depending on the regularity of the noise, we prove that any Markov solution has the strong Feller property for regular initial conditions. We give also a few consequences of these facts, together with a new sufficient condition for well-posedness.

Subfields: Applied Mathematics, Control and Systems Engineering, Finance

View Paper: https://www.semanticscholar.org/paper/0776170773e8505245a6d63c73e959f4380b2b5e

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Citations in the corpus

Stochastic 3D tamed Navier-Stokes equations: Existence, uniqueness and small time large deviation principles (2012, background): “Existence of Markov selections was proved in [14,10,16].”
Large Deviations for Stochastic Tamed 3D Navier-Stokes Equations (2010, background): “[ 12 , 13]). We would like to emphasize that the deterministic tamed equation has the following feature: if there exists a bounded solution to the classical 3D Navier-Stokes equation, then this solution also solves the tamed equation.”
On Ill‐ and Well‐Posedness of Dissipative Martingale Solutions to Stochastic 3D Euler Equations (2020, methodology): “We refer to [30] for more details on this step.”
Global-in-time probabilistically strong and Markov solutions to stochastic 3D Navier–Stokes equations: Existence and nonuniqueness (2021, methodology): “Note that [FR08] only introduced almost sure supermartingales with respect to the canonical filtration (B t )t>0.”
Local and global well-posedness of SPDE with generalized coercivity conditions☆ (2012, background): “[31, 13, 24] and the references therein), the results for the three dimensional case are still quite incomplete due to the lack of uniqueness (cf.[11, 12, 15, 16, 19, 20, 33, 34]).”
High mode transport noise improves vorticity blow-up control in 3D Navier–Stokes equations (2019, methodology): “reached due to regularity problems of the solution to the Kolmogorov equation; (3) Markov selections with the Strong Feller property - elaborating a preliminary result of [10] - have been constructed [24], proving a continous dependence result on initial conditions, due to noise, which has no counte”
Local martingale and pathwise solutions for an abstract fluids model (2010, methodology): “with our methods. 1 2 A.Debussche,N.Glatt-HoltzandR.Temam Let us recall at this point that the theory of the related stochastic Navier-Stokesequations haveundergone substantial developments; see e.g. [3, 48, 9, 8, 17, 12, 2, 5, 6, 19, 10, 36, 37, 45, 18, 23]. However to emphasizethe notable diﬀerenc”
Martingale solutions and Markov selections for stochastic partial differential equations (2009, methodology): “Our construction of almost sure Markov selections differs from that in [8] in the following ways: Our abstract Markov selection theorem is stated in a Polish space so that it can be used to deal with more general stochastic equations.”
Stochastic tamed 3D Navier–Stokes equations: existence, uniqueness and ergodicity (2008, background): “In the case of two dimension, they also obtained the existence and pathwise uniqueness ofL2-continuous adapted solutions.”