Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one

Mohan, Manil T.

doi:10.1016/j.spa.2020.01.007

Cited by 22 publications

(12 citation statements)

References 33 publications

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“…In this section, we study the LDP for the solutions to the system (3.2) for short time, which in the finite dimensional case is the celebrated Varadhan's large deviation estimate. Short time LDP for solution of the stochastic quasigeostrophic equation with multiplicative noise is obtained in [50], stochastic generalized porous media equations is established in [44] and stochastic 2D Oldroyd models is derived in [33]. We discuss the large deviations for the family {u(ε 2 t) : ε ∈ (0, 1]} of solutions to (3.2)…”

Section: Large Deviations For Short Timementioning

confidence: 99%

Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

Mohan

2020

Stochastics

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This work addresses some asymptotic behavior of solutions to the stochastic convective Brinkman-Forchheimer (SCBF) equations perturbed by multiplicative Gaussian noise in bounded domains. Using a weak convergence approach of Budhiraja and Dupuis, we establish the Laplace principle for the strong solution to the SCBF equations in a suitable Polish space. Then, the Wentzell-Freidlin large deviation principle is derived using the well known results of Varadhan and Bryc. The large deviations for short time are also considered in this work. Furthermore, we study the exponential estimates on certain exit times associated with the solution trajectory of the SCBF equations. Using contraction principle, we study these exponential estimates of exit times from the frame of reference of Freidlin-Wentzell type large deviations principle. This work also improves several LDP results available in the literature for the tamed Navier-Stokes equations as well as Navier-Stokes equations with damping in bounded domains.

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Section: Large Deviations For Short Timementioning

confidence: 99%

Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

Mohan

2020

Stochastics

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“…2 + 1 and r ≥ 2, since the operator G(u) = −µA(u) + B(u) + βC(u) satisfies the local monotonicity condition (2.26) as well as demicontinuity condition (Lemma 2.7), one can establish the existence of a strong solution by a localized version (stochastic generalization) of the Minty-Browder technique (see [46,59], etc for similar techniques). In order to establish the energy equality (Itô's formula) (3.4) for 2 ≤ r ≤ pd d−p (2 ≤ r < ∞ for d = 2), one can use the recent result in Theorem 2.1, [26].…”

Section: Proof For P ≥ Dmentioning

confidence: 99%

“…Large deviation theory gained its attention in the past few decades due to its wide range of applications in the areas like mathematical finance, risk management, fluid mechanics, statistical mechanics, quantum physics, etc (cf. [7,11,13,16,17,20,22,46,49,50,55,57,59,61,64,65,66], etc and the references therein). The small time large deviation principle (LDP) examines the asymptotic behavior of the tails of a family of probability measures at a given point in space when the time is very small.…”

Section: Introductionmentioning

confidence: 99%

On the small time asymptotics of stochastic Ladyzhenskaya-Smagorinsky equations with damping perturbed by multiplicative noise

Mohan

2021

Preprint

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The Ladyzhenskaya-Smagorinsky equations model turbulence phenomena, and are given byIn this work, we consider the stochastic Ladyzhenskaya-Smagorinsky equations with the damping αu+β|u| r−2 u, for r ≥ 2 (α, β ≥ 0), subjected to multiplicative Gaussian noise. We show the local monotoincity (p ≥ d 2 + 1, r ≥ 2) as well as global monotonicity (p ≥ 2, r ≥ 4) properties of the linear and nonlinear operators, which along with an application of stochastic version of Minty-Browder technique imply the existence of a unique pathwise strong solution. Then, we discuss the small time asymptotics by studying the effect of small, highly nonlinear, unbounded drifts (small time large deviation principle) for the stochastic Ladyzhenskaya-Smagorinsky equations with damping.

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“…Apart from these, there are several articles on this model involving both semidiscrete and fully discrete analysis, based on, for example, large-time numerical schemes, asymptotic analysis, stability analysis of various implicit/explicit fully discrete schemes, projection methods, stabilized methods and penalty methods, see [1, 14, 17, 26-28, 33, 34] and references therein. Very recently, we see works on the stochastic model [20,21] and on modified characteristic FEM [32] for the problem.…”

Section: Introductionmentioning

confidence: 99%

On a three step two‐grid finite element method for the Oldroyd model of order one

Bir

Goswami

2021

Z Angew Math Mech

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In this work, an optimal error analysis of a three step two-grid method for the equations of motion arising in the 2𝐷 Oldroyd model of order one is discussed. The model, which can be thought of as an integral perturbation of Navier-Stokes equations (NSE), represents linear viscoelastic fluid flows. This non-linear model is analyzed here using a three step numerical scheme. In the first step the problem is solved on a coarse grid, and we use this course grid solution to linearize the problem and solve it in second step, on a finer grid. Third step is a correcting step done on the finer grid. Optimal error estimates for the velocity in 𝐿 ∞ (𝐋 2 ) and 𝐿 ∞ (𝐇 1 )-norms and for the pressure in 𝐿 ∞ (𝐿 2 )-norm in the semidiscrete case are established. Estimates are shown to be uniform under the uniqueness assumption. Then, based on backward Euler method, a completely discrete scheme is analyzed and optimal a priori error estimates are derived. All the analysis are carried out for the non-smooth initial data. Finally we present some numerical results to validate our theoretical results. These examples show that the three step two-grid method is efficient than solving a nonlinear problem directly, as is expected. K E Y W O R D Sbackward Euler method, Oldroyd model of order one, optimal and uniform error estimates, two-grid method, uniform estimates 1

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Well posedness, large deviations and ergodicity of the stochastic 2D Oldroyd model of order one

Cited by 22 publications

References 33 publications

Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

Moderate deviation principle for the 2D stochastic convective Brinkman–Forchheimer equations

On the small time asymptotics of stochastic Ladyzhenskaya-Smagorinsky equations with damping perturbed by multiplicative noise

On a three step two‐grid finite element method for the Oldroyd model of order one

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