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Utpal Manna

Publications and source records attributed to Utpal Manna.

17 recordsLinked to original sources
arXiv2026

Small-Time Asymptotic Behavior of the Stochastic Landau--Lifshitz--Baryakhtar Equation

We establish a small-time large deviation principle for the stochastic Landau--Lifshitz--Baryakhtar equation using the framework of exponential equivalence. This result characterizes the asymptotic behavior of the solution on very short time scales. In particular, it shows that, as the stochastic thermal fluctuations become small, the magnetization remains exponentially concentrated near its initial state, reflecting the short-time stability of the magnetization dynamics. The associated rate function provides a quantitative measure of deviations from the initial state and the resulting short-time stability.

math.AP↗
arXiv2023

Stochastic control of the Landau-Lifshitz-Gilbert equation

We consider the stochastic Landau-Lifshitz-Gilbert equation in dimension 1. A control process is added to the effective field. We show the existence of a weak martingale solution for the resulting controlled equation. The proof uses the classical Faedo-Galerkin approximation, along with the Jakubowski version of the Skorohod Theorem. We then show pathwise uniqueness for the obtained solution, which is then coupled with the theory of Yamada and Watanabe to give the existence of a unique strong solution. We then show, using some semigroup techniques that the obtained solution satisfies the maximum regularity. We then show the existence of an optimal control. A main ingredient of the proof is using the compact embedding of a space into itself, albeit with the weak topology.

math.PR↗
arXiv2023

Optimal control of the stochastic Landau-Lifshitz-Bloch equation

We consider the stochastic Landau-Lifshitz-Bloch equation in dimensions 1,2,3, perturbed by a real-valued Wiener process. We consider a Suslin space-valued control process with a general control operator, which can depend on both the control and the corresponding solution. We reduce the equation to a more general (relaxed) form, where the concept of Young measures is used. We then show the existence of a weak martingale solution to the controlled equation (relaxed). In the second part of the work, we show that for a general lower semicontinuous cost functional, the problem admits a weak relaxed optimal control. This is done using the theory of Young measures. Moreover, pathwise uniqueness is shown (for dimensions 1,2), which implies the existence of a strong solution.

math.PR↗
arXiv2023

Well-posedness and large deviations for the stochastic Landau Lifshitz Bloch equation

The stochastic Landau-Lifshitz-Bloch equation in dimensions 1; 2; and 3 perturbed by pure jump noise is considered in the Marcus canonical form. A proof for existence of a martingale solution is given. The proof uses the Faedo-Galerkin approximation; which is followed by compactness and tightness arguments. This is followed by use of the Jakubowski's version of the Skorohod Theorem. Pathwise uniqueness and the theory of Yamada and Watanabe give the existence of a strong solution for dimensions 1 and 2. A weak convergence method is later used to establish a Wentzell-Freidlin type large deviation principle for the small noise asymptotic of solutions for dimensions 1 and 2.

math.PR↗
arXiv2018

Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise

We construct a martingale solution of the stochastic nonlinear Schrödinger equation with a multiplicative noise of jump type in the Marcus canonical form. The problem is formulated in a general framework that covers the subcritical focusing and defocusing stochastic NLS in $H^1$ on compact manifolds and on bounded domains with various boundary conditions. The proof is based on a variant of the Faedo-Galerkin method. In the formulation of the approximated equations, finite dimensional operators derived from the Littlewood-Paley decomposition complement the classical orthogonal projections to guarantee uniform estimates. Further ingredients of the construction are tightness criteria in certain spaces of cadlag functions and Jakubowski's generalization of the Skorohod-Theorem to nonmetric spaces.

math.PR↗
arXiv2017

Stochastic Control of Tidal Dynamics Equation with Levy Noise

In this work we first present the existence, uniqueness and regularity of the strong solution of the tidal dynamics model perturbed by Lévy noise. Monotonicity arguments have been exploited in the proofs. We then formulate a martingale problem of Stroock and Varadhan associated to an initial value control problem and establish existence of optimal controls.

math.PR↗
arXiv2017

Internal Stabilization of a Class of Parabolic Integro-Differential Equations: Application to Viscoelastic Fluids

In this paper, we prove the stabilizability of abstract Parabolic Integro-Differential Equations (PIDE) in a Hilbert space with decay rate $e^{-γt} $ for certain $γ> 0,$ by means of a finite dimensional controller in the feedback form. We determine a linear feedback law which is obtained by solving an algebraic Riccati equation. To prove the existence of the Riccati operator, we consider a linear quadratic optimal control problem with unbounded observation operator. The abstract theory of stabilization developed here is applied to specific problems related to viscoelastic fluids, e.g. Oldroyd B model and Jeffreys model.

math.OC↗
arXiv2017

Higher Order Regularity and Blow-up Criterion for Semi-dissipative and Ideal Boussinesq Equations

In this paper we establish local-in-time existence and uniqueness of strong solutions in $H^s$ for $s > \frac{n}{2}$ to the viscous, zero thermal-diffusive Boussinesq equations in $\mathbb{R}^n , n = 2,3$. Beale-Kato-Majda type blow-up criterion has been established in three-dimensions with respect to the $BMO$-norm of the vorticity. We further prove the local-in-time existence and blow-up criterion for non-viscous and fully ideal Boussinesq systems. Commutator estimates due to Kato and Ponce (1988) \cite {KP} and Fefferman et. al. (2014) \cite {Fe} play important roles in the calculations.

math.AP↗
arXiv2017

On The Two and Three Dimensional Ideal Magnetic Bénard Problem - Local Existence and Blow-up Criterion

In this paper, we consider the ideal magnetic Bénard problem in both two and three dimensions and prove local-in-time existence and uniqueness of strong solutions in $H^s$ for $s > \frac{n}{2}+1, n = 2,3$. In addition, a necessary condition is derived for singularity development with respect to the $BMO$-norm of the vorticity and electrical current, generalising the Beale-Kato-Majda condition for ideal hydrodynamics.

math.AP↗
arXiv2017

Strong Solutions of Stochastic Models for Viscoelastic Flows of Oldroyd Type

In this work we study stochastic Oldroyd type models for viscoelastic fluids in $\mathbb{R}^d, d= 2, 3$. We show existence and uniqueness of strong local maximal solutions when the initial data are in $H^s$ for $s>d/2, d= 2, 3$. Probabilistic estimate of the random time interval for the existence of a local solution is expressed in terms of expected values of the initial data.

math.PR↗
arXiv2017

Existence of weak martingale solution of Nematic Liquid Crystals driven by Pure Jump Noise

In this work we consider a stochastic evolution equation which describes the system governing the nematic liquid crystals driven by a pure jump noise. The existence of a martingale solution is proved for both 2D and 3D cases. The construction of the solution is based on the classical Faedo-Galerkin approximation, compactness method and the Jakubowski's version of the Skorokhod representation theorem for non-metric spaces. We prove the solution is pathwise unique and further establish the existence of a strong solution in the 2D case.

math.PR↗
arXiv2014

Stochastic Navier-Stokes Equations in Unbounded Channel Domains

In this paper we prove the existence and uniqueness of path-wise strong solution to stochastic viscous flow in unbounded channels with multiple outlets using local monotonicity arguments. We devise a construction for solvability using a stochastic basic vector field.

math.PR↗
arXiv2010

Shell Model of Turbulence Perturbed by Lévy Noise

In this work we prove the existence and uniqueness of the strong solution of the shell model of turbulence perturbed by Lévy noise. The local monotonicity arguments have been exploited in the proofs.

math.PR↗
arXiv2008

Lyapunov Functionals and Local Dissipativity for the Vorticity Equation in L^p and Besov Spaces

In this paper we establish the local Lyapunov property of certain L^p and Besov norms of the vorticity fields. We have resolved in part, a certain open problem posed by Tosio Kato for the three dimensional Navier Stokes equation by studying the vorticity equation. The local dissipativity of the sum of linear and non-linear operators of the vorticity equation is established. One of the main techniques used here is Littlewood-Paley analysis.

math.AP↗
arXiv2008

Stochastic 2-D Navier-Stokes Equation with Artificial Compressibility

In this paper we study the stochastic Navier-Stokes equation with artificial compressibility. The main results of this work are the existence and uniqueness theorem for strong solutions and the limit to incompressible flow. These results are obtained by utilizing a local monotonicity property of the sum of the Stokes operator and the nonlinearity.

math.PR↗