The asymptotic analysis of a class of stochastic partial differential equations (SPDEs) with full... more The asymptotic analysis of a class of stochastic partial differential equations (SPDEs) with fully locally monotone coefficients covering a large variety of physical systems, a wide class of quasilinear SPDEs and a good number of fluid dynamic models is carried out in this work. The aim of this work is to develop the large deviation theory for small Gaussian as well as Poisson noise perturbations of the above class of SPDEs. We establish a Wentzell-Freidlin type large deviation principle for the strong solutions to such SPDEs perturbed by Lévy noise in a suitable Polish space using a variational representation (based on a weak convergence approach) for nonnegative functionals of general Poisson random measures and Brownian motions. The well-posedness of an associated deterministic control problem is established by exploiting pseudo-monotonicity arguments and the stochastic counterpart is obtained by an application of Girsanov's theorem.
The Laplace principle for the strong solution of the stochastic shell model of turbulence perturb... more The Laplace principle for the strong solution of the stochastic shell model of turbulence perturbed by Lévy noise is established in a suitable Polish space using weak convergence approach. The large deviation principle is proved using the well known results of Varadhan and Bryc.
In this paper, we consider an inverse problem for three dimensional viscoelastic fluid flow equat... more In this paper, we consider an inverse problem for three dimensional viscoelastic fluid flow equations, which arises from the motion of Kelvin-Voigt fluids in bounded domains (a hyperbolic type problem). This inverse problem aims to reconstruct the velocity and kernel of the memory term simultaneously, from the measurement described as the integral over determination condition. By using the contraction mapping principle in an appropriate space, a local in time existence and uniqueness result for the inverse problem of Kelvin-Voigt fluids are obtained. Furthermore, using similar arguments, a global in time existence and uniqueness results for an inverse problem of Oseen type equations are also achieved.
The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous... more The convective Brinkman-Forchheimer (CBF) equations describe the motion of incompressible viscous fluids through a rigid, homogeneous, isotropic, porous medium and is given by \begin{document}$ \partial_t{\boldsymbol{u}}-\mu \Delta{\boldsymbol{u}}+({\boldsymbol{u}}\cdot\nabla){\boldsymbol{u}}+\alpha{\boldsymbol{u}}+\beta|{\boldsymbol{u}}|^{r-1}{\boldsymbol{u}}+\nabla p = {\boldsymbol{f}},\ \nabla\cdot{\boldsymbol{u}} = 0. $\end{document} In this work, we consider some distributed optimal control problems like total energy minimization, minimization of enstrophy, etc governed by the two dimensional CBF equations with the absorption exponent \begin{document}$ r = 1,2 $\end{document} and \begin{document}$ 3 $\end{document}. We show the existence of an optimal solution and the first order necessary conditions of optimality for such optimal control problems in terms of the Euler-Lagrange system. Furthermore, for the case \begin{document}$ r = 3 $\end{document}, we show the second order n...
The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic... more The long time behavior of Wong-Zakai approximations of 2D as well as 3D non-autonomous stochastic convective Brinkman-Forchheimer (CBF) equations with nonlinear diffusion terms on bounded and unbounded (R for d = 2, 3) domains is discussed in this work. To establish the existence of random pullback attractors, the concept of asymptotic compactness (AC) is used. In bounded domains, AC is proved via compact Sobolev embeddings. In unbounded domains, due to the lack of compact embeddings, the ideas of energy equations and uniform tail estimates are exploited to prove AC. In the literature, CBF equations are also known as Navier-Stokes equations (NSE) with damping, and it is interesting to see that the modification in NSE by linear and nonlinear damping provides better results than that available for NSE. The presence of linear damping term helps to establish the results in the whole domain R. The nonlinear damping term supports to obtain better results in 3D and also for a large class o...
A Pontryagin maximum principle for an optimal control problem in three dimensional linearized com... more A Pontryagin maximum principle for an optimal control problem in three dimensional linearized compressible viscous flows is established using the Ekeland variational principle. The controls are distributed over a bounded domain, while the state variables are subject to a set of constraints and governed by the linearized compressible Navier-Stokes equations. The maximum principle is of integral-type and obtained for minimizers of a tracking-type integral cost functional.
In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and ad... more In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and address some optimal control problems like total energy minimization, minimization of dissipation of energy of the flow, etc. We also examine an another control problem which is similar to that of data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is tidal dynamics, using optimal control techniques. For all these cases, different distributed optimal control problems are formulated as the minimization of suitable cost functionals subject to the controlled two dimensional tidal dynamics system. The existence of an optimal control as well as the Pontryagin maximum principle for such systems is established and the optimal control is characterized via adjoint variable. The Pontryagin's maximum principle gives the first-order necessary conditions of optimality. We also establish the uniqueness of optimal control in small time inte...
In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and ad... more In this work, we consider the two dimensional tidal dynamics equations in a bounded domain and address some optimal control problems like total energy minimization, minimization of dissipation of energy of the flow, etc. We also examine an another interesting control problem which is similar to that of the data assimilation problems in meteorology of obtaining unknown initial data, when the system under consideration is the tidal dynamics, using optimal control techniques. For these cases, different distributed optimal control problems are formulated as the minimization of suitable cost functionals subject to the controlled two dimensional tidal dynamics system. The existence of an optimal control as well as the first order necessary conditions of optimality for such systems are established and the optimal control is characterized via the adjoint variable. We also establish the uniqueness of optimal control in small time interval.
Journal of Dynamics and Differential Equations, 2021
The large time behavior of the deterministic and stochastic three dimensional convective Brinkman... more The large time behavior of the deterministic and stochastic three dimensional convective Brinkman-Forchheimer (CBF) equations ∂ t u − µ∆u + (u • ∇)u + αu + β|u| r−1 u + ∇p = f , ∇ • u = 0, for r ≥ 3 (r > 3, for any µ and β, and r = 3 for 2βµ ≥ 1), in periodic domains is carried out in this work. Our first goal is to prove the existence of global attractors for the 3D deterministic CBF equations. Then, we show the existence of random attractors for the 3D stochastic CBF equations perturbed by small additive smooth noise. Finally, we establish the upper semicontinuity of random attractor for the 3D stochastic CBF equations (stability of attractors), when the coefficient of random perturbation approaches to zero.
In this work we address the analysis of the stationary generalized Burgers-Huxley equation (a non... more In this work we address the analysis of the stationary generalized Burgers-Huxley equation (a nonlinear elliptic problem with anomalous advection) and propose conforming, nonconforming and discontinuous Galerkin finite element methods for its numerical approximation. The existence, uniqueness and regularity of weak solutions is discussed in detail using a Faedo-Galerkin approach and fixed-point theory, and a priori error estimates for all three types of numerical schemes are rigorously derived. A set of computational results are presented to show the efficacy of the proposed methods.
The three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt ... more The three-dimensional viscoelastic fluid flow equations, arising from the motion of Kelvin-Voigt (Kelvin-Voight) fluids in bounded domains is considered in this work. We investigate the long-term dynamics of such viscoelastic fluid flow equations with "fading memory" (non-autonomous). We first establish the existence of an absorbing ball in appropriate spaces for the semigroup defined for the Kelvin-Voigt fluid flow equations of order one with "fading memory" (transformed autonomous coupled system). Then, we prove that the semigroup is asymptotically compact, and hence we establish the existence of a global attractor for the semigroup. We provide estimates for the number of determining modes for both asymptotic as well as for trajectories on the global attractor. Once the differentiability of the semigroup with respect to initial data is established, we show that the global attractor has finite Hausdorff as well as fractal dimensions. We also show the existence o...
ESAIM: Control, Optimisation and Calculus of Variations, 2020
In this work, we consider the controlled two dimensional tidal dynamics equations in bounded doma... more In this work, we consider the controlled two dimensional tidal dynamics equations in bounded domains. A distributed optimal control problem is formulated as the minimization of a suitable cost functional subject to the controlled 2D tidal dynamics equations. The existence of an optimal control is shown and the dynamic programming method for the optimal control of 2D tidal dynamics system is also described. We show that the feedback control can be obtained from the solution of an infinite dimensional Hamilton-Jacobi equation. The non-differentiability and lack of smoothness of the value function forced us to use the method of viscosity solutions to obtain a solution of the infinite dimensional Hamilton-Jacobi equation. The Bellman principle of optimality for the value function is also obtained. We show that a viscosity solution to the Hamilton-Jacobi equation can be used to derive the Pontryagin maximum principle, which give us the first order necessary conditions of optimality. Fina...
In this work we prove the existence and uniqueness of the strong solution to the two-dimensional ... more In this work we prove the existence and uniqueness of the strong solution to the two-dimensional stochastic magneto-hydrodynamic system perturbed by Lévy noise. The local monotonicity arguments have been exploited in the proofs. The existence of a unique invariant measures has been proved using the exponential stability of solutions.
In this article we formulate an optimization problem of minimizing the distance from the uniform ... more In this article we formulate an optimization problem of minimizing the distance from the uniform van der Waerden matrices to orthostochastic matrices of different orders. We find a lower bound for the number of stationary points of the minimization problem, which is connected to the number of possible partitions of a natural number. The existence of Hadamard matrices ensures the existence of global minimum orthostochastic matrices for such problems. The local minimum orthostochastic matrices have been obtained for all other orders except for 11 and 19. We explore the properties of Hadamard, conference and weighing matrices to obtain such minimizing orthostochastic matrices.
Stochastics and Partial Differential Equations: Analysis and Computations, 2018
We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for th... more We establish the existence and uniqueness of a local smooth solution to the Cauchy problem for the Hall-magnetohydrodynamics system that is inviscid, resistive, and forced by multiplicative Lévy noise in the three dimensional space. Moreover, when the initial data is sufficiently small, we prove that the solution exists globally in time in probability; that is, the probability of the global existence of a unique smooth solution may be arbitrarily close to one given the initial data of which its expectation in a certain Sobolev norm is sufficiently small. The proofs go through for the two and a half dimensional case as well. To the best of the authors' knowledge, an analogous result is absent in the deterministic case due to the lack of viscous diffusion, exhibiting the regularizing property of the noise. Our result may also be considered as a physically meaningful special case of the extension of work of Kim (
We formulate a control problem for a distributed parameter system where the state is governed by ... more We formulate a control problem for a distributed parameter system where the state is governed by the compressible Navier-Stokes equations. Introducing a suitable cost functional, the existence of an optimal control is established within the framework of strong solutions in three dimensions.
In this work we prove the existence and uniqueness of pathwise solutions up to a stopping time to... more In this work we prove the existence and uniqueness of pathwise solutions up to a stopping time to the stochastic Euler equations perturbed by additive and multiplicative Lévy noise in two and three dimensions. The existence of a unique maximal solution is also proved.
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Papers by Manil T Mohan