arXiv subjects

Kush Kinra

Publications and source records attributed to Kush Kinra.

At least 19 recordsLinked to original sources

arXiv2026

Boundary control for optimal mixing by two-dimensional second-grade fluids

We study optimal boundary mixing of a nondiffusive scalar transported by a two-dimensional incompressible second-grade fluid. The control is the tangential traction in a Navier-slip boundary condition, and the principal objective is the terminal $(H^1(\Omega))'$ mix-norm, supplemented by a quadratic control cost and an optional enstrophy reward. The second-grade constitutive law introduces a spatially filtered acceleration and a generalized vorticity; after lifting the nonhomogeneous boundary data, both $\mathbf{g}$ and $\partial_t\mathbf{g}$ enter the state equation, where $\mathbf{g}$ is the boundary control variable. We prove global state well-posedness for a passive scalar and well-posedness on a common, control-independent local interval for an active scalar. We then establish existence of an optimal control, directional differentiability of the control-to-state map in the topology required by the terminal objective, and a weak backward adjoint formulation. A duality identity yields the first-order variational inequality. In the absence of the friction coefficient $\beta$ and enstrophy reward weight $\zeta$ (that is, $\beta=\zeta=0$), the additional adjoint regularity implies uniqueness of the optimal control for all sufficiently large control penalties $\gamma$.

math.OC↗

arXiv2026

On Continuous Data Assimilation for a class of 2D and 3D stochastic non-Newtonian fluids of differential type

Continuous data assimilation (CDA) techniques, most notably the nudging approach proposed by Azouani, Olson, and Titi (AOT), have been shown to be very successful in deterministic frameworks for achieving long-time synchronization between an approximate state and true state. In this note, we develop and study a CDA scheme for a class of stochastic non-Newtonian fluids, namely third-grade fluids, subject to either additive or multiplicative Gaussian stochastic forcing in both two- and three-dimensional settings. We establish sufficient criteria on the nudging gain and the observational mesh size that guarantee convergence of the assimilated state toward the underlying stochastic solution. Convergence is proved in the mean-square sense, and, in the case of additive noise, we further obtain almost sure (pathwise) convergence.

math.PR↗

arXiv2026

A note on continuous data assimilation for stochastic convective Brinkman-Forchheimer equations in 2D and 3D

Continuous data assimilation (CDA) methods, such as the nudging algorithm introduced by Azouani, Olson, and Titi (AOT), have proven to be highly effective in deterministic settings for asymptotically synchronizing approximate solutions with observed dynamics. In this note, we introduce and analyze an algorithm for CDA for the two- and three-dimensional stochastic convective Brinkman-Forchheimer equations (CBFEs) driven by either additive or multiplicative Gaussian noise. The model is believed to provide an accurate description when the flow velocity exceeds the regime of validity of Darcy's law and the porosity remains moderately large. We derive sufficient conditions on the nudging parameter and the spatial resolution of observations that ensure convergence of the assimilated solution to the true stochastic flow. We demonstrate convergence in the mean-square sense, and additionally establish pathwise convergence in the presence of additive noise. The CBFEs, also known as Navier-Stokes equations with damping, exhibit enhanced stability properties due to the presence of nonlinear damping term. In particular, we show that nonlinear damping not only enables the implementation of CDA in three dimensions but also yields improved convergence results in two dimensions when compared to the classical Navier-Stokes equations.

math.PR↗

arXiv2026

Rate of convergence of random attractors towards deterministic singleton attractor for a class of non-Newtonian fluids of differential type

In this article, we investigate the long-term dynamics of a class of two- and three-dimensional non-Newtonian fluids of differential type, known as third-grade fluids. We first show that when the external forcing is sufficiently small, the global attractor of the underlying system (which characterizes its asymptotic behavior) reduces to a single point. We then consider the system under stochastic perturbations, specifically infinite-dimensional additive white noise. In this random setting, we do not find conclusive evidence that the corresponding random attractor remains a single point, as in the deterministic case. However, we are able to estimate the rate at which the random attractor approaches the deterministic singleton attractor as the intensity of the stochastic noise tends to zero.

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arXiv2026

Large time behaviour for a class of 2D and 3D stochastic non-Newtonian fluids of differential types: Attractors and invariant measures

This study investigates a stochastic version of a class of non-Newtonian fluids governed by third-grade fluid equations, which exhibit complex and highly nonlinear dynamics. In particular, we address the random dynamics and asymptotic behavior of stochastic third-grade fluid equations (STGFEs) driven by a \emph{linear multiplicative It\^o-type white noise} on general domains $\mathbb{Q}\subseteq\mathbb{R}^d$, $d\in\{2,3\}$. We first prove that the non-autonomous STGFEs generate a continuous non-autonomous random dynamical system $\Phi$, and we establish the existence of a pullback absorbing set. Using compact Sobolev embeddings on bounded domains and uniform tail estimates on unbounded domains, we show the pullback asymptotic compactness of $\Phi$, which leads to the existence of pullback random attractors that are compact and attracting in $\mathbb{L}^2(\mathbb{Q})$. As a consequence, we demonstrate the existence of an invariant measure associated with the STGFEs and, exploiting the linear multiplicative structure of the noise along with the exponential stability of solutions, we prove uniqueness of the invariant measure in the case of zero external forcing. These results are entirely new for STGFEs on general domains, and, in particular, the existence of pullback random attractors with linear multiplicative noise is obtained here for the first time. We further note that, unlike Stratonovich noise, which is widely used in the literature to study random attractors, It\^o noise is more appropriate for domains that do not satisfy the Poincar\'e inequality. Overall, this work resolves several open problems regarding random attractors, invariant measures, and ergodicity for stochastic third-grade fluids on general unbounded domains $\mathbb{Q}\subseteq\mathbb{R}^d$, $d\in\{2,3\}$.

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arXiv2026

Optimal control problem associated with three-dimensional critical convective Brinkman-Forchheimer equations

In this article, we are concerned about the velocity tracking optimal control problem for 3D critical convective Brinkman-Forchheimer equations defined on a simply connected bounded domain $\mathbb{D}\subset\mathbb{R}^3$ with $\mathrm{C}^2$-boundary $\partial\mathbb{D}$. The control is introduced through an external force. The objective is to optimally minimize a velocity tracking cost functional, for which the velocity vector field is oriented towards a target velocity. Most importantly, we are concerned about the first-order necessary optimality conditions for above-mentioned optimal control problem which is the main challenging task of this article. To overcome the difficulties related to the differentiability of the control-to-state mapping, consequence of the lack of regularity of the state variable on bounded domains, we first establish some intermediate optimality conditions and then pass to the limit.

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arXiv2025

Global-in-time optimal control of stochastic third-grade fluids with additive noise

In this article, we address the velocity tracking control problem for a class of stochastic non-Newtonian fluids. More precisely, we consider the stochastic third-grade fluid equation perturbed by infinite-dimensional additive white noise and defined on the two-dimensional torus $\mathbb{T}^2$. The control acts as a distributed random external force. Taking an \emph{infinite-dimensional Ornstein-Uhlenbeck process}, the stochastic system is converted into an equivalent pathwise deterministic one, which allows to show the well-posedness of the original stochastic system globally in time. The state being a stochastic process with sample paths in $\mathrm{L}^\infty(0,T;\mathbb{H}^3(\mathbb{T}^2))$ and finite moments can be controlled in an optimal way. Namely, we establish the existence and uniqueness of solutions to the corresponding linearized state and adjoint equations. Furthermore, we derive an appropriate stability result for the state equation and verify that the G\^ateaux derivative of the control-to-state mapping coincides with the solution of the linearized state equation. Finally, we establish the first-order optimality conditions and prove the existence of an optimal solution.

math.PR↗

arXiv2025

Random dynamics and invariant measures for a class of non-Newtonian fluids of differential type on 2D and 3D Poincar\'e domains

In this article, we consider a class of incompressible stochastic third-grade fluids (non-Newtonian fluids) equations on two- as well as three-dimensional Poincar\'e domains $\mathcal{O}$ (which may be bounded or unbounded). Our aims are to study the well-posedness and asymptotic analysis for the solutions of the underlying system. Firstly, we prove that the underlying system defined on $\mathcal{O}$ has a unique weak solution (in the analytic sense) under Dirichlet boundary condition and it also generates random dynamical system $\Psi$. Secondly, we consider the underlying system on bounded domains. Using the compact Sobolev embedding $\mathbb{H}^1(\mathcal{O}) \hookrightarrow\mathbb{L}^2(\mathcal{O})$, we prove the existence of a unique random attractor for the underlying system on bounded domains with external forcing in $\mathbb{H}^{-1}(\mathcal{O})+\mathbb{W}^{-1,\frac{4}{3}}(\mathcal{O})$. Thirdly, we consider the underlying system on unbounded Poincar\'e domains with external forcing in $\mathbb{L}^{2}(\mathcal{O})$ and show the existence of a unique random attractor. In order to obtain the existence of a unique random attractor on unbounded domains, due to the lack of compact Sobolev embedding $\mathbb{H}^1(\mathcal{O}) \hookrightarrow\mathbb{L}^2(\mathcal{O})$, we use the uniform-tail estimates method which helps us to demonstrate the asymptotic compactness of $\Psi$. Note that due to the presence of several nonlinear terms in the underlying system, we are not able to use the energy equality method to obtain the asymptotic compactness of $\Psi$ in unbounded domains, which makes the analysis of this work in unbounded domains more difficult and interesting. Finally, as a consequence of the existence of random attractors, we address the existence of invariant measures for underlying system.

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arXiv2025

Inverse problem for wave equation of memory type with acoustic boundary conditions: Global solvability

In this article, we study the one-dimensional inverse problem of determining the memory kernel by the integral overdetermination condition for the direct problem of finding the velocity potential and the displacement of boundary points. A wave equation with initial and acoustic boundary conditions in media with dispersion is used as a mathematical model. The inverse problem is reduced to an equivalent problem with homogeneous boundary conditions for the system of integro-differential equations. Using the technique of estimating integral equations and the contraction mappings principle in Sobolev spaces, the global existence and uniqueness theorem for the inverse problem is proved.

math.AP↗

arXiv2024

Random dynamics of solutions for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded Poincar\'e domains

In this article, we consider a novel version of three-dimensional (3D) globally modified Navier-Stokes (GMNS) system introduced by [Caraballo et. al., Adv. Nonlinear Stud. (2006), 6:411-436], which is very significant from the perspective of deterministic as well as stochastic partial differential equations. Our focus is on examining a stochastic version of the suggested 3D GMNS equations that are perturbed by an infinite-dimensional additive noise. We can consider a rough additive noise (Lebesgue space valued) with this model, which is not appropriate to consider with the system presented in [Caraballo et. al., Adv. Nonlinear Stud. (2006), 6:411-436]. One of the technical problems associated with the rough noise is overcome by the use of the corresponding Cameron-Martin (or reproducing kernel Hilbert) space. This article aims to accomplish three objectives. Firstly, we establish the existence and uniqueness of weak solutions (in the analytic sense) of the underlying stochastic system using a Doss-Sussman transformation and a primary ingredient Minty-Browder technique. Secondly, we demonstrate the existence of random attractors for the underlying stochastic system in the natural space of square integrable divergence-free functions. Finally, we show the existence of an invariant measure for the underlying stochastic system for any viscosity coefficient $ \nu > 0 $ and uniqueness of invariant measure for sufficiently large $\nu$ by using the exponential stability of solutions. A validation of the proposed version of 3D GMNS equations has also been discussed in the appendix by establishing that the sequence of weak solutions of 3D GMNS equations converges to a weak solution of 3D Navier-Stokes equations as the modification parameter goes to infinity.

math.PR↗

arXiv2024

Non-uniqueness of H\"older continuous solutions for stochastic Euler and Hypodissipative Navier-Stokes equations

We construct infinitely many H\"older continuous, global-in-time, and stationary solutions to the stochastic Euler equations and the hypodissipative Navier-Stokes equations, taking values in the space $C(\mathbb{R};C^{\vartheta})$. For the Euler case, the H\"older exponent $\vartheta$ satisfies $0<\vartheta<\frac{5}{7}\beta$ with $0<\beta< \frac{1}{200}$, while for the hypodissipative Navier-Stokes equations, $\beta$ must additionally satisfy $0<\beta< \min\left\{ \frac{2(1-2\alpha)}{21}, \frac{1}{200}\right\}$. The construction relies on a modified stochastic convex integration scheme, which is central to the analysis. This scheme incorporates Beltrami flows as building blocks and carefully tracks inductive estimates, both pathwise and in expectation. These refinements allow us to achieve improved H\"older regularity for solutions to the underlying stochastic equations, advancing the scope of convex integration techniques in the stochastic setting.

math.AP↗

arXiv2024

Existence and asymptotic autonomous robustness of random attractors for three-dimensional stochastic globally modified Navier-Stokes equations on unbounded domains

In this article, we discuss the existence and asymptotically autonomous robustness (AAR) (almost surely) of random attractors for 3D stochastic globally modified Navier-Stokes equations (SGMNSE) on Poincar\'e domains (which may be bounded or unbounded). Our aim is to investigate the existence and AAR of random attractors for 3D SGMNSE when the time-dependent forcing converges to a time-independent function under the perturbation of linear multiplicative noise as well as additive noise. The main approach is to provide a way to justify that, on some uniformly tempered universe, the usual pullback asymptotic compactness of the solution operators is uniform across an infinite time-interval $(-\infty,\tau]$. The backward uniform ``tail-smallness'' and ``flattening-property'' of the solutions over $(-\infty,\tau]$ have been demonstrated to achieve this goal. To the best of our knowledge, this is the first attempt to establish the existence as well as AAR of random attractors for 3D SGMNSE on unbounded domains.

math.PR↗

arXiv2023

Feedback stabilization of Convective Brinkman-Forchheimer Extended Darcy equations

In this article, the following controlled convective Brinkman-Forchheimer extended Darcy (CBFeD) system is considered in a $d$-dimensional torus: \begin{align*} \frac{\partial\boldsymbol{y}}{\partial t}-\mu \Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta\vert \boldsymbol{y}\vert ^{r-1}\boldsymbol{y}+\gamma\vert \boldsymbol{y}\vert ^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\ \nabla\cdot\boldsymbol{y}=0, \end{align*} where $d\in\{2,3\}$, $\mu,\alpha,\beta>0$, $\gamma\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$. We prove the exponential stabilization of CBFeD system by finite- and infinite-dimensional feedback controllers. The solvability of the controlled problem is achieved by using the abstract theory of $m$-accretive operators and density arguments. As an application of the above solvability result, by using infinite-dimensional feedback controllers, we demonstrate exponential stability results such that the solution preserves an invariance condition for a given closed and convex set. By utilizing the unique continuation property of controllability for finite-dimensional systems, we construct a finite-dimensional feedback controller which exponentially stabilizes CBFeD system locally, where the control is localized in a smaller subdomain. Furthermore, we establish the local exponential stability of CBFeD system via proportional controllers.

math.OC↗

arXiv2023

Wong-Zakai approximation for a class of SPDEs with fully local monotone coefficients and its application

In this article, we establish the \textsl{Wong-Zakai approximation} result for a class of stochastic partial differential equations (SPDEs) with fully local monotone coefficients perturbed by a multiplicative Wiener noise. This class of SPDEs encompasses various fluid dynamic models and also includes quasi-linear SPDEs, the convection-diffusion equation, the Cahn-Hilliard equation, and the two-dimensional liquid crystal model. It has been established that the class of SPDEs in question is well-posed, however, the existence of a unique solution to the associated approximating system cannot be inferred from the solvability of the original system. We employ a Faedo-Galerkin approximation method, compactness arguments, and Prokhorov's and Skorokhod's representation theorems to ensure the existence of a \textsl{probabilistically weak solution} for the approximating system. Furthermore, we also demonstrate that the solution is pathwise unique. Moreover, the classical Yamada-Watanabe theorem allows us to conclude the existence of a \textsl{probabilistically strong solution} (analytically weak solution) for the approximating system. Subsequently, we establish the Wong-Zakai approximation result for a class of SPDEs with fully local monotone coefficients. We utilize the Wong-Zakai approximation to establish the topological support of the distribution of solutions to the SPDEs with fully local monotone coefficients. Finally, we explore the physically relevant stochastic fluid dynamics models that are covered by this work's functional framework.

math.PR↗

arXiv2023

Backward uniqueness of 2D and 3D convective Brinkman-Forchheimer equations and its applications

In this work, we consider the two- and three-dimensional convective Brinkman-Forchheimer (CBF) equations (or damped Navier--Stokes equations) on a torus $\mathbb{T}^d,$ $d\in\{2,3\}$: $$ \frac{\partial \boldsymbol{u}}{\partial t}-\mu \Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0,$$ where $\mu,\alpha,\beta>0$ and $r\in[1,\infty)$ is the absorption exponent. For $d=2,r\in[1,\infty)$ and $d=3,r\in[3,\infty)$ ($2\beta\mu\geq 1$ for $d=r=3$), we first show the backward uniqueness of deterministic CBF equations by exploiting the logarithmic convexity property and the global solvability results available in the literature. As a direct consequence of the backward uniqueness result, we first derive the approximate controllability with respect to the initial data (viewed as a start controller). Secondly, we apply the backward uniqueness results in the attractor theory to show the zero Lipschitz deviation of the global attractors for 2D and 3D CBF equations. By an application of log-Lipschitz regularity, we prove the uniqueness of Lagrangian trajectories in 2D and 3D CBF flows and the continuity of Lagrangian trajectories with respect to the Eulerian initial data. Finally, we consider the stochastic CBF equations with a linear multiplicative Gaussian noise. For $d=2,r\in[1,\infty)$ and $d=3,r\in[3,5]$ ($2\beta\mu\geq 1$ for $d=r=3$), we show the pathwise backward uniqueness as well as approximate controllability via starter controller results.

math.AP↗

arXiv2023

Theory of weak asymptotic autonomy of pullback stochastic weak attractors and its applications to 2D stochastic Euler equations driven by multiplicative noise

The two dimensional stochastic Euler equations (EE) perturbed by a linear multiplicative noise of It\^o type on the bounded domain $\mathcal{O}$ have been considered in this work. Our first aim is to prove the existence of \textsl{global weak (analytic) solutions} for stochastic EE when the divergence free initial data $\boldsymbol{u}^*\in\mathbb{H}^1(\mathcal{O})$, and the external forcing $\boldsymbol{f}\in\mathrm{L}^2_{\mathrm{loc}}(\mathbb{R};\mathbb{H}^1(\mathcal{O}))$. In order to prove the existence of weak solutions, a vanishing viscosity technique has been adopted. In addition, if $\mathrm{curl}\ \boldsymbol{u}^*\in\mathrm{L}^{\infty}(\mathcal{O})$ and $\mathrm{curl}\ \boldsymbol{f}\in\mathrm{L}^{\infty}_{\mathrm{loc}}(\mathbb{R};\mathrm{L}^{\infty}(\mathcal{O}))$, we establish that the global weak (analytic) solution is unique. This work appears to be the first one to discuss the existence and uniqueness of global weak (analytic) solutions for stochastic EE driven by linear multiplicative noise. Secondly, we prove the existence of a \textsl{pullback stochastic weak attractor} for stochastic \textsl{non-autonomous} EE using the abstract theory available in the literature. Finally, we propose an abstract theory for \textsl{weak asymptotic autonomy} of pullback stochastic weak attractors. Then we consider the 2D stochastic EE perturbed by a linear multiplicative noise as an example to discuss that how to prove the weak asymptotic autonomy for concrete stochastic partial differential equations. As EE do not contain any dissipative term, the results on attractors (deterministic and stochastic) are available in the literature for dissipative (or damped) EE only. Since we are considering stochastic EE without dissipation, all the results of this work for 2D stochastic EE perturbed by a linear multiplicative noise are totally new.

math.PR↗

arXiv2023

2D and 3D convective Brinkman-Forchheimer equations perturbed by a subdifferential and applications to control problems

The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential \begin{equation*} \frac{\partial \boldsymbol{y}}{\partial t}-\mu \Delta\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+\alpha\boldsymbol{y}+\beta|\boldsymbol{y}|^{r-1}\boldsymbol{y}+\nabla p+\Psi(\boldsymbol{y})\ni\boldsymbol{g},\ \nabla\cdot\boldsymbol{y}=0, \end{equation*} in a $d$-dimensional torus is considered in this work, where $d\in\{2,3\}$, $\mu,\alpha,\beta>0$ and $r\in[1,\infty)$. For $d=2$ with $r\in[1,\infty)$ and $d=3$ with $r\in[3,\infty)$ ($2\beta\mu\geq 1$ for $d=r=3$), we establish the existence of \textsf{\emph{a unique global strong solution}} for the above multi-valued problem with the help of the \textsf{\emph{abstract theory of $m$-accretive operators}}. %for nonlinear differential equations of accretive type in Banach spaces. Moreover, we demonstrate that the same results hold \textsf{\emph{local in time}} for the case $d=3$ with $r\in[1,3)$ and $d=r=3$ with $2\beta\mu<1$. We explored the $m$-accretivity of the nonlinear as well as multi-valued operators, Yosida approximations and their properties, and several higher order energy estimates in the proofs. For $r\in[1,3]$, we {quantize (modify)} the Navier-Stokes nonlinearity $(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}$ to establish the existence and uniqueness results, while for $r\in[3,\infty)$ ($2\beta\mu\geq1$ for $r=3$), we handle the Navier-Stokes nonlinearity by the nonlinear damping term $\beta|\boldsymbol{y}|^{r-1}\boldsymbol{y}$. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.

math.OC↗

arXiv2022

Asymptotically autonomous robustness in Probability of non-autonomous random attractors for stochastic convective Brinkman-Forchheimer equations on $\mathbb{R}^3$

This article is concerned with the \emph{asymptotically autonomous robustness} (almost surely and in probability) of non-autonomous random attractors for two stochastic versions of 3D convective Brinkman-Forchheimer (CBF) equations defined on the whole space $\mathbb{R}^3$: $$\frac{\partial\boldsymbol{v}}{\partial t}-\mu \Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v} +\alpha\boldsymbol{v}+ \beta|\boldsymbol{v}|^{r-1}\boldsymbol{v}+\nabla p=\boldsymbol{f}(t)+``\mbox{stochastic terms}",\quad \nabla\cdot\boldsymbol{v}=0,$$ with initial and boundary vanishing conditions, where $\mu,\alpha,\beta >0$, $r\geq1$ and $\boldsymbol{f}(\cdot)$ is a given time-dependent external force field. By the asymptotically autonomous robustness of a non-autonomous random attractor $ \mathscr{A}=\{ \mathscr{A}(\tau,\omega): \tau\in\mathbb{R}, \omega\in\Omega\}$ we mean its time-section $\mathscr{A}(\tau,\omega)$ is robust to a time-independent random set as time $\tau$ tends to negative infinity according to the Hausdorff semi-distance of the underlying space. Our goal is to study this topic, almost surely and in probability, for the non-autonomous 3D CBF equations when the stochastic term is a linear multiplicative or additive noise, and the time-dependent forcing converges towards a time-independent function. Our main results contain two cases: i) $r\in(3,\infty)$ with any $\beta,\mu>0$; ii) $r=3$ with $2\beta\mu\geq1$. The main procedure to achieve our goal is how to justify that the usual pullback asymptotic compactness of the solution operators is uniform on some \emph{uniformly} tempered universes over an \emph{infinite} time-interval $(-\infty,\tau]$. This can be done by a method based on Kuratowski's measure of noncompactness.

math.PR↗