arXiv subjects

Sagar Gautam

Publications and source records attributed to Sagar Gautam.

14 recordsLinked to original sources

arXiv2026

Divergence-Free Approximation in Sobolev and Lebesgue Spaces on General Unbounded Domains with Applications to Energy Equality in Fluid Dynamics

We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovski\u{\i}\ operators to construct such approximations, thereby extending the approximation theory developed for smooth bounded domains and the simultaneous approximation framework introduced by \emph{Fefferman, Hajduk, and Robinson, {Proc. Lond. Math. Soc.} (3) \ {125} (2022), no.~4, 759-777}. As an application, we establish energy equality for Leray-Hopf weak solutions of the incompressible convective Brinkman-Forchheimer (CBF) equations on $\mathbb{R}^d$, $d\in\{2,3\}$ covering both the critical and supercritical regimes. For general unbounded domains, we employ the resolvent operator associated with the Stokes operator, developed by \emph{Farwig, Kozono and Sohr, {Acta Math.}, {195} (2005), 21-53}, to obtain simultaneous approximation results. We also establish a generalized version of the classical Lions-Magenes lemma, which is of independent interest. Finally, by combining this result with the simultaneous approximation framework for unbounded domains, we establish energy equality for weak solutions of the CBF equations on general unbounded domains.

math.AP↗

arXiv2026

A Verification Theorem for an Optimal Control Problem Governed by the Convective Brinkman--Forchheimer Equations

This article establishes a verification theorem for an optimal control problem governed by the two- and three-dimensional convective Brinkman--Forchheimer equations on the $d$-dimensional torus, $d\in\{2,3\}$: $$\frac{\partial\mathfrak{u}}{\partial t} -\mu\Delta\mathfrak{u} +(\mathfrak{u}\cdot\nabla)\mathfrak{u} +\alpha\mathfrak{u} +\beta|\mathfrak{u}|^{r-1}\mathfrak{u} +\nabla\mathfrak{p} =\boldsymbol{f}, \qquad \nabla\cdot\mathfrak{u}=0,$$ where $\mu,\alpha,\beta>0$ and $r\in[1,\infty)$. We derive the Pontryagin maximum principle and develop a verification framework for the associated control problem, a topic that has received comparatively little attention for fluid models of Navier--Stokes type. A major challenge in establishing the verification theorem and the corresponding feedback characterization for the CBF system is that the analysis requires a substantially different regularity framework from that used for the two-dimensional Navier--Stokes equations. In particular, the present approach relies on strong solution theory, a delicate treatment of the nonlinear absorption term, novel estimates in negative-order Sobolev spaces, and continuous dependence estimates in stronger topologies, especially in the three-dimensional setting. A distinctive feature of the present work is that the verification framework is developed not only in two dimensions, but also in the three-dimensional supercritical regime, corresponding to $r\in(3,5]$, and in the critical case $r=3$ under the condition $2\beta\mu\geq1$. Consequently, the feedback characterization and verification arguments can be rigorously justified in both two and three dimensions.

math.OC↗

arXiv2026

Kolmogorov equations for stochastic convective Brinkman-Forchheimer equations forced by L\'evy Noise and its application to infinite horizon problems

This article examines the Kolmogorov equation corresponding to the following stochastic two- and three-dimensional incompressible ($\nabla\cdot\boldsymbol{u}=0$) convective Brinkman-Forchheimer equations, also known as the damped Navier-Stokes equations, driven by L\'evy noise on the torus: \begin{align*} \mathrm{d}\boldsymbol{u}+[-\mu\Delta\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p]\mathrm{d} t =\sqrt{\mathrm{Q}}\mathrm{d}\mathrm{W}+\int_{Z}\sigma(t,z)\widetilde{\pi}(\mathrm{d} t,\mathrm{d} z), \end{align*} where $\mu,\alpha,\beta>0$ are physical constants; $\mathrm{Q}$ is a non-negative, trace-class operator; $\mathrm{W}$ is a cylindrical Wiener process on a Hilbert space; $\sigma$ represents the jump-noise coefficient; $(Z,\mathscr{B}(Z))$ is a measurable space; $\pi$ is a time-homogeneous Poisson random measure; and $\widetilde{\pi}$ denotes its compensator. The main contribution of this work is the establishment of the essential $m$-dissipativity of the corresponding Kolmogorov operator, a property that has received limited attention in the existing literature for systems driven by jump-type noise. \emph{Our main innovation is that, in contrast to traditional techniques which crucially depend on exponential moment estimates, we utilize the intrinsic structure of the absorption term $\alpha\boldsymbol{u}+\beta|\boldsymbol{u}|^{r-1}\boldsymbol{u}$ to dispense with these requirements. This allows us to establish the essential $m$-dissipativity of the Kolmogorov operator without the need for exponential moments.} We apply the developed framework to an infinite-horizon stochastic optimal control problem, demonstrating the solvability of the associated infinite-dimensional Hamilton-Jacobi-Bellman (integro-differential) equation.

math.PR↗

arXiv2026

A domain hemivariational inequality for 2D and 3D convective Brinkman-Forchheimer extended Darcy equations

This paper investigates domain hemivariational inequality problems arising from the non-stationary two- and three-dimensional convective Brinkman-Forchheimer extended Darcy (CBFeD) equations, which describe the flow of viscous incompressible fluids through saturated porous media in bounded domains. These equations may be regarded as generalized Navier-Stokes systems incorporating both damping and pumping mechanisms. For all admissible absorption exponents $r \ge 1 $ and effective viscosity $μ> 0 $, the existence of weak solutions to the non-stationary 2D and 3D CBFeD equations with hemivariational inequalities is established via a regularized Galerkin approximation scheme, based on a suitable regularization of the Clarke subdifferential. A noteworthy aspect of the analysis is that the existence results extend to the three-dimensional non-stationary Navier-Stokes equations. Moreover, under appropriate conditions on the absorption exponent, specifically, $r \ge 1 $ in two dimensions and $ r \ge 3 $ in three dimensions, it is shown that weak solutions satisfy the energy equality. In addition, uniqueness of solutions is proved for $ r \ge 1$ in 2D and $r \ge 3$ in 3D, with the additional requirement $2βμ> 1 $ in the critical case $r = 3 $.

math.AP↗

arXiv2025

A viscosity solution approach to the large deviation principle for stochastic convective Brinkman-Forchheimer equations

This article develops the viscosity solution approach to the large deviation principle for the following two- and three-dimensional stochastic convective Brinkman-Forchheimer equations on the torus $\mathbb{T}^d,\ d\in\{2,3\}$ with small noise intensity: \begin{align*} \mathrm{d}\boldsymbol{u}_n+[-μΔ\boldsymbol{u}_n+ (\boldsymbol{u}_n\cdot\nabla)\boldsymbol{u}_n +α\boldsymbol{u}_n+β|\boldsymbol{u}_n|^{r-1}\boldsymbol{u}_n+\nabla p_n]\mathrm{d} t=\boldsymbol{f}\mathrm{d} t+\frac{1}{\sqrt{n}}\mathrm{Q}^{\frac12}\mathrm{d}\mathrm{W}, \ \nabla\cdot\boldsymbol{u}_n=0, \end{align*} where $μ,α,β>0$, $r\in[1,\infty)$, $\mathrm{Q}$ is a trace class operator and $\mathrm{W}$ is Hilbert-valued calendrical Wiener process. We build our analysis on the framework of Varadhan and Bryc, together with the techniques of [J. Feng et.al., Large Deviations for Stochastic Processes, American Mathematical Society (2006) vol. \textbf{131}]. By employing the techniques from the comparison principle, we identify the Laplace limit as the convergence of the viscosity solution of the associated second-order singularly perturbed Hamilton-Jacobi-Bellman equation. A key advantage of this method is that it establishes a Laplace principle without relying on additional sufficient conditions such as Bryc's theorem, which the literature commonly requires. For $r>3$ and $r=3$ with $2βμ\geq1$, we also derive the exponential moment bounds without imposing the classical orthogonality condition $((\boldsymbol{u}_n\cdot\nabla)\boldsymbol{u}_n,\mathrm{A}\boldsymbol{u}_n)=0$, where $\mathrm{A}=-Δ$, in both two-and three-dimensions. We first establish the large deviation principle in the Skorohod space. Then, by using the $\mathrm{C}-$exponential tightness, we finally establish the large deviation principle in the continuous space.

math.PR↗

arXiv2025

Well-posedness of a boundary hemivariational inequality for stationary and non-stationary 2D and 3D convective Brinkman-Forchheimer equations

This paper investigates boundary hemivariational inequality problems associated with both stationary and non-stationary two and three-dimensional convective Brinkman-Forchheimer equations (or Navier-stokes equations with damping), which model the flow of viscous incompressible fluids through saturated porous media. The governing equations are nonlinear in both velocity and pressure and are subject to nonstandard boundary conditions. Specifically, we impose the no-slip condition along with a Clarke subdifferential relation between pressure and the normal velocity components. For the stationary case, we establish the existence and uniqueness of weak solutions using a surjectivity theorem for pseudomonotone operators. The existence of weak solutions to the non-stationary hemivariational inequality is established via a limiting process applied to a temporally semi-discrete scheme, where the time derivative is approximated using the backward Euler method-commonly referred to as the Rothe method. It is demonstrated that the discrete problem admits solutions, which possess a weakly convergent subsequence as the time step tends to zero, and that any such weak limit satisfies the original hemivariational inequality. A novel outcome of this paper is that the existence results obtained in this work is applicable to 3D non-stationary Navier-Stokes equations also. Moreover, under appropriate conditions on the absorption exponent, we show that Leray-Hopf weak solutions satisfies the energy equality, the solution is shown to be unique and to depend continuously on the given data.

math.AP↗

arXiv2025

Error estimates for viscous Burgers' equation using deep learning method

The article focuses on error estimates as well as stability analysis of deep learning methods for stationary and non-stationary viscous Burgers equation in two and three dimensions. The local well-posedness of homogeneous boundary value problem for non-stationary viscous Burgers equation is established by using semigroup techniques and fixed point arguments. By considering a suitable approximate problem and deriving appropriate energy estimates, we prove the existence of a unique strong solution. Additionally, we extend our analysis to the global well-posedness of the non-homogeneous problem. For both the stationary and non-stationary cases, we derive explicit error estimates in suitable Lebesgue and Sobolev norms by optimizing a loss function in a Deep Neural Network approximation of the solution with fixed complexity. Finally, numerical results on prototype systems are presented to illustrate the derived error estimates.

math.NA↗

arXiv2025

Optimal control of convective Brinkman-Forchheimer equations: Dynamic programming equation and Viscosity solutions

It has been pointed out in the work [F. Gozzi et.al., \emph{Arch. Ration. Mech. Anal.} {163}(4) (2002), 295--327] that the existence and uniqueness of viscosity solutions to the first-order Hamilton-Jacobi-Bellman equation (HJBE) associated with the three-dimensional Navier-Stokes equations (NSE) have not been resolved due to the lack of global solvability and continuous dependence results. However, by adding a damping term to NSE, the so-called \emph{damped Navier-Stokes equations} fulfills the requirement of existence and uniqueness of global strong solutions. In this work, we address this issue in the context of the following two- and three-dimensional convective Brinkman-Forchheimer (CBF) equations (damped NSE) in $\mathbb{T}^d,\ d\in\{2,3\}$: \begin{align*} \frac{\partial\boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0, \end{align*} where $μ,α,β>0$, $r\in[1,\infty)$. We first prove the existence of a viscosity solution to the infinite-dimensional HJBE in the supercritical regime. For spatial dimension $d=2$, we consider the nonlinearity exponent $r\in(3,\infty)$, while for $d=3$, due to some technical difficulty, we focus on $r\in(3,5]$. In the case $r=3$, we require the condition $2βμ\geq 1$ for both $d=2$ and $d=3$. Next, we derive a comparison principle for the HJB equation covering the ranges $r\in(3,\infty)$ and $r=3$ with $2βμ\geq 1$ in $d\in\{2,3\}$. It ensures the uniqueness of the viscosity solution.

math.OC↗

arXiv2025

Hamilton-Jacobi-Bellman equation and Viscosity solutions for an optimal control problem for stochastic convective Brinkman-Forchheimer equations

In this work, we consider the following two- and three-dimensional stochastic convective Brinkman-Forchheimer (SCBF) equations in torus $\mathbb{T}^d,\ d\in\{2,3\}$: \begin{align*} \mathrm{d}\boldsymbol{u}+\left[-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p\right]\mathrm{d}t=\mathrm{d}\mathrm{W}, \ \nabla\cdot\boldsymbol{u}=0, \end{align*} where $μ,α,β>0$, $r\in[1,\infty)$ and $\mathrm{W}$ is a Hilbert space valued $\mathrm{Q}-$Wiener process. The above system can be considered as damped stochastic Navier-Stokes equations. Using the dynamic programming approach, we study the infinite-dimensional second-order Hamilton-Jacobi equation associated with an optimal control problem for SCBF equations. For the supercritical case, that is, $r\in(3,\infty)$ for $d=2$ and $r\in(3,5)$ for $d=3$ ($2βμ\geq 1$ for $r=3$ in $d\in\{2,3\}$), we first prove the existence of a viscosity solution for the infinite-dimensional HJB equation, which we identify with the value function of the associated control problem. By establishing a comparison principle for $r\in(3,\infty)$ and $r=3$ with $2βμ\geq1$ in $d\in\{2,3\}$, we prove that the value function is the unique viscosity solution and hence we resolve the global unique solvability of the HJB equation in both two and three dimensions.

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arXiv2024

On the convective Brinkman-Forchheimer equations

The convective Brinkman--Forchheimer equations or the Navier--Stokes equations with damping in bounded or periodic domains $\subset\mathbb{R}^d$, $2\leq d\leq 4$ are considered in this work. The existence and uniqueness of a global weak solution in the Leray-Hopf sense satisfying the energy equality to the system: $$\partial_t\boldsymbol{u}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f},\ \nabla\cdot\boldsymbol{u}=0,$$ (for all values of $β>0$ and $μ>0$, whenever the absorption exponent $r>3$ and $2βμ\geq 1$, for the critical case $r=3$) is proved. We exploit the monotonicity as well as the demicontinuity properties of the linear and nonlinear operators and the Minty-Browder technique in the proofs. Finally, we discuss the existence of global-in-time strong solutions to such systems in periodic domains.

math.AP↗

arXiv2024

Kolmogorov equations for 2D stochastic convective Brinkman-Forchheimer equations: Analysis and Applications

In this work, we consider the following 2D stochastic convective Brinkman-Forchheimer (SCBF) equations in a bounded smooth domain $\mathcal{O}$: \begin{align*} \mathrm{d}\boldsymbol{u}+\left[-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p\right]\mathrm{d}t=\sqrt{\mathrm{Q}}\mathrm{W}, \ \nabla\cdot\boldsymbol{u}=0, \end{align*} where $μ,α,β>0$, $r\in\{1,2,3\}$, $\mathrm{Q}$ is a non-negative operator of trace class, $\mathrm{W}$ is a cylindrical Wiener process in a Hilbert space $\mathbb{H}$. Under the following assumption on the viscosity co-efficient $μ$ and the Darcy co-efficient $α$: for some positive constant $γ_1$, \begin{equation*} μ(μ+α)^2>γ_1\max\{4\mathrm{Tr}(\mathrm{Q}),\mathrm{Tr}(\mathrm{A}^{2δ}\mathrm{Q})\}, \end{equation*} where $\mathrm{A}$ is the Stokes operator and $δ\in(0,\frac{1}{2})$, our primary goal is to solve the corresponding Kolmogorov equation in the space $\mathbb{L}^2(\mathbb{H};η),$ where $η$ is the unique invariant measure associated with 2D SCBF equations. Then, we establish the well-known ``carré du champs'' identity. Some sharp estimates on the derivatives of the solution constitute the key component of the proofs. We take into consideration two control problems from the application point of view. The first is an infinite horizon control problem for which we establish the existence of a solution for the Hamilton-Jacobi-Bellman equation associated with it. Finally, by exploiting $m$-accretive theory, we demonstrate the existence of a unique solution for an obstacle problem associated with the Kolmogorov operator corresponding to the stopping-time problem for 2D SCBF equations.

math.OC↗

arXiv2024

Approximate controllability and Irreducibility of the transition semigroup associated with 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 $\mathbb{T}^d$: \begin{align*} \frac{\partial\boldsymbol{y}}{\partial t}-μΔ\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+α\boldsymbol{y}+β\vert \boldsymbol{y}\vert^{r-1}\boldsymbol{y}+γ\vert \boldsymbol{y}\vert ^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\ \nabla\cdot\boldsymbol{y}=0, \end{align*} where $d\in\{2,3\}$, $μ,α,β>0$, $γ\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$ and $\boldsymbol{u}$ is the control. For the super critical ($r>3$) and critical ($r=3$ with $2βμ>1$) cases, we first show the approximate controllability of the above system in the usual energy space (divergence-free $\mathbb{L}^2(\mathbb{T}^d)$ space). As an application of the approximate controllability result, we establish the irreducibility of the transition semigroup associated with stochastic CBFeD system perturbed by non-degenerate Gaussian noise in the usual energy space by exploiting the regularity of solutions, smooth approximation of the multi-valued map $\mathrm{sgn}(\cdot)$ a density argument and monotonicity properties of the linear and nonlinear operators.

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}-μΔ\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+α\boldsymbol{y}+β\vert \boldsymbol{y}\vert ^{r-1}\boldsymbol{y}+γ\vert \boldsymbol{y}\vert ^{q-1}\boldsymbol{y}+\nabla p=\boldsymbol{g}+\boldsymbol{u},\ \nabla\cdot\boldsymbol{y}=0, \end{align*} where $d\in\{2,3\}$, $μ,α,β>0$, $γ\in\mathbb{R}$, $r,q\in[1,\infty)$ 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

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}-μΔ\boldsymbol{y}+(\boldsymbol{y}\cdot\nabla)\boldsymbol{y}+α\boldsymbol{y}+β|\boldsymbol{y}|^{r-1}\boldsymbol{y}+\nabla p+Ψ(\boldsymbol{y})\ni\boldsymbol{g},\ \nabla\cdot\boldsymbol{y}=0, \end{equation*} in a $d$-dimensional torus is considered in this work, where $d\in\{2,3\}$, $μ,α,β>0$ and $r\in[1,\infty)$. For $d=2$ with $r\in[1,\infty)$ and $d=3$ with $r\in[3,\infty)$ ($2βμ\geq 1$ 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 $2βμ<1$. 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 $r\in[3,\infty)$ ($2βμ\geq1$ for $r=3$), we handle the Navier-Stokes nonlinearity by the nonlinear damping term $β|\boldsymbol{y}|^{r-1}\boldsymbol{y}$. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.

math.OC↗