arXiv subjects

Manil T. Mohan

Publications and source records attributed to Manil T. Mohan.

At least 19 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

Global Well-posedness and Asymptotic Analysis of a Damped Nonlinear Wave Equation with a Codimension-One Constraint

We prove the global existence and uniqueness of strong solutions to a constrained version of the damped nonlinear wave equation $$ \vartheta_{tt}+\gamma \vartheta_t-\Delta\vartheta+|\vartheta|^{p-2}\vartheta=0 $$ on a smooth bounded domain $\varOmega\subset\mathbb{R}^d$, where the evolution is projected onto the tangent space of the Hilbert manifold $$\mathcal{M} = \left\{ \vartheta\in L^2(\varOmega):\|\vartheta\|_{L^2(\varOmega)}=1 \right\}, $$ which is the unit sphere in $L^2(\varOmega)$. We assume that $$p\in[2,\infty)\ \text{ for }\ d=1,2, \ \text{ while }\ 2\leq p\leq \frac{2(d-1)}{d-2} \ \text{ for }\ d\geq3.$$ By employing the Faedo--Galerkin approximation method, together with suitable a priori estimates and compactness arguments, we establish the global well-posedness of the problem. In particular, we show that the Hilbert manifold $\mathcal{M}$ is invariant under the flow, and hence the $L^2$-constraint is preserved throughout the evolution. Using the \emph{Lusternik--Schnirelmann theory}, we show that the corresponding stationary problem possesses at least countably many stationary solutions. We further investigate the long-time behaviour of solutions and prove that, along a subsequence, every solution of the constrained problem converges to a stationary solution by invoking \emph{Webb's theorem} and \emph{Barbalat's lemma}. When the initial data are sufficiently close to the first eigenfunction of the associated stationary problem, we show that the unique strong solution converges in $H_0^1(\varOmega)$ to the unique positive ground-state solution.

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arXiv2026

Global existence and uniqueness of solutions in a chemotaxis system with density-suppressed motility, a $\theta-$logistic source, and indirect signal production under a strong Allee effect

We consider the following chemotaxis-growth system, featuring density-suppressed motility and a $\theta$-logistic growth term that incorporates a strong Allee effect: \begin{equation*} \left\{ \begin{aligned} u_t &= \Delta(\gamma(v)u) + \mu u(1-u^\theta)(u-A), & x \in \Omega, \quad t > 0, v_t &= \Delta v - v+ w^\beta, & x \in \Omega,\quad t > 0, w_t &= -\delta w +u, & x \in \Omega, \quad t > 0, \end{aligned} \right. \end{equation*} subject to homogeneous Neumann boundary conditions in a bounded domain $\Omega\subset \mathbb{R}^d (d\geq 2)$ with smooth boundary. Here $\mu\in \mathbb{R} , \delta,\beta>0, \theta\geq1$, and the positive motility function $\gamma(v) \in C^3([0,\infty))$ fulfills $\gamma'(v) \leq 0$ for all $v\geq 0$. The main objective of this paper is to establish the global existence and boundedness of classical solutions to the proposed problem. Our analysis combines the Schauder fixed point theorem for proving local-in-time existence with the extensibility criterion, fundamental energy estimates, $L^p-$bounds, the Gagliardo-Nirenberg interpolation inequality, Young's inequality, semigroup estimates, and a Moser-type iteration scheme to derive uniform-in-time $L^{\infty}-$bounds, thereby ensuring global bounded classical solutions. More precisely, we establish that the chemotaxis-growth system admits a unique globally bounded classical solution whenever $\beta < \frac{2(\theta+2)}{d}$. Furthermore, if the logarithmic derivative of the motility function is assumed to be uniformly bounded on $[0,\infty)$, that is, $\frac{\gamma'(v)}{\gamma(v)} \in L^{\infty}([0,\infty))$, then the above restriction on $\beta$ can be weakened to $\beta < \frac{d(\theta+1)+2(\theta+2)}{2d}$, provided that $\theta > \frac{4-d}{d-2}$.

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arXiv2026

A Quasi-Variational--Hemivariational Inequality for the Convective Brinkman--Forchheimer Extended Darcy Equations with Bingham Fluids

This paper is devoted to the analysis of a quasi-variational--hemivariational inequality associated with the convective Brinkman--Forchheimer extended Darcy (CBFeD) equations for Bingham fluids in both two and three spatial dimensions. The considered model describes incompressible fluid flow through porous media while incorporating convection, nonlinear damping effects, and Forchheimer-type resistance, together with non-smooth and non-convex slip boundary conditions. We first derive an appropriate weak formulation of the problem, which leads naturally to a Bingham-type quasi-variational--hemivariational inequality with a velocity-dependent constraint set. By employing the Kakutani--Ky Fan fixed point theorem, we establish the existence of weak solutions for the resulting multivalued quasi-variational inequality formulation of the CBFeD system. Furthermore, we prove that every weak solution of the associated quasi-variational inequality is also a solution of the corresponding quasi-variational--hemivariational inequality. The analysis presented in this work provides a rigorous mathematical framework for CBFeD models with Bingham fluids under non-monotone and non-smooth boundary interactions.

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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

Averaging principle for a slow-fast stochastic nonlinear fractional Schr\"odinger equation

We establish an averaging principle for a structural multiscale stochastic nonlinear fractional Schr\"odinger system on the one-dimensional torus driven by a multiplicative Wiener noise. The slow component is governed by a fractional Schr\"odinger operator with a general polynomial nonlinearity, while the fast component evolves on a shorter time scale and exhibits dissipative diffusion, nonlinear interactions, and stochastic forcing. Under suitable dissipative assumptions, we have shown that, as the scale separation parameter tends to zero, the slow component converges strongly to an effective stochastic fractional Schr\"odinger equation. The effective drift is obtained by averaging the coupling term with respect to the unique invariant measure of the frozen fast dynamics. The proof relies on uniform a priori estimates, ergodicity of the fast equation, H\"older time regularity of the slow component obtained via a vanishing viscosity method, and a Khasminskii-type time discretization argument adapted to fractional dispersive operators. The analysis is technically challenging due to limited smoothing of the fractional Schr\"odinger semigroup and the presence of general polynomial nonlinearities, which are handled through refined estimates and viscosity approximation.

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arXiv2026

Non-uniqueness for the stochastic incompressible Euler equations with a passive tracer

In this work we investigate the phenomenon of pathwise non-uniqueness for the stochastic incompressible Euler equations with a passive tracer on the whole Euclidean space. The stochastic perturbations are interpreted as a transport noise and a linear multiplicative noise in the Stratonovich sense. In both cases, via classical transformations, we convert the SPDEs into PDEs with random coefficients. Using the Baire category method developed by De Lellis and Sz\'ekelyhidi Jr., we then construct infinitely many global-in-time weak solutions to the random PDEs in any spatial dimension greater than or equal to two. By applying the inverse transformations, we obtain pathwise non-uniqueness for the original SPDEs. Finally, we present an application of our result to the three-dimensional stochastic ideal MHD equations. This study can be regarded as a stochastic counterpart of Bronzi et al.~[\textit{Commun.~Math.~Sci.},~2015]. In particular, our non-uniqueness result in the random setting extends theirs from a constant energy profile equal to one to arbitrary positive, bounded, and continuous time-dependent profiles, originally established in two dimensions via the convex integration technique.

math.PR↗

arXiv2026

Existence and uniqueness results of a stochastic nonlinear heat equation with a constraint of codimension one

In this work, we investigate the well-posedness of a stochastic heat equation with an arbitrary (but polynomial) nonlinearity in any dimension $d\geq 1$ perturbed by a multiplicative white noise in the Stratonovich form, subject to an $L^2-$norm constraint on the solution. In bounded smooth domains, we establish the existence of a martingale solution taking values in $H_0^1 \cap L^p$ for arbitrary $2 \le p < \infty$, using a modified Faedo-Galerkin scheme. By utilizing a sequence of self-adjoint operators which are bounded in $L^p$ for any $2 \le p < \infty$, we provide a novel proof of an It\^o formula for the $L^p-$norm of the solution. Together with pathwise uniqueness of the martingale solution, the Yamada-Watanabe result then yields the existence of a strong solution and uniqueness in law.

math.PR↗

arXiv2026

Rate of convergence of a nonlinear heat equation with a constraint of codimension one

We consider a nonlinear constrained heat flow evolving on the manifold $\mathcal{M}=\{v\in L^{2}:\|v\|_{L^{2}}=1\}$ over bounded smooth domains. It is known that the solution corresponding to any nonnegative initial datum remains on $\mathcal{M}$ and converges to the unique positive ground state of the associated stationary problem. In this work, we first establish certain time-regularity estimates and then use these to derive explicit exponential rates of convergence for the energy, the solution in the $L^2, H^1$ and $H^2-$norms, and the associated nonlinear eigenvalue, thereby proving a sharp exponential stability of the ground state. Moreover, using the \L{}ojasiewicz-Simon inequality, we obtain decay rates for locally stabilized solutions toward a stationary state in the $L^2$ and $H^1-$norms, where the rate depends on the corresponding \L{}ojasiewicz-Simon exponent. Our results are new, and the approach relies on spectral analysis of the linearized operator, uniform higher-order estimates, and the compactness of solution trajectories.

math.AP↗

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 $\mu > 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 $2\beta \mu > 1 $ in the critical case $r = 3 $.

math.AP↗

arXiv2026

Long-time behaviour of a nonlocal stochastic fractional reaction--diffusion equation arising in tumour dynamics

We introduce a stochastic nonlocal reaction--diffusion model arising in tumour dynamics. Spatial dispersal is described by the fractional Laplacian, accounting for anomalous diffusion and long--range relocation events. The system is perturbed by multiplicative fractional Brownian motion (fBm) with Hurst parameter $H>1/2$, which we interpret as temporally correlated fluctuations in the tumour microenvironment and host response. We first establish well--posedness and identify parameter regimes leading to global--in--time solutions or finite--time blow--up under general multiplicative fractional noise. We then focus on linear multiplicative noise and, via a Doss--Sussmann transformation, derive sharper results: explicit lower and upper bounds for the blow--up time together with quantitative estimates of the blow--up probability, clarifying how noise intensity can accelerate progression or, on favourable paths, enhance suppression consistent with extinction (loss of viability). Finally, one--dimensional simulations illustrate the interplay between anomalous diffusion, fractional noise, and the nonlocal reaction mechanism in shaping the long--time dynamics.

math.AP↗

arXiv2026

Existence and uniqueness of time-periodic solutions of the 2D and 3D convective Brinkman-Forchheimer extended Darcy equations

In this work, we investigate the existence and uniqueness of solutions to the following 2D and 3D convective Brinkman-Forchheimer extended Darcy equations defined on a bounded smooth domain $\Omega\subset\mathbb{R}^d$, $d\in\{2,3\}$, \begin{align*}\frac{\partial\boldsymbol{v}}{\partial t}-\mu \Delta\boldsymbol{v}+(\boldsymbol{v}\cdot\nabla)\boldsymbol{v}+\alpha\boldsymbol{v}+\beta\vert \boldsymbol{v}\vert^{r-1}\boldsymbol{v}+\gamma\vert \boldsymbol{v}\vert ^{q-1}\boldsymbol{v}+\nabla p=\boldsymbol{g},\ \nabla\cdot\boldsymbol{v}=0, \end{align*} where $\mu,\alpha,\beta>0$, $\gamma\in\mathbb{R}$, $r,q\in[1,\infty)$ with $r>q\geq 1$ and $\boldsymbol{g}$ is an external forcing term. For $r \geq 1 $, under periodic forcing, we establish the \emph{existence of time-periodic global weak solutions} to the system by employing \emph{Faedo-Galerkin approximations}, together with the \emph{Banach-Alaoglu theorem}, the \emph{Aubin-Lions-Simon compactness lemma}, and the \emph{Lions-Magenes lemma}. The \emph{existence of periodic solutions} for the Faedo-Galerkin approximated problem is obtained via \emph{Brouwer's fixed point theorem}. In the \emph{supercritical} case $(r>3)$ and the \emph{critical} case ($r=3$), we prove the \emph{uniqueness of the global weak solution} without imposing any smallness condition on the external forcing. This constitutes a new result compared to the classical 2D Navier-Stokes equations with periodic inputs, for which the \emph{uniqueness of strong solutions} typically requires \emph{smallness assumptions} on the external force.

math.AP↗

arXiv2026

Optimal existence of weak solutions for the generalised Navier-Stokes-Voigt equations

In this study, we investigate the incompressible generalised Navier-Stokes-Voigt equations within a bounded domain $\Omega \subset \mathbb{R}^d$, where $d \geq 2$. The governing momentum equation is expressed as: $$ \partial_t(\boldsymbol{v}- \kappa \Delta \boldsymbol{v}) + \nabla \cdot (\boldsymbol{v} \otimes \boldsymbol{v}) + \nabla \pi - \nu \nabla \cdot \left( |\mathbf{D}(\boldsymbol{v})|^{p-2} \mathbf{D}(\boldsymbol{v}) \right) = \boldsymbol{f}. $$ Here, for $d \in \{2,3,4\}$, $\boldsymbol{v}$ represents the velocity field, $\pi$ denotes the pressure, and $\boldsymbol{f}$ is the external forcing term. The constants $\kappa$ and $\nu$ correspond to the relaxation time and kinematic viscosity, respectively. The parameter $p \in (1, \infty)$ characterizes the fluid's flow behavior, and $\mathbf{D}(\boldsymbol{v})$ denotes the symmetric part of the velocity gradient $\nabla \boldsymbol{v}$. For power-law exponents satisfying $p>1$ when $2\leq d\leq 3$, and $p> \frac{2d}{d+2}$ for $d=4$, we establish the existence of weak solutions to the generalised Navier-Stokes-Voigt system. Moreover, we prove uniqueness of the weak solution for the same ranges of $p$. The results are optimal in the sense that $p>1$ is minimal for $2 \leq d \leq 3$. Moreover, for $p>\frac{2d}{d+2}$ with $d>3$, the framework uses a Gelfand triple, allowing the Aubin--Dubinski\u{\i} lemma to yield strong convergence of approximate solutions. This convergence is essential for the existence proof and holds precisely for $p>\frac{2d}{d+2}$ when $d=4$.

math.AP↗

arXiv2025

Well-posedness and the \L{}ojasiewicz-Simon inequality in the asymptotic analysis of a nonlinear heat equation with constraints of finite codimension

We establish the global well-posedness of the $D(A)-$valued strong solution to a nonlinear heat equation with constraints on a \textit{Poincar\'e domain} $\bO\subset \R^d$ whose boundary is of class $C^2$. Consider the following nonlinear heat equation \begin{align*} \frac{\partial u}{\partial t} - \Delta u + |u|^{p-2}u = 0, \end{align*} projected onto the tangent space $T_u\bM$, where $\mathcal{M}:=\left\{u\in L^2(\bO):\|u\|_{L^2(\bO)}=1\right\}$ is a submanifold of $L^2(\bO)$. The nonlinearity exponent satisfies $2\le p < \infty$ for $1\leq d\leq 4$ and $2 \le p \le \frac{2d-4}{d-4}$ for $d \ge 5$. The solution is constrained to lie within $\mathcal{M}$ which encodes the norm-preserving constraint. By modifying the nonlinearity and exploiting the abstract theory for \textit{$m-$accretive }evolution equations, we prove the existence of a global strong solution. Using {resolvent-idea } and the \textit{Yosida approximation} method, we derive regularity results. In the asymptotic analysis, $\bO$ is restricted to bounded domains with even $p$ and $1\le d \le 3$. For any initial data in $D(A) \cap \mathcal{M}$, we apply the \textit{\L{}ojasiewicz-Simon gradient inequality} on a Hilbert submanifold [F. Rupp, \textit{J. Funct. Anal.}, 279(8), 2020], to demonstrate that the unique global strong solution converges in $W^{2,q}(\bO) \cap W^{1,q}_0(\bO)$ to a stationary state, where $2 \le q < \frac{2d}{d + 4 - 4\beta}$ and $1 < \beta < \frac{3}{2}$. This work proposes an alternative method for establishing the global existence and analyzing long-term behavior of the unique strong solution to an $L^2-$norm preserving nonlinear heat equation.

math.AP↗

arXiv2025

Finite element analysis of a nonlinear heat Equation with damping and pumping effects

We study the following nonlinear heat equation with damping and pumping effects (a reaction-diffusion equation) posed on a bounded simply connected convex domain $\Omega \subset \mathbb{R}^d$, $d \geq 1$ with Lipschitz boundary $\partial\Omega$: $$ \frac{\partial u(t)}{\partial t} - \nu \Delta u(t) + \alpha |u(t)|^{p-2}u(t) - \sum_{\ell=1}^M \beta_{\ell} |u(t)|^{q_{\ell}-2}u(t) = f(t), \quad t>0, $$ subject to homogeneous Dirichlet boundary conditions and the initial condition $u(0)=u_0$, where $2 \leq p < \infty$ and $2 \leq q_{\ell} < p$ for $1 \leq \ell \leq M$. For $u_0 \in L^2(\Omega)$ and $f \in L^2(0,T;H^{-1}(\Omega))$, we establish the existence and uniqueness of a weak solution for all dimensions $d \in \mathbb{N}$ and damping exponents $2 \leq p < \infty$. Furthermore, for $u_0 \in H^2(\Omega) \cap H_0^1(\Omega)$ and $f \in H^1(0,T;H^1(\Omega))$, we obtain regularity results: these hold for every $2 \leq p < \infty$ when $1 \leq d \leq 4$, and for $2 \leq p \leq \frac{2d-6}{d-4}$ when $d \geq 5$. We further conduct finite element analysis using conforming, nonconforming, and discontinuous Galerkin methods, deriving a priori error estimates for both semi- and fully discrete schemes, supported by numerical results. To relax restrictions on $p$ in the semidiscrete analysis, we use appropriate projection/interpolation operators: the Ritz projection in the conforming case ($2 \le p \le \frac{2d}{d-2}$), the Scott-Zhang interpolation for $\frac{2d}{d-2} < p \le \frac{2d-6}{d-4}$, the Cl\'ement interpolation in the nonconforming setting, and the $L^2$-projection in the DG framework. In the fully discrete case, error estimates hold for the above $p$-range under $u_0 \in D(A^{3/2})$ and $f \in H^1(0,T;H^1(\Omega))$.

math.NA↗

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+[-\mu\Delta\boldsymbol{u}_n+ (\boldsymbol{u}_n\cdot\nabla)\boldsymbol{u}_n +\alpha\boldsymbol{u}_n+\beta|\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 $\mu,\alpha,\beta>0$, $r\in[1,\infty)$, $\mathrm{Q}$ 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 $2\beta\mu\geq1$, 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 $\mathrm{A}=-\Delta$, 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

Optimal Control of a Hemivariational Inequality of Stationary Convective Brinkman-Forchheimer Extended Darcy equations with Numerical Approximation

This paper studies an optimal control problem for a stationary convective Brinkman-Forchheimer extended Darcy (CBFeD) hemivariational inequality in two and three dimensions, subject to control constraints, and develops its numerical approximation. The hemivariational inequality provides the weak formulation of a stationary incompressible fluid flow through a porous medium, governed by the CBFeD equations, which account for convection, damping, and nonlinear resistance effects. The problem incorporates a non-leak boundary condition and a subdifferential friction-type condition. We first analyze the stability of solutions with respect to perturbations in the external force density and the superpotential. Next, we prove the existence of a solution to the optimal control problem, where the external force density acts as the control variable. We then propose a numerical scheme for solving the optimal control problem and establish its convergence. For concreteness, the numerical method is implemented using finite element discretization. Finally, we provide some numerical examples to validate the theory developed.

math.NA↗