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Manil T. Mohan

Publications and source records attributed to Manil T. Mohan.

At least 37 records · Page 2Linked to original sources

arXiv2025

Well-posedness and the Łojasiewicz-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{Poincaré 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} - Δu + |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{Łojasiewicz-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 $2 \le q < \frac{2d}{d + 4 - 4β}$ and $1 < β< \frac{3}{2}$. 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.

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arXiv2025

Martingale solution, invariant measure and ergodicity for stochastic convective Brinkman-Forchheimer equations on general domains in $\mathbb{R}^d$

The convective Brinkman-Forchheimer equations (CBFEs) \[ \frac{\partial \boldsymbol{X}}{\partial t} - μΔ\boldsymbol{X} + (\boldsymbol{X}\cdot\nabla)\boldsymbol{X} + α\boldsymbol{X} + β|\boldsymbol{X}|^{r-1}\boldsymbol{X} + \nabla p = \mathbf{F}, \qquad \nabla\cdot\boldsymbol{X}=0, \] with parameters $μ,α,β>0$ and $r\in[1,\infty)$ describe incompressible fluid motion in saturated porous media. In the stochastic setting, for $d=2,3$ and $r\in[3,\infty)$ (with $2βμ\geq 1$ when $r=3$), strong pathwise solutions on general domains are already known, hence weak martingale solutions exist as well. In the same parameter regime, invariant probability measures on bounded domains have also been obtained. The present work complements and significantly extends these results. More precisely, on general domains in $\mathbb{R}^d$ (bounded or unbounded), for all $d\in\{2,3\}$, we prove the existence of a weak martingale solution to the stochastic CBFEs for every exponent $r\in[1,\infty)$, which includes the regimes where no strong solution theory is available. For $d=2$, $r\in[1,\infty)$, and for $d=3$, $r\in[3,\infty)$, we further show that the martingale solutions satisfy the energy equality (Itô's formula) and possess $\mathbb{H}$-valued continuous trajectories almost surely. In this regularity regime (excluding $2βμ< 1$ when $r=3$), we establish pathwise uniqueness and thereby, via the Yamada-Watanabe argument, obtain the existence of strong solutions and uniqueness in law, thereby recovering, in particular, the known results. Finally, for $d=2$, $r\in[1,\infty)$, and for $d=3$, $r\in[3,\infty)$ (with $2βμ\geq 1$ when $r=3$), we prove the existence of an invariant probability measure for the associated Markov semigroup, while for $d=2,3$ with $r\in[3,\infty)$ (and with $2βμ\geq 1$ for $r=3$), we show that at most one invariant measure can exist.

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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 $Ω\subset \mathbb{R}^d$, $d \geq 1$ with Lipschitz boundary $\partialΩ$: $$ \frac{\partial u(t)}{\partial t} - νΔu(t) + α|u(t)|^{p-2}u(t) - \sum_{\ell=1}^M β_{\ell} |u(t)|^{q_{\ell}-2}u(t) = f(t), \quad t>0, $$ 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 $u_0 \in L^2(Ω)$ and $f \in L^2(0,T;H^{-1}(Ω))$, 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 $u_0 \in H^2(Ω) \cap H_0^1(Ω)$ and $f \in H^1(0,T;H^1(Ω))$, 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 Clément 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 $f \in H^1(0,T;H^1(Ω))$.

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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.

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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.

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arXiv2025

Well-posedness of stationary 2D and 3D convective Brinkman-Forchheimer extended Darcy Hemivariational inequalities

This study addresses the well-posedness of a hemivariational inequality derived from the convective Brinkman-Forchheimer extended Darcy (CBFeD) model in both two and three dimensions. The CBFeD model describes the behavior of incompressible viscous fluid flow through a porous medium, incorporating the effects of convection, damping, and nonlinear resistance. The mathematical framework captures steady-state flow conditions under a no-slip boundary assumption, with a non-monotone boundary condition that links the total fluid pressure and the velocity's normal component through a Clarke subdifferential formulation. To facilitate the analysis, we introduce an auxiliary hemivariational inequality resembling a nonlinear Stokes-type problem with damping and pumping terms, which serves as a foundational tool in establishing the existence and uniqueness of weak solutions for the CBFeD model. The analytical strategy integrates techniques from convex minimization theory with fixed-point methods, specifically employing either the Banach contraction mapping principle or Schauder's fixed point theorem. The Banach-based approach, in particular, leads to a practical iterative algorithm that solves the original nonlinear hemivariational inequality by sequentially solving Stokes-type problems, ensuring convergence of the solution sequence. Additionally, we derive equivalent variational formulations in terms of minimization problems. These formulations lay the groundwork for the design of efficient and stable numerical schemes tailored to simulate flows governed by the CBFeD model.

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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

Global well-posedness and small time asymptotics of stochastic Ladyzhenskaya-Smagorinsky equations with damping on unbounded domains

The Ladyzhenskaya-Smagorinsky equations model turbulence phenomena, and are given by $$\frac{\partial \boldsymbol{u}}{\partial t}-μ\mathrm{div}\left(\left(1+|\nabla\boldsymbol{u}|^2\right)^{\frac{p-2}{2}}\nabla\boldsymbol{u}\right)+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+\nabla p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0,$$ for $p\geq 2.$ In this work, we consider the stochastic Ladyzhenskaya-Smagorinsky equations with the damping $α\boldsymbol{u}+β|\boldsymbol{u}|^{r-2}\boldsymbol{u},$ for $r\geq 2$ ($α,β\geq 0$), subjected to multiplicative Gaussian noise in a Poincaré domain (which may be bounded or unbounded) $\mathcal{O}\subset\mathbb{R}^d$ ($2\leq d\leq 4$). We show the local monotonicity ($p\geq \frac{d}{2}+1,\ r\geq 2$) as well as global monotonicity ($p\geq 2,\ r\geq 4$) properties of the linear and nonlinear operators, which along with an application of a stochastic version of the Minty-Browder technique imply the existence of a unique pathwise strong solution satisfying the energy equality (Itô formula), which is proved with the help of the methodology developed in [Krylov, \emph{Probab. Theory Related Fields}, {\bf 147} (2010), 583--605.] Then, we discuss the small time asymptotics by studying the effect of small, highly nonlinear, unbounded drifts (small time large deviation principle) for the stochastic Ladyzhenskaya-Smagorinsky equations with damping.

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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.

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arXiv2025

Stochastic convective Brinkman-Forchheimer equations on general Unbounded Domains

The stochastic convective Brinkman-Forchheimer (SCBF) equations in an open connected set $\mathcal{O}\subseteq\mathbb{R}^d$ ($d\in \{2,3,4\}$) or torus are considered in this work. We show the existence of a pathwise unique strong solution (in the probabilistic sense) satisfying the energy equality (Itô formula) to SCBF equations perturbed by multiplicative Gaussian noise. We exploited a monotonicity property of the linear and nonlinear operators as well as a stochastic generalization of the Minty-Browder technique in the proofs. The energy equality is obtained by approximating the solution using approximate functions constituting the elements of eigenspaces of a compact operator in such a way that the approximations are bounded and converge in both Sobolev and Lebesgue spaces simultaneously. We further discuss the global in time regularity results of such strong solutions on the torus. The exponential stability results (in mean square and pathwise sense) for the stationary solutions is also established in this work for large effective viscosity. Moreover, a stabilization result of the stochastic convective Brinkman-Forchheimer equations by using a multiplicative noise is obtained. Finally, when $\mathcal{O}$ is a bounded domain, we establish the existence of a unique invariant measure for the SCBF equations with multiplicative Gaussian noise, which is both ergodic and strongly mixing, using the exponential stability of strong solutions.

math.AP↗

arXiv2025

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 $\mathbb{T}^d,$ $d\in\{2,3\}$: $$ \frac{\partial \boldsymbol{u}}{\partial t}-μΔ\boldsymbol{u}+(\boldsymbol{u}\cdot\nabla)\boldsymbol{u}+α\boldsymbol{u}+β|\boldsymbol{u}|^{r-1}\boldsymbol{u}+\nabla p=\boldsymbol{f}, \ \nabla\cdot\boldsymbol{u}=0,$$ where $μ,α,β>0$ and $r\in[1,\infty)$ is the absorption exponent. For $d=2,r\in[1,\infty)$ and $d=3,r\in[3,\infty)$ ($2βμ\geq 1$ 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 $d=3,r\in[3,5]$ ($2βμ\geq 1$ for $d=r=3$), we show the pathwise backward uniqueness as well as approximate controllability via starter controller results.

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arXiv2025

Boundary Stabilizability of Generalized Burgers-Huxley Equation with Memory

In this paper, we study a generalized Burgers-Huxley equation with memory, subject to nonhomogeneous Dirichlet boundary conditions. We construct a linear, finite-dimensional Dirichlet boundary feedback controller aimed at stabilizing the stationary solution corresponding to the homogeneous boundary condition. This controller is designed using eigenfunctions of the Laplace operator. We begin by analyzing the stabilization of a linear system under the proposed feedback law. Subsequently, we demonstrate that the same controller also stabilizes the full nonlinear system by applying the Banach fixed point theorem. Finally, we provide a remark on the stabilization of the generalized Burgers-Huxley equation with memory around the zero solution under nonhomogeneous Dirichlet boundary conditions.

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arXiv2025

Mixed Finite Element Method for a Hemivariational Inequality of Stationary convective Brinkman-Forchheimer Extended Darcy equations

This paper presents the formulation and analysis of a mixed finite element method for a hemivariational inequality arising from the stationary convective Brinkman-Forchheimer extended Darcy (CBFeD) equations. This model extends the incompressible Navier-Stokes equations by incorporating both damping and pumping effects. The hemivariational inequality describes the flow of a viscous, incompressible fluid through a saturated porous medium, subject to a nonsmooth, nonconvex friction-type slip boundary condition. The incompressibility constraint is handled via a mixed variational formulation. We establish the existence and uniqueness of solutions by utilizing the pseudomonotonicity and coercivity properties of the underlying operators and provide a detailed error analysis of the proposed numerical scheme. Under suitable regularity assumptions, the method achieves optimal convergence rates with low-order mixed finite element pairs. The scheme is implemented using the $\text{P1b/P1}$ element pair, and numerical experiments are presented to validate the theoretical results and confirm the expected convergence behavior.

math.NA↗

arXiv2025

Global well-posedness and Asymptotic analysis of a nonlinear heat equation with constraints of finite codimension

We prove the global existence and the uniqueness of the $L^p\cap H_0^1-$valued ($2\leq p < \infty$) strong solutions of a nonlinear heat equation with constraints over bounded domains in any dimension $d\geq 1$. Along with the \textit{Faedo-Galerkin} approximation method and the compactness arguments, we utilize the monotonicity and the hemicontinuity properties of the nonlinear operators to establish the well-posedness results. In particular, we show that a Hilbertian manifold $\mathbb{M}$, which is the unit sphere in $L^2$ space, describing the constraint is invariant. Finally, in the asymptotic analysis, we generalize the recent work of [P. Antonelli, et. al. \emph{Calc. Var. Partial Differential Equations}, 63(4), 2024] to any bounded smooth domain in $\mathbb{R}^d$, $d\geq1$, when the corresponding nonlinearity is a damping. In particular, we show that, for positive initial datum and any $2\le p < \infty$, the unique positive strong solution of the above mentioned nonlinear heat equation with constraints converges in $L^p\cap H_0^1$ to the unique positive ground state.

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arXiv2025

Optimal Control of Stationary Doubly Diffusive Flows on Lipschitz Domains

In this work, we study the control constrained distributed optimal control of a stationary doubly diffusive flow model. For the control problem, we use a well-posedness analysis based on minimal assumptions on data and domain. We show the existence of an optimal control with quadratic type cost functional, study the Fréchet differentiability properties of the control-to-state map and establish the first-order necessary optimality conditions corresponding to the optimal control problem. Expanding on this we prove the local optimality of a reference control using second-order sufficient optimality condition for the control problem.

math.OC↗

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

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↗

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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