2025/05/11 by Gautam, Sagar, Mohan, Manil T.
#FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2505.07095
It has been pointed out in the work [F. Gozzi et.al., 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 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 \mathbbTd, d∈\2,3\: \frac∂\boldsymbolu∂ t-μΔ\boldsymbolu+(\boldsymbolu⋅∇)\boldsymbolu+α\boldsymbolu+β|\boldsymbolu|r-1\boldsymbolu+∇ p=\boldsymbolf, ∇⋅\boldsymbolu=0, where μ,α,β>0, r∈[1,∞). 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∈(3,∞), while for d=3, due to some technical difficulty, we focus on r∈(3,5]. In the case r=3, we require the condition 2βμ≥ 1 for both d=2 and d=3. Next, we derive a comparison principle for the HJB equation covering the ranges r∈(3,∞) and r=3 with 2βμ≥ 1 in d∈\2,3\. It ensures the uniqueness of the viscosity solution.