Hamilton-Jacobi-Bellman equation and viscosity solutions for the optimal control problem of stochastic convective Brinkman-Forchheimer equations

Abstract

This work is devoted to the infinite-dimensional second-order Hamilton-Jacobi-Bellman equation associated, via the dynamic programming approach, with an optimal control problem for the two- and three-dimensional stochastic convective Brinkman-Forchheimer equations on the torus, driven by an additive Hilbert space valued [Formula: see text]-Wiener process. In the two-dimensional setting, many existing results (see [F. Gozzi et.al., Comm. Pure Appl. Math. 58(5) (2005), 671–700]) rely essentially on the identity [Formula: see text], which is fundamental in closing the estimates. This cancellation fails in three dimensions, where the real analytical challenge arises. Our contribution is precisely a framework that does not rely on it. The damping term [Formula: see text] together with the dissipative term [Formula: see text] allows us to control [Formula: see text] effectively, so that our analysis extends naturally to three dimensions, which cannot be handled for the Navier-Stokes equations. In contrast, for the classical Navier-Stokes equations, the lack of suitable Sobolev embeddings prevents control of the convective term in three dimensions, thereby obstructing the closure of estimates. For the supercritical case, that is, [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text] ([Formula: see text] for [Formula: see text] in [Formula: see text]) we first prove the existence of a viscosity solution of the Hamilton-Jacobi-Bellman equation, identified with the value function of the control problem. By establishing a comparison principle for [Formula: see text] and [Formula: see text] with [Formula: see text] in [Formula: see text], we then prove uniqueness.

The authors' abstract, as published at the source. Communications in Contemporary Mathematics, 2026 · DOI ↗