2D and 3D convective Brinkman-Forchheimer equations perturbed by a subdifferential and applications to control problems

Abstract

The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential), we establish the existence of a unique global strong solution for the above multi-valued problem with the help of the abstract theory of maccretive operators. Moreover, we demonstrate that the same results hold local in time for the case d = 3 with r ∈ [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 ∈ [1, 3], we quantize (modify) the Navier-Stokes nonlinearity (y • ∇)y to establish the existence and uniqueness results, while for r ∈ [3, ∞) (2βµ ≥ 1 for r = 3), we handle the Navier-Stokes nonlinearity by the nonlinear damping term β|y| r−1 y. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.