2023/01/04 by Sagar Gautam, Gautam, Sagar, Kush Kinra +3
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Advanced Mathematical Physics Problems #FOS: Mathematics #Numerical methods in inverse problems #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.2301.01527
openalex publication_date 2023/01/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The following convective Brinkman-Forchheimer (CBF) equations (or damped Navier-Stokes equations) with potential \frac∂ \boldsymboly∂ t-μΔ\boldsymboly+(\boldsymboly⋅∇)\boldsymboly+α\boldsymboly+β|\boldsymboly|r-1\boldsymboly+∇ p+Ψ(\boldsymboly)\ni\boldsymbolg, ∇⋅\boldsymboly=0, in a d-dimensional torus is considered in this work, where d∈\2,3\, μ,α,β>0 and r∈[1,∞). For d=2 with r∈[1,∞) and d=3 with r∈[3,∞) (2βμ≥ 1 for d=r=3), we establish the existence of \textsfa unique global strong solution for the above multi-valued problem with the help of the \textsfabstract theory of m-accretive operators. %for nonlinear differential equations of accretive type in Banach spaces. Moreover, we demonstrate that the same results hold \textsflocal 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 (\boldsymboly⋅∇)\boldsymboly 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|\boldsymboly|r-1\boldsymboly. Finally, we discuss the applications of the above developed theory in feedback control problems like flow invariance, time optimal control and stabilization.