2007/02/12 by Nassif Ghoussoub, Ghoussoub, Nassif, Abbas Moameni +1 · 1 citation
Mathematics · #Advanced Mathematical Physics Problems #Analysis of PDEs (math.AP) #FOS: Mathematics #Navier-Stokes equation solutions #Nonlinear Partial Differential Equations
paper · pdf · doi:10.48550/arxiv.math/0702339
openalex publication_date 2007/02/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
The nonlinear selfdual variational principle established in a preceeding paper [8] -- though good enough to be readily applicable in many stationary nonlinear partial differential equations -- did not however cover the case of nonlinear evolutions such as the Navier-Stokes equations. One of the reasons is the prohibitive coercivity condition that is not satisfied by the corresponding selfdual functional on the relevant path space. We show here that such a principle still hold for functionals of the form I(u)= ∫0T [ L (t, u(t), u(t)+Λu(t)) +< Λu(t), u(t) > ] dt +ℓ (u(0)- u(T), \frac u(T)+ u(0)2) where L (resp., ℓ) is an anti-selfdual Lagrangian on state space (resp., boundary space), and Λ is an appropriate nonlinear operator on path space. As a consequence, we provide a variational formulation and resolution to evolution equations involving nonlinear operators such as the Navier-Stokes equation (in dimensions 2 and 3) with various boundary conditions. In dimension 2, we recover the well known solutions for the corresponding initial-value problem as well as periodic and anti-periodic ones, while in dimension 3 we get Leray solutions for the initial-value problems, but also solutions satisfying u(0)=αu(T) for any given α in (-1,1). Our approach is quite general and does apply to many other situations.