Anti-selfdual Hamiltonians: Variational resolutions for Navier-Stokes and other nonlinear evolutions
Nassif Ghoussoub
TL;DR
The paper develops a variational framework based on anti-selfdual Lagrangians and their associated Hamiltonians to solve nonlinear stationary and evolution equations, with Navier–Stokes and hydrodynamic-type systems as primary applications. By exploiting zeros of the Lagrangian and path-space liftings, it provides existence (and sometimes uniqueness) results for PDEs that are not Euler–Lagrange equations in the classical sense. It introduces nonlinear variational principles, nonlinear Lax–Milgram-type results, and path-space formulations that handle boundary conditions via self-dual boundary Lagrangians, as well as unbounded operators through $\lambda$-regularization. The approach yields variational resolutions for a broad class of nonlinear evolutions, including Navier–Stokes, transport-type operators, and coupled PDE systems, offering a versatile tool for existence proofs in fluid dynamics and related fields.
Abstract
The theory of anti-selfdual (ASD) Lagrangians developed in \cite{G2} allows a variational resolution for equations of the form $\Lambda u+Au +\partial \phi (u)+f=0$ where $\phi$ is a convex lower-semi-continuous function on a reflexive Banach space $X$, $f\in X^*$, $A: D(A)\subset X\to X^*$ is a positive linear operator and where $\Lambda: D(\Lambda)\subset X\to X^{*}$ is a non-linear operator that satisfies suitable continuity and anti-symmetry properties. ASD Lagrangians on path spaces also yield variational resolutions for nonlinear evolution equations of the form $\dot u (t)+\Lambda u(t)+Au(t) +f\in -\partial \phi (u(t))$ starting at $u(0)=u_{0}$. In both stationary and dynamic cases, the equations associated to the proposed variational principles are not derived from the fact they are critical points of the action functional, but because they are also zeroes of the Lagrangian itself.The approach has many applications, in particular to Navier-Stokes type equations and to the differential systems of hydrodynamics, magnetohydrodynamics and thermohydraulics.