Mathematics – Analysis of PDEs
Scientific paper
2005-03-24
Mathematics
Analysis of PDEs
32 pages. For the most updated version of this paper, please visit http://www.pims.math.ca/~nassif/pims_papers.html
Scientific paper
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.
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