{
  "id": "math/0702339",
  "version": "v1",
  "published": "2007-02-12T18:36:10.000Z",
  "updated": "2007-02-12T18:36:10.000Z",
  "title": "Anti-symmetric Hamiltonians (II): Variational resolutions for Navier-Stokes and other nonlinear evolutions",
  "authors": [
    "Nassif Ghoussoub",
    "Abbas Moameni"
  ],
  "comment": "25 pages. Updated versions --if any-- of this author's papers can be downloaded at http://pims.math.ca/~nassif/",
  "categories": [
    "math.AP"
  ],
  "abstract": "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)= \\int_0^T \\Big [ L (t, u(t),\\dot {u}(t)+\\Lambda u(t)) +< \\Lambda u(t), u(t) > \\Big ] dt +\\ell (u(0)- u(T), \\frac {u(T)+ u(0)}{2}) where $L$ (resp., $\\ell$) is an anti-selfdual Lagrangian on state space (resp., boundary space), and $\\Lambda$ 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 $u(0)=\\alpha u(T)$ for any given $\\alpha$ in $(-1,1)$. Our approach is quite general and does apply to many other situations.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-02-12T18:36:10.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "nonlinear evolutions",
      "anti-symmetric hamiltonians",
      "variational resolutions",
      "selfdual variational principle",
      "stationary nonlinear partial differential equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 25,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007math......2339G"
    }
  }
}