Bounded Domain Limit for Navier-Stokes and Euler Equations
Analytical and Stochastic Fluid Dynamics
October 11, 2005 04:45 PM to 05:15 PM
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Abstract: |
Suppose we have well-posedness for some PDE both in all of R^d and in a bounded domain U of R^d. Imagine we take the initial data for the whole space and restrict it to U, modifying it slightly to satisfy any required boundary conditions. If we scale U by r, does the solution on rU approach the solution in R^d as r goes to infinity in some appropriate sense? We answer this question for weak solutions to the Navier-Stokes and Euler equations in two dimensions, showing that strong convergence occurs in some norms and weak convergence in other norms of interest. Critical to this is showing that the initial velocity can be modified to match the appropriate boundary conditions while changing its norms by a vanishingly small amount as r goes to infinity. A secondary goal is to establish, for initial velocity that decays at infinity as slowly as possible, the same existence, uniqueness, and regularity results as for classical (that is, finite energy) solutions.
The original motivation for this work was to establish the existence and uniqueness of statistical solutions to the Navier-Stokes and Euler equations in the entire plane with infinite energy and, for the Euler equations, unbounded initial vorticities, by taking advantage of the well developed theory of statistical solutions in a bounded domain of the plane.
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