Abstract
We introduce a variational approach to treat the regularity of the Navier–Stokes equations both in dimensions 2 and 3. Though the method allows the full treatment in dimension 2, we seek to precisely stress where it breaks down for dimension 3. The basic feature of the procedure is to look directly for strong solutions, by minimizing a suitable error functional that measures the departure of feasible fields from being a solution of the problem. By considering the divergence-free property as part of feasibility, we are able to avoid the explicit analysis of the pressure. Two main points in our analysis are:
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1
Coercivity for the error functional is achieved by looking at scaling.
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2
Global minimizers of the error are shown to have zero error (and thus they are solutions of the problem) by looking at optimality conditions, which lead to investigate the linearized problem.
The method can also be applied to other situations, and numerical approximations may be pursued within this framework.
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Communicated by I. Straskraba
Research supported by MTM2010–19739 of the MICINN (Spain), PCI08-0084-0424 of the JCCM (Castilla-La Mancha), and by grant PR2008-0148 of the MICINN (Spain) for a sabbatical in the Mathematics Department of Princeton University.
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Pedregal, P. A Variational Approach for the Navier–Stokes System. J. Math. Fluid Mech. 14, 159–176 (2012). https://doi.org/10.1007/s00021-011-0058-x
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DOI: https://doi.org/10.1007/s00021-011-0058-x