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Well-posedness of The Prandtl Equation in Sobolev Spaces
We develop a new approach to study the well-posedness theory of the Prandtl
equation in Sobolev spaces by using a direct energy method under a monotonicity
condition on the tangential velocity field instead of using the Crocco
transformation. Precisely, we firstly investigate the linearized Prandtl
equation in some weighted Sobolev spaces when the tangential velocity of the
background state is monotonic in the normal variable. Then to cope with the
loss of regularity of the perturbation with respect to the background state due
to the degeneracy of the equation, we apply the Nash-Moser-Hormander iteration
to obtain a well-posedness theory of classical solutions to the nonlinear
Prandtl equation when the initial data is a small perturbation of a monotonic
shear flow
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