Well-posedness for the Prandtl system without analyticity or monotonicity

Abstract

It has been thought for a while that the Prandtl system is only well-posed under the Oleinik monotonicity assumption or under an analyticity assumption. We show that the Prandtl system is actually locally well-posed for data that belong to the Gevrey class 7/4 in the horizontal variable x. Our result improves the classical local well-posedness result for data that are analytic in x (that is Gevrey class 1). The proof uses new estimates, based on non-quadratic energy functionals.

Landau Damping: Paraproducts and Gevrey Regularity

“…Norms such as G λ(t),σ ;s are common when dealing with analytic or Gevrey regularity, for example, see the works [10,21,22,25,27,45,50,67]. The Sobolev correction σ is included mostly for technical convenience.…”

“…To gain from the cancellations inherent in transport we follow the commutator trick used in (for example) [10,25,45,50] by applying the identity, 27) to write,…”

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

“…The verification of the instability in [48] motivates the work of [50], where the local solvability is established for a set of initial data without monotonicity, but belonging to the Gevrey class 7 4 in the x 1 variable. The key condition for the initial data in [50] is that the monotonicity is absent only on a single smooth curve but in a non-degenerate manner. More precisely, in [50] it is assumed that u E = p E = 0 and u P 0,1 is periodic in x 1 with Gevrey 7 4 regularity, and that…”

Vanishing viscosity limit of Navier–Stokes Equations in Gevrey class

“…Since the verification of the Prandtl boundary layer theory meet the major obstacle in the setting of the Sobolev space, it will be interesting to expect the vanishing viscosity limit for the incompressible Navier-Stokes equations in the setting of Gevery space as sub-space of Sobolev space, see a series of works in this direction [11,22,23]. In fact, Gevrey space is an intermediate space between the space of analytic functions and the Sobolev space.…”

“…So far the rigorous verification of the Prandtl boundary layer theory was achieved only for some specific settings, cf. [1,7,11,13,22,28,34] for instance, not to mention the convergence to Prandtl's equation and Euler equations. Several partial results on the inviscid limits, in the case of half-space, were given in [33] by imposing analyticity on the initial data, and in [26] for vorticity admitting compact support which is away from the boundary.…”