Gevrey regularity for the Navier–Stokes in a half-space

Well-Posedness of the Prandtl Equations Without Any Structural Assumption

“…Let us stress that the pressure in the Prandtl model is independent of y: its value is given by the pressure in the Euler flow at z = 0. This explains the right-hand side of (1), which depends only on t, x, and is coherent with the third boundary condition in (2).…”

“…Moreover, u ǫ remains (real) analytic in (x, y) as long as the Sobolev norm of u ǫ does not blow up, that is on (0, T ǫ,max ). This property, related to the analytic regularization of the heat kernel is well-known, even in the more difficult context of the Navier-Stokes equation: see [7,25,2] and references therein.…”

“…To relax the analyticity condition is much harder. In the special case where u in has for each value of x a single non-degenerate critical point in y, the first author and N. Masmoudi proved the local well-posedness of system (1)- (2) for data that are in Gevrey class 7/4 with respect to x [12]. Well-posedness was extended to Gevrey class 2 in article [23], for data that are small perturbations of a shear flow with a single non-degenerate critical point.…”

Identification of the initial population of a nonlinear predator-prey system backwards in time

“…To illustrate the ill-posedness of the backward problem given by equations (4), (7) and 9, we consider a simpler version without the convection terms, K 1 = K 2 = 0, and with globally Lipschitz reactions (F, G) satisfying F (0, 0) = G(0, 0) = 0 and max { F (u, v), G(u, v) } ≤ K( u + v ) for some K ≥ 0. First, we consider the forward problem given by equations (1)- (4). Let u 0 = v 0 = 0 and u n 0 = v n 0 = nφ n , where φ n is the n−th eigenfunction of −∆ in Ω with homogeneous Dirichlet boundary conditions on ∂Ω.…”

“…The homogeneous Dirichlet boundary conditions (4) express that the physical system is self-contained and that no populations at the boundary ∂Ω exist for any time t ∈ (0, T ), [20]. Other types of boundary conditions can also be considered.…”