Marco Cannone

Chapter 3 - Harmonic Analysis Tools for Solving the Incompressible Navier–Stokes Equations

Introduction Section 1: Preliminaries 1.1 The Navier-Stokes equations 1.2 Classical, mild and weak solutions 1.3 Navier meets Fourier Section 2: Functional setting of the equations 2.1 The Littlewood-Paley decomposition 2.2 The Besov spaces 2.3 The paraproduct rule 2.4 The wavelet decomposition 2.5 Other useful function spaces Section 3: Existence theorems 3.1 The fixed point theorem 3.2 Scaling invariance 3.3 Super-critical case

Well-Posedness of the Boundary Layer Equations

We consider the mild solutions of the Prandtl equations on the half space. Requiring analyticity only with respect to the tangential variable, we prove the short time existence and the uniqueness of the solution in the proper function space. Theproof is achieved applying the abstract Cauchy--Kowalewski theorem to the boundary layer equations once the convection-diffusion operator is explicitly inverted. This improves the result of [M. Sammartino and R. E. Caflisch, Comm. Math. Phys., 192 (1998), pp. 433--461], as we do not require analyticity of the data with respect to the normal variable.

A losing estimate for the ideal mhd equations with application to blow-up criterion

In this paper we study the blow‐up criterion of smooth solution to the ideal MHD equations in

STRONG SOLUTIONS TO THE INCOMPRESSIBLE NAVIER-STOKES EQUATIONS IN THE HALF-SPACE

\R^n

Well-posedness of Prandtl equations with non-compatible data

. By means of the Fourier frequency localization and Bony paraproduct decomposition, we show a losing estimate for the ideal MHD equations and apply it to establish an improved blow‐up criterion of smooth solutions. As a special case, we recover a previous result of Planchon for the incompressible Euler equations.

Existence and uniqueness for the Prandtl equations

We derive an exact formula for solutions to the Stokes equations in the half space with an external forcing term. This formula is used to establish local and global existence and uniqueness in a suitable Besov space for solutions to the Navier Stokes equations. In particular, wellposedness is proved for initial data in L3(R3 +).

Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing

In this paper we shall be concerned with Prandtls equations with incompatible data, i.e. with initial data that, in general, do not fulfil the boundary conditions imposed on the solution. Under the hypothesis of analyticity in the streamwise variable, we shall prove that Prandtls equations, on the half-plane or on the half-space, are well posed for a short time.

Remarks on self-similar solutions for the surface quasi-geostrophic equation and its generalization

Abstract Under the hypothesis of analyticity of the data with respect to the tangential variable we prove the existence and uniqueness of the mild solution of Prandtl boundary layer equation. This can be considered an improvement of the results of [8] as we do not require analyticity with respect to the normal variable.

On the validity of the Picard algorithm for nonlinear parabolic equations

We consider the 2D quasi-geostrophic equation with supercritical dis- sipation and dispersive forcing in the whole space. When the dispersive amplitude parameter is large enough, we prove the global well-posedness of strong solution to the equation with large initial data. We also show the strong convergence result as the amplitude parameter goes to 1. Both results rely on the Strichartz-type estimates for the corresponding linear equation.

Smooth or singular solutions to the Navier–Stokes system ?

where Ri,γ = ∂xi(−Δ) γ 2 , i = 1, 2, γ ∈ [1, 2] are pseudo-differential operators which generalize the usual Riesz transform. When γ = 1, (1.1) is just the surface quasi-geostrophic equation which arises from the geostrophic fluids and is viewed as a two-dimensional model of the 3D Euler system (cf. [7,12]). When γ = 2, (1.1) corresponds to the classical 2D Euler equations in vorticity form. (1.1) in the case 1 < γ < 2 is the intermediate toy model introduced by Constantin et al. [6]. It is well-known that the 2D Euler equations preserve the global regularity for the smooth data (e.g. [1, 15]) by using the L∞-norm conservation of θ, while for the SQG equation and its generalization with 1 ≤ γ < 2, although they are of very simple form and have been intensely studied, it still remains open whether the solutions blow up at finite time or not. The equation (1.1) is invariant under the scaling transformations that for α > −1, θ(t, x) → θλ(t, x) = λα+γ−1θ(λα+1t, λx), ∀λ > 0. We say a solution is self-similar if θ = θλ for all λ > 0. The self-similar blowup singularity is an important scenario that may occur in the evolution of θ and is the main concern in this note. More precisely, we assume there exists (t∗, x∗) ∈ R×R such that the solution θ develops a self-similar singularity at (t∗, x∗) of the form