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We consider the semilinear focusing wave equation

The authors consider the energy super critical semilinear heat equation \\partial _{t}u=\\Delta u+u^{p}, x\\in \\mathbb{R}^3, p>5. The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.

We consider the energy critical semilinear heat equation ∂ t u = Δ u + | u | 4 d - 2 u , x ∈ R d and give a complete classification of the flow near the ground state solitary wave Q ( x ) = 1 1 + | x | 2 d ( d - 2 ) d - 2 2 in dimension d ≥ 7 , in the energy critical topology and without radial symmetry assumption. Given an initial data Q + ε 0 with ‖ ∇ ε 0 ‖ L 2 ≪ 1 , the solution either blows up in the ODE type I regime, or dissipates, and these two open sets are separated by a codimension one set of solutions asymptotically attracted by the solitary wave. In particular, non self similar type II blow up is ruled out in dimension d ≥ 7 near the solitary wave even though it is known to occur in smaller dimensions (Schweyer, J Funct Anal 263(12):3922–3983, 2012 ). Our proof is based on sole energy estimates deeply connected to Martel et al. (Acta Math 212(1):59–140, 2014 ) and draws a route map for the classification of the flow near the solitary wave in the energy critical setting. A by-product of our method is the classification of minimal elements around Q belonging to the unstable manifold.

We consider the energy supercritical d+1-dimensional semi-linear heat equation \\begin{equation*} \\partial _tu=\\Delta u+u^{p}, \\ \\ x\\in \\mathbb{R}^{d+1}, \\ \\ p\\geq 3, \\ d\\geq 14. \\end{equation*} A fundamental open problem on this canonical nonlinear model is to understand the possible blow-up profiles appearing after renormalisation of a singularity. We exhibit in this paper a new scenario corresponding to the first example of a strongly anisotropic blow-up bubble: the solution displays a completely different behaviour depending on the considered direction in space. A fundamental step of the analysis is to solve the reconnection problem in order to produce finite energy solutions which is the heart of the matter. The corresponding anistropic mechanism is expected to be of fundamental importance in other settings in particular in fluid mechanics. The proof relies on a new functional framework for the construction and stabilisation of type II bubbles in the parabolic setting using energy estimates only, and allows us to exhibit new unexpected blow-up speeds.

We consider the inviscid unsteady Prandtl system in two dimensions, motivated by the fact that it should model to leading order separation and singularity formation for the original viscous system. We give a sharp expression for the maximal time of existence of regular solutions, showing that singularities only happen at the boundary or on the set of zero vorticity, and that they correspond to boundary layer separation. We then exhibit new Lagrangian formulae for backward self-similar profiles, and study them also with a different approach that was initiated by Elliott–Smith–Cowley and Cassel–Smith–Walker. One particular profile is at the heart of the so-called Van-Dommelen and Shen singularity, and we prove its generic appearance (that is, for an open and dense set of blow-up solutions) for any prescribed Eulerian outer flow. We comment on the connection between these results and the full viscous Prandtl system. This paper combines ideas for transport equations, such as Lagrangian coordinates and incompressibility, and for singularity formation, such as self-similarity and renormalisation, in a novel manner, and designs a new way to study singularities for quasilinear transport equations.

We consider the two-dimensional unsteady Prandtl system. For a special class of outer Euler flows and solutions of the Prandtl system, the trace of the tangential derivative of the tangential velocity along the transversal axis solves a closed one-dimensional equation. First, we give a precise description of singular solutions for this reduced problem. A stable blow-up pattern is found, in which the blow-up point is ejected to infinity in finite time, and the solutions form a plateau with growing length. Second, in the case where, for a general analytic solution, this trace of the derivative on the axis follows the stable blow-up pattern, we show persistence of analyticity around the axis up to the blow-up time, and establish a universal lower bound of (T - t)^7/4 for its radius of analyticity.

We show global asymptotic stability of solitary waves of the nonlinear Schrödinger equation in space dimension 1. Furthermore, the radiation is shown to exhibit long range scattering if the nonlinearity is cubic at the origin, or standard scattering if it is higher order. We handle a general nonlinearity without any vanishing condition, but requiring that the linearized operator around the solitary wave has neither nonzero eigenvalues, nor threshold resonances. Initial data are chosen in a neighborhood of the solitary waves in the natural space H^1 L^2,1 (where the latter is a weighted L^2 space). The proof combines for the first time modulation and renormalization techniques with the distorted Fourier transform.

We consider the kinetic wave equation arising in wave turbulence to describe the Fourier spectrum of solutions to the cubic Schrödinger equation. This equation has two Kolmogorov–Zakharov steady states corresponding to out-of-equilibrium cascades transferring, for the first solution mass from ∞ to 0 (small spatial scales to large scales), and for the second solution energy from 0 to ∞. After conjecturing the generic development of the two cascades, we verify it partially in the isotropic case by proving the nonlinear stability of the mass cascade in the stationary setting. This constructs non-trivial out-of-equilibrium steady states with a direct energy cascade as well as an indirect mass cascade.

We show global asymptotic stability of solitary waves of the nonlinear Schrödinger equation in space dimension 1. Furthermore, the radiation is shown to exhibit long range scattering if the nonlinearity is cubic at the origin, or standard scattering if it is higher order. We handle a general nonlinearity without any vanishing condition, requiring that the linearized operator around the solitary wave has neither nonzero eigenvalues, nor threshold resonances. Initial data are chosen in a neighborhood of the solitary waves in the natural space H 1 ∩ L 2,1 (where the latter is the weighted L 2 space). The proof combines for the first time modulation and renormalization techniques with the distorted Fourier transform.

The primitive equations (PEs) model large-scale dynamics of the oceans and the atmosphere. While it is by now well known that the three-dimensional viscous PEs are globally well posed in Sobolev spaces, and that there are solutions to the inviscid PEs (also called the hydrostatic Euler equations) that develop singularities in finite time, the qualitative description of the blowup still remains undiscovered. In this paper, we provide a full description of two blow-up mechanisms, for a reduced PDE that is satisfied by a class of particular solutions to the PEs. In the first one a shock forms, and pressure effects are subleading, but in a critical way: they localize the singularity closer and closer to the boundary near the blow-up time (with a logarithmic-in-time law). This first mechanism involves a smooth blow-up profile and is stable among smooth enough solutions. In the second one the pressure effects are fully negligible; this dynamics involves a two-parameter family of nonsmooth profiles, and is stable only by smoother perturbations.