Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation

By Carlos E. Kenig, F. Merle. 2006

519 citations recorded by the source corpus.

We study the energy-critical focusing non-linear wave equation, with data in the energy space, in dimensions 3, 4 and 5. We prove that for Cauchy data of energy smaller than the one of the static solution W which gives the best constant in the Sobolev embedding, the following alternative holds. If the initial data has smaller norm in the homogeneous Sobolev space H1 than the one of W, then we have global well-posedness and scattering. If the norm is larger than the one of W, then we have break-down in finite time.

Subfields: Applied Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/970f77bc2aa64c76a7b0d48da036c071d43248ba

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Citations in the corpus

Scattering for the radial 3D cubic wave equation (2014, methodology): “s, i.e., T+ <+∞ and T− >−∞. 3. Concentration compactness In thissectionwebegin the proofofTheorem1.1. We will followthe concentrationcompactness/rigidity method introduced by Kenig and Merle in [21, 22]. The concentration compactness part of the argument, which is based on the proﬁle decomposi”
Soliton resolution for equivariant wave maps to the sphere (2013, methodology): “We now do an induction on the energy in the spirit of the Kenig Merle concentration compactness argument [15, 16].”
Scattering threshold for the focusing nonlinear Klein–Gordon equation (2010, background): “5[26] is restricted to the dimensions d ≤ 5 for simplicity of the perturbation argument, but the elimination of critical elements works in any higher dimensions.”
Energy Dispersed Large Data Wave Maps in 2 + 1 Dimensions (2009, methodology): “In the case where the target M is a negatively curved Riemann surface, Krieger and Schlag [16] provide global regularity and scattering via a modification of the Kenig-Merle method [6], which uses as a key component suitably defined Bahouri-Gerard [1] type decompositions.”
Global regularity of wave maps III. Large energy from $\R^{1+2}$ to hyperbolic spaces (2008, background): “emark 1.17. Analogous claims to Claim 1.16 have been established for the energy-critical NLS in [2], [3], [7], [44], [77], [17], for the masscritical NLS in [19], [71], for the energy-critical NLW in [18] (see also [16]), and for the energy-critical and mass-critical Hartree equations in [35], [36],”
Regularity of Wave-Maps in Dimension 2 + 1 (2009, methodology): “These bounds are also essentially identical to the “rigidity estimate” of Kenig-Merle [12].”
The focusing energy-critical nonlinear Schr\"odinger equation in dimensions five and higher (2008, methodology): “uency cascade. In Section 8, we preclude the soliton. To achieve this, we ﬁrst use the negative regularity proved in Section 6 to deduce compactness properties for uin L2 x . Secondly, we argue as in [10, 19] to show that a minimal kinetic energy blowup solution must have zero momentum. Notice that ”
Global well-posedness and scattering for the mass critical nonlinear Schr{\"o}dinger equation with mass below the mass of the ground state (2011, unknown): “This method has since been applied to the focusing energy critical Schrödinger problem ([24], [31]) as well as the focusing energy critical wave equation, see [25].”
Slow blow-up solutions for the $H^1({\mathbb R}^3)$ critical focusing semilinear wave equation (2009, background): “1) ∂ttu−∆u− u = 0 there has been some recent activity, see [9], [8], [7], which we now describe in more detail.”