Concentration Compactness for Critical Radial Wave Maps

The concentration-compactness/rigidity method for critical dispersive and wave equations

Concentration Compactness for the Critical Maxwell-klein-gordon Equation

We prove global regularity, scattering and a priori bounds for the energy critical MaxwellKlein-Gordon equation relative to the Coulomb gauge on (1 + 4)-dimensional Minkowski space. The proof is based upon a modified Bahouri-Gérard profile decomposition [1] and a concentration compactness/rigidity argument by Kenig-Merle [5], following the method developed by the first author and Schlag [10] in...

Relative Compactness and Product Stable Quotient Maps

Concentration of Local Energy for Two-dimensional Wave Maps

We construct some particular kind of solution to the two dimensional equivariant wave map problem in space-time domain of type Ωα(t) = {x ∈ R2 : |x|α < t}, where α ∈ (0, 1], with a source term. More precisely, taking the initial data (u0, u1) in time T to belongs to the Sobolev space H1+ε × Hε with some ε > 0, the source term is in L1((0, T ); Hε (Ωα(t))), we show that the H1+ε norm of the solu...

Optimal polynomial blow up range for critical wave maps

We prove that the critical Wave Maps equation with target S2 and origin R2+1 admits energy class blow up solutions of the form u(t ,r ) =Q(λ(t )r )+ε(t ,r ) where Q : R2 → S2 is the ground state harmonic map and λ(t) = t−1−ν for any ν > 0. This extends the work [17], where such solutions were constructed under the assumption ν> 2 . Also in the later chapter, we give the necessary remarks and ke...