The paper is devoted to the study of shock formation of the 3-dimensional quasilinear wave equation − 1 + 3G (0)(∂tφ) 2 ∂ 2 t φ + ∆φ = 0, ( ) where G (0) is a non-zero constant. We will exhibit a family of smooth initial data and show that the foliation of the incoming characteristic hypersurfaces collapses. Similar to 1-dimensional conservational laws, we refer this specific type breakdown of smooth solutions as shock formation. Since ( ) satisfies the classical null condition, it admits global smooth solutions for small data. Therefore, we will work with large data (in energy norm). Moreover, no symmetry condition is imposed on the initial datum.We emphasize the geometric perspectives of shock formations in the proof. More specifically, the key idea is to study the interplay between the following two objects:(1) the energy estimates of the linearized equations of ( );(2) the differential geometry of the Lorentzian metric g = − 1 (1 + 3G (0)(∂tφ) 2 ) dt 2 +dx 2 1 +dx 2 2 +dx 2 3 . Indeed, the study of the characteristic hypersurfaces (implies shock formation) is the study of the null hypersurfaces of g.The techniques in the proof are inspired by the work [5] in which the formation of shocks for 3dimensional relativistic compressible Euler equations with small initial data is established. We also use the short pulse method which is introduced in the study of formation of black holes in general relativity in [6] and generalized in [13].

Abstract. Magnetohydrodynamics (MHD) studies the dynamics of magnetic fields in electrically conducting fluids. In addition to the sound wave and electromagnetic wave behaviors, magneto-fluids also exhibit an interesting phenomenon: They can produce the Alfvén waves, which were first described in a physics paper by Hannes Alfvén in 1942. Subsequently, Alfvén was awarded the Nobel prize for his fundamental work on MHD with fruitful applications in plasma physics, in particular the discovery of Alfvén waves.This work studies (and constructs) global solutions for the three dimensional incompressible MHD systems (with or without viscosity) in strong magnetic backgrounds. We present a complete and selfcontained mathematical proof of the global nonlinear stability of Alfvén waves. Specifically, our results are as follows:• We obtain asymptotics for global solutions of the ideal system (i.e.,viscosity µ = 0) along characteristics; in particular, we have a scattering theory for the system.• We construct the global solutions (for small viscosity µ) and we show that as µ → 0, the viscous solutions converge in the classical sense to the zero-viscosity solution. Furthermore, we have estimates on the rate of the convergence in terms of µ.• We explain a linear-driving decay mechanism for viscous Alfvén waves with arbitrarily small diffusion. More precisely, for a given solution, we exhibit a time Tn 0 (depending on the profile of the datum rather than its energy norm) so that at time Tn 0 the H 2 -norm of the solution is small compared to µ (therefore the standard perturbation approach can be applied to obtain the convergence to the steady state afterwards).The results and proofs have the following main features and innovations:• We do not assume any symmetry condition on initial data. The size of initial data (and the a priori estimates) does not depend on viscosity µ. The entire proof is built upon the basic energy identity.• The Alfvén waves do not decay in time: the stable mechanism is the separation (geometrically in space) of left-and right-traveling Alfvén waves. The analysis of the nonlinear terms are analogous to the null conditions for non-linear wave equations.• We use the (hyperbolic) energy method. In particular, in addition to the use of usual energies, the proof relies heavily on the energy flux through characteristic hypersurfaces.• The viscous terms are the most difficult terms since they are not compatible with the hyperbolic approach. We obtain a new class of space-time weighted energy estimates for (weighted) viscous terms. The design of weights is one of the main innovations and it unifies the hyperbolic energy method and the parabolic estimates.• The approach is 'quasi-linear' in nature rather than a linear perturbation approach: the choices of the coordinate systems, characteristic hypersurfaces, weights and multiplier vector fields depend on the solution itself. Our approach is inspired by Christodoulou-Klainerman's proof of the nonlinear stability of Minkowski space-time in general relativity.

It is well-known that in dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data. It is also known that in dimension two such result still holds for a certain class of quasi-linear wave equations with null conditions. The proofs are based on the decay mechanism of linear waves. However, in one dimension, waves do not decay. Nevertheless, we will prove that small data still lead to global solutions if the null condition is satisfied.

Abstract. We show the existence of complete, asymptotically flat Cauchy initial data for the vacuum Einstein field equations, free of trapped surfaces, whose future development must admit a trapped surface. Moreover, the datum is exactly a constant time slice in Minkowski space-time inside and exactly a constant time slice in Kerr space-time outside.The proof makes use of the full strength of Christodoulou's work on the dynamical formation of black holes and Corvino-Schoen's work on the constructions of initial data set.

Abstract. For semi-linear wave equations with null form non-linearities on R 3+1 , we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future.We also exhibit a set of localized data for which the corresponding solutions are strongly focused, which in geometric terms means that a wave travels along an specific incoming null geodesic in such a way that almost all of the energy is confined in a tubular neighborhood of the geodesic and almost no energy radiating out of this tubular neighborhood.