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].

Catalytic hydrogenation of nitrate (NO3 −) in water on Pd−Cu ensembles has been denoted as a promising denitridation method, but its hydrogenation selectivity remains challenging. In this study, the hydrogenation selectivity of nitrate on the Pd−Cu/TiO2 systems was discussed mainly concerning the size effect of Pd−Cu ensembles in a gas−liquid cocurrent flow system. Demonstrated by their TEM images, homogeneous morphologies as well as narrow size distributions of Pd−Cu ensembles on titania have been prepared by a facile photodeposition process, and the size of the ensembles was controlled and varied with the total metal loadings. The different XRD patterns and XPS spectra of Pd−Cu/TiO2 catalysts from their corresponding monometallic counterparts suggested the formation of Pd−Cu complex on the surface of TiO2. It is first indicated that the hydrogenation selectivity of nitrate generally depends on the size of active phase with critical size of approximately 3.5 nm, below which NO2 − becomes the predominant product instead of nitrogen. Ammonium production was increasing slowly throughout the reaction, but this can be efficiently restrained by bubbling CO2. The optimal catalytic activity and nitrogen selectivity of 99.9% and 98.3% respectively could be achieved on the Pd−Cu/TiO2 catalyst with average size of 4.22 nm under the modification of CO2 after approximately 30 min reduction. The catalytic activities of nitrite on several Pd−Cu bimetallic catalysts were examined to strongly depend on the size of active metal, as is well responsible for the observed distinct hydrogenation selectivity of nitrate.

We prove peeling estimates for the small data solutions of the Maxwell Klein Gordon equations with non-zero charge and with a non-compactly supported scalar field, in (3 + 1) dimensions. We obtain the same decay rates as in an earlier work by Lindblad and Sterbenz, but giving a simpler proof. In particular we dispense with the fractional Morawetz estimates for the electromagnetic field, as well as certain spacetime estimates. In the case that the scalar field is compactly supported we can avoid fractional Morawetz estimates for the scalar field as well. All of our estimates are carried out using the double null foliation and in a gauge invariant manner.

Abstract. We consider the motion of the interface separating a vacuum from an inviscid, incompressible, and irrotational fluid, subject to the self-gravitational force and neglecting surface tension, in two space dimensions. The fluid motion is described by the Euler-Poission system in moving bounded simply connected domains. A family of equilibrium solutions of the system are the perfect balls moving at constant velocity. We show that for smooth data which are small perturbations of size ǫ of these static states, measured in appropriate Sobolev spaces, the solution exists and remains of size ǫ on a time interval of length at least cǫ −2 , where c is a constant independent of ǫ. This should be compared with the lifespan O(ǫ −1 ) provided by local well-posdness. The key ingredient of our proof is finding a nonlinear transformation which removes quadratic terms from the nonlinearity. An important difference with the related gravity water waves problem is that unlike the constant gravity for water waves, the self-gravity in the Euler-Poisson system is nonlinear. As a first step in our analysis we also show that the Taylor sign condition always holds and establish local well-posedness for this system.

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