Global well-posedness below the charge norm for the Dirac-Klein-Gordon system in one space dimension

We prove global well-posedness below the charge norm (i.e., the $L^2$ norm of the Dirac spinor) for the Dirac-Klein-Gordon system of equations (DKG) in one space dimension. Adapting a method due to Bourgain, we split off the high frequency part of the initial data for the spinor, and exploit nonlinear smoothing effects to control the evolution of the high frequency part. To prove the nonlinear smoothing we rely on the null structure of the DKG system, and bilinear estimates in Bourgain-Klainerman-Machedon spaces

An Aconceptual View of Mind and World

In Mind and World (1994/1996), John McDowell follows Donald Davidson in claiming that the world is a conceptually laden structure. A (conceptual) language and tradition constitutes the world, and our (conceptual) "openness to the world� (ibid, p.155). This means that the condition for access to the world is a clear subject – object split, and a clear split between content and the way the content is presented. With this view as the basis he criticizes the idea of a non-conceptual1 experience and non-conceptual content, starting from the demand that (conceptual) thinking must be constrained by, and rationally answerable to the empirical world (ibid p.xii)

Bilinear Estimates and Applications to Nonlinear Wave Equations

We undertake a systematic review of some results concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we provide a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates

Anisotropic bilinear L2L^2 estimates related to the 3d wave equation

We first review the $L^2$ bilinear generalizations of the $L^4$ estimate of Strichartz for solutions of the homogeneous 3D wave equation, and give a short proof based solely on an estimate for the volume of intersection of two thickened spheres. We then go on to prove a number of new results, the main theme being how additional, anisotropic Fourier restrictions influence the estimates. Moreover, we prove some refinements which are able to simultaneously detect both concentrations and nonconcentrations in Fourier space.Comment: 41 pages, 1 figur

On an estimate for the wave equation and applications to nonlinear problems

We prove estimates for solutions of the Cauchy problem for the inhomogeneous wave equation on $\R^{1+n}$ in a class of Banach spaces whose norms only depend on the size of the space-time Fourier transform. The estimates are local in time, and this allows one, essentially, to replace the symbol of the wave operator, which vanishes on the light cone in Fourier space, with an inhomogeneous symbol, which can be inverted. Our result improves earlier estimates of this type proved by Klainerman-Machedon. As a corollary, one obtains a rather general result concerning local well-posedness of nonlinear wave equations

On the radius of spatial analyticity for the quartic generalized KdV equation

Lower bound on the rate of decrease in time of the uniform radius of spatial analyticity of solutions to the quartic generalized KdV equation is derived, which improves an earlier result by Bona, Gruji\'c and Kalisch.Comment: 11 pages. arXiv admin note: substantial text overlap with arXiv:1706.0465

Finite-energy global well-posedness of the Maxwell-Klein-Gordon system in Lorenz gauge

It is known that the Maxwell-Klein-Gordon system (M-K-G), when written relative to the Coulomb gauge, is globally well-posed for finite-energy initial data. This result, due to Klainerman and Machedon, relies crucially on the null structure of the main bilinear terms of M-K-G in Coulomb gauge. It appears to have been believed that such a structure is not present in Lorenz gauge, but we prove here that it is, and we use this fact to prove finite-energy global well-posedness in Lorenz gauge. The latter has the advantage, compared to Coulomb gauge, of being Lorentz invariant, hence M-K-G in Lorenz gauge is a system of nonlinear wave equations, whereas in Coulomb gauge the system has a less symmetric form, as it contains also a nonlinear elliptic equation.Comment: 25 page

Global well-posedness of the Chern-Simons-Higgs equations with finite energy

We prove that the Cauchy problem for the Chern-Simons-Higgs equations on the (2+1)-dimensional Minkowski space-time is globally well posed for initial data with finite energy. This improves a result of Chae and Choe, who proved global well-posedness for more regular data. Moreover, we prove local well-posedness even below the energy regularity, using the the null structure of the system in Lorenz gauge and bilinear space-time estimates for wave-Sobolev norms.Comment: 16 page

Bilinear Fourier restriction estimates related to the 2d wave equation

We study bilinear $L^2$ Fourier restriction estimates which are related to the 2d wave equation in the sense that we restrict to subsets of thickened null cones. In an earlier paper we studied the corresponding 3d problem, obtaining several refinements of the Klainerman-Machedon type estimates. The latter are bilinear generalizations of the $L^4$ estimate of Strichartz for the 3d wave equation. In 2d there is no $L^4$ estimate for solutions of the wave equation, but as we show here, one can nevertheless obtain $L^2$ bilinear estimates for thickened null cones, which can be viewed as analogues of the 3d Klainerman-Machedon type estimates. We then prove a number of refinements of these estimates, analogous to those we obtained earlier in 3d. The main application we have in mind is the Maxwell-Dirac system.Comment: 19 page

Almost optimal local well-posedness of the Maxwell-Klein-Gordon equations in 1+4 dimensions

We prove that the Maxwell-Klein-Gordon equations on $\R^{1+4}$ relative to the Coulomb gauge are locally well-posed for initial data in $H^{1+\epsilon}$ for all $\epsilon > 0$. This builds on previous work by Klainerman and Machedon who proved the corresponding result for a model problem derived from the Maxwell-Klein-Gordon system by ignoring the elliptic features of the system, as well as cubic terms