Bilinear space-time estimates for homogeneous wave equations

Abstract

In this paper, we pursue a systematic treatment of the regularity theory for products and bilinear forms of solutions of the homogeneous wave equation. We discuss necessary and sufficient conditions for the validity of bilinear estimates, based on L2 norms in space and time, of derivatives of products of solutions. Also, we give necessary conditions and formulate some conjectures for similar estimates based on LqtLxr norms

Restricted convolution inequalities, multilinear operators and applications

“…We have the following estimate for products of solutions to these; see [10] for similar results with an additional average in time.…”

Almost optimal local well-posedness for the (3+1)-dimensional Maxwell–Klein–Gordon equations

“…2 As a simple example, take a generic system of equations of the form 2Ψ I = Ψ I ∇Ψ I . It can be shown that these will not be locally well posed for regularities at the level of H 1 or below (see [24]), even though they have the same scale transformations as (5)- (6). Furthermore, smooth solutions to these generic equations will blow up in finite time even for small initial data.…”

“…This is a huge blow to any usual type of iteration procedure, because dispersive type space-time estimates of the form 5 L q (L r ) play a central role in obtaining inductive estimates. 6 Nevertheless, it turns out that the dangerous part of the curvature F MKG which does not exhibit any extra space-time estimates miraculously cancels itself where it appears in the equation (7) for the scalar field φ! This allows one to recover enough space-time estimates for solutions to that equation so that one may prove local existence and uniqueness (again, in the appropriate sense) for the system (2) with initial data taken in the Sobolev spaces H s for any 1 2 < s. This provides the first example of an equation coming from (3 + 1)-dimensional gauge field theory where (part of) the local well-posedness conjecture contained in [13] can be verified.…”

“…Using the fact that (curl)∇ x ≡ 0 and that the resulting equation for the {A i } now forces them to be divergence free (so we can drop (12d)), we can rewrite the above system 5 These are mixed Lebesgue spaces to be defined in a moment. 6 Even worse, this failure of dispersion for the gauge potentials raises serious questions as to what happens in the case of nonabelian gauge fields. It would seem at first glance that this could cause the Yang-Mills equations to be ill-posed close to the scale invariant Sobolev space (alsȯ…”

“…Using certain standard eccentric initial data sets (see the next subsection and the work [6]), it is possible to show that (20) can only be true in the range 2 This failure is a serious setback to setting up an iteration procedure for the model equations, because it means that the first iterate to the corresponding time cutoff system is not in any subspace of the wave-Sobolev spaces X s, 1 2 for s close to the scaling. It is important to notice that this failure is not just a problem for the model; in fact, one sees immediately that the integral (20) would come up when trying to put the first iterate A (…”