Rough invariant waves

Description

The project is concerned with the fixed-time 𝐿𝑝 regularity of wave equations with rough coefficients, and its applications. Recently, new tools have been developed for the fixed-time 𝐿𝑝 regularity of wave equations with rough coefficients, whereas before only sharp results about the 𝐿𝑝 regularity of wave equations with smooth coefficients were known. These new tools include Hardy spaces for Fourier integral operators, denoted by 𝐻𝐹𝐼𝑂𝑝(ℝ𝑛) for 1≤𝑝≤∞, which facilitate iterative constructions involving Fourier integral operators that were not previously available on 𝐿𝑝(ℝ𝑛) for 𝑝≠2. Indeed, for 𝑛>1 and 𝑝≠2 it has long been known that the solution operators to wave equations are typically not bounded on 𝐿𝑝(ℝ𝑛), and the optimal result is that these operators “lose” 𝑠(𝑝)=(𝑛−1)|12−1𝑝| derivatives on 𝐿𝑝(ℝ𝑛) for 1<𝑝<∞. Such a loss prohibits the use of iterative constructions on 𝐿𝑝 to solve either wave equations with rough coefficients by approximation, or nonlinear equations by linearization. By contrast, the solution operators to wave equations with smooth coefficients are bounded on the Hardy spaces for Fourier integral operators. Moreover, these spaces satisfy the Sobolev embeddings 𝑊𝑠(𝑝)2,𝑝(ℝ𝑛)⊆𝐻𝐹𝐼𝑂𝑝(ℝ𝑛)⊆𝑊−𝑠(𝑝)2,𝑝(ℝ𝑛) for all 1<𝑝<∞, allowing one to recover the optimal 𝐿𝑝 regularity by working on 𝐻𝐹𝐼𝑂𝑝(ℝ𝑛). In this sense, the loss of 𝐿𝑝 regularity for wave equations is only apparent, and iterative constructions are possible after all, by first iterating on 𝐻𝐹𝐼𝑂𝑝(ℝ𝑛) and then afterwards applying the Sobolev embeddings. The goal of this project is to push the 𝐿𝑝 theory for wave equations with rough coefficients to its limit, by determining how rough the coefficients of a wave equation can be before the 𝐿𝑝 regularity theory breaks down. Concurrently, the project will apply the available results on 𝐿𝑝 regularity for linear wave equations to nonlinear wave equations with rough initial data.

Summary of project results

The project was concerned with determining, in a quantitative sense, how rough an object needs to be before it becomes difficult to predict the behavior of waves on that object. This roughness can be quantified by assigning a number to the object, the critical smoothness number.

Before the project commenced, the principal investigator and a collaborator showed that, for a specific type of wave behavior called regularity and for two-dimensional objects called surfaces, the critical smoothness number is not larger than two. One of the main goals of the project was to determine whether the critical smoothness number of surfaces is in fact two, or less than two. One could ask the same question for objects in higher dimensions.

Another goal of the project was to develop tools to study the regularity of more complex types of waves, called nonlinear waves, that do not form isolated systems but involve complicated feedback mechanisms.

The principal investigator first conducted a literature review, comparing his earlier results to the literature on analogous problems for which the critical smoothness number had already been determined.
Techniques used for those problems were modified and adapted to the setting of the project.It then turned out that a modification of existing tools could be used to attack the second goal of the project, whereas for the first goal it would be more beneficial to try a different approach. Instead of attempting to directly determine the critical smoothness number associated with  regularity, it seems to be more insightful to study the critical smoothness number associated with a different phenomenon called local smoothing.Subsequently, part of the project shifted its focus to the study of local smoothing, which is also of great independent interest.

New tools were developed to study the  regularity of the more complicated systems called nonlinear waves, thereby achieving the second main goal of the project. Such tools can in turn be used by other researchers in the field, to create a greater understanding of these complex wave equations.
The new tools were also used, by the principal investigator and a collaborator, to show that the critical smoothness number of objects in higher dimensions is not larger than two. For now, it remains an open question whether the critical smoothness number of either surfaces or higher-dimensional objects is two, or less than two.On the other hand, the principal investigator obtained results that led to a greater understanding of the phenomenon called local smoothing, which benefits researchers working on this separate problem. It is also expected that the study of local smoothing will provide an alternative approach to the problem of determining the critical smoothness number of various objects.The results of the project will benefit both researchers working in the same area as the principal investigator, and in other parts of mathematics. In turn, since waves of various kinds are ubiquitous in the universe, a greater understanding of wave equations also leads to a better understanding of the world around us.