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.