Rough invariant waves

Project data:

POLS grant 2020/37/K/ST1/02765.
Financed by the Norwegian Financial Mechanism 2014-2021, as part of the Basic Research Programme. Operated by the National Science Centre under the Norway Grants, and prepared in cooperation with the Research Council of Norway.
01-04-2021 to 31-03-2024.
131,231 EUR (576,250 PLN).
 

Project goal:

The project will explore the fixed-time L^p regularity of wave equations with very rough coefficients, and it will develop tools to apply this theory to nonlinear wave equations with rough initial data.
For a more accessible description see here.
 

Research output:

1. Local smoothing and Hardy spaces for Fourier integral operators on manifolds. J. Funct. Anal. 286 (2024), no. 2, Paper No. 110221, 72 pp. With N. Liu, L. Song and L. Yan.
2. Operator-valued (L^p,L^q) Fourier multipliers and stability theory for evolution equations. Indag. Math. 34 (2023), no. 1, 1-36. Preprint available here.
3. Nonlinear wave equations with slowly decaying initial data. J. Differential Equations 350 (2023), 152-188. With R. Schippa. Preprint available here.
4. Local smoothing and Hardy spaces for Fourier integral operators. J. Funct. Anal. 283 (2022), no. 12, Paper No. 109721, 22 pp. Preprint available here.
5. Rough pseudodifferential operators on Hardy spaces for Fourier integral operators II. J. Fourier Anal. Appl. 28 (2022), no. 4, Paper No. 65, 27 pp. Partially supported by funding from the grant. Preprint available here.
6. Rough pseudodifferential operators on Hardy spaces for Fourier integral operators. J. Anal. Math. 149 (2023), no. 1, 135-165. Partially supported by funding from the grant. Preprint available here.

Progress:

The project has been concluded, and most of the initial goals of the project have been achieved, although for part of the research it turned out to be more fruitful to explore a different set of ideas. 
More precisely, publication 5, which improves upon publication 6, allowed Hassell and myself to extend our results on wave equations with rough coefficients to lower levels of regularity in dimensions 3 and higher (see here), thereby already achieving part of the first goal of the project. On the other hand, both for applications and due to their inherent interest, a significant chunk of the project ended up being devoted to local smoothing estimates, cf. publications 1 and 4. These estimates also have applications to nonlinear wave equations, as was demonstrated in publication 3. Moreover, publications 5 and 6 provide one with possible tools for further applications to nonlinear equations. 
Publication 2 is a survey on a tangential and slightly related topic.
 

Presentations:

• Geometric Function and Mapping Theory seminar, IMPAN. November 2023.
• Sun Yat-Sen University, China. October 2023.
• Kiel University, Germany. April 2023.
• AustMS Annual Meeting 2022. Sydney, Australia. December 2022.
• Analysis and PDE seminar. Australian National University, Australia. November 2022.
• Analysis seminar. Monash University, Australia. November 2022.
• Analysis seminar. Macquarie University, Australia. November 2022.
• PDE seminar. Georgia Institute of Technology, United States. September 2022. Video presentation. 
• Analysis seminar. Delft University of Technology, Netherlands. June 2022.
• Rajchman, Zygmund, Marcinkiewicz, anniversary conference. IMPAN, Poland. October 2021. 
• Ghent Methusalem Junior Seminar. Ghent University, Belgium. October 2021. Video presentation. 
• Rough Wave Equations. Mathematisches Forschungsinstitut Oberwolfach, Germany. September 2021. 

Travels:

• Sun Yat-Sen University, China. 4-17 October 2023.
• Kiel University, Germany. 17-21 April 2023.
• AustMS Annual Meeting 2022. Sydney, Australia. 6-9 December 2022. 
• Australian National University, Australia. 13 November to 4 December 2022.
• Sydney, Canberra and Melbourne, Australia. 28 October to 12 November 2022.
• Workshop on Microlocal Analysis & PDEs. University College London, United Kingdom. 20-23 July 2022.
• New Challenges in Operator Semigroups. Oxford University, United Kingdom. 17-19 July 2022.
• Fourier Analysis @200. University of Edinburgh, United Kingdom. 26 June to 2 July 2022.
• Delft University of Technology, Netherlands. 8-23 June 2022. 
• Rough Wave Equations. Mathematisches Forschungsinstitut Oberwolfach, Germany. 13-18 September 2021.