Project A9 • Spectral methods for dispersive equations (7/2015 - 6/2019)

Principal investigators

Project summary

In recent years, dispersive equations, such as the nonlinear Schrödinger equation or the nonlinear wave equation, have been studied on increasingly more general domains, manifolds, and graphs. Methods from spectral theory were therefore needed to support the classical harmonic analysis arguments. Furthermore, by randomizing the spectral decomposition of the Laplacian, J. Bourgain introduced probabilistic methods to prove wellposedness results in supercritical cases and to construct invariant (Gibbs) measures. In order to advance the theory of dispersive equations in random environments, these methods are currently being adapted to stochastic Schrödinger and wave equations by extending the recent progress on deterministic equations to their stochastic counterpart.

Based on our work on spectral theoretical formulations of the Littlewood–Paley theory, on random sum techniques in operator theory, and on our extension of stochastic analysis estimates such as Burkholder–Davis–Gundy inequalities and estimates for stochastic convolutions in the infinite dimensional \(L^p\)-setting, which are also based on functional calculus techniques, we will pursue the following goals:

• Investigate further the possibility of obtaining spectrally localized dispersive estimates directly from the spectral data of a selfadjoint operator \(A\), Gaussian bounds, and oscillatory integral representations for operator-valued functions. This will lead to Strichartz estimates, possibly with "loss of derivatives", and a corresponding wellposedness theory for nonlinear equations in this framework.

• Extend the random Sobolev embeddings for compact Riemmanian manifolds of N. Burq and G. Lebeau (initiated by J. Bourgain's work) to further classes of selfadjoint operators, particularly in "infinite volume" settings. Then prove wellposedness results for a large set of rough initial values.

• Adapt the very versatile method of M. Keel and T. Tao for proving Strichartz estimates to the stochastic setting and apply it to the wellposedness theory of stochastic dispersive equations, such as nonlinear Schrödinger and wave equations.

In investigating these problems, we aim to provide a unified approach to the various dispersive equations also considered in other projects of this proposal.

Publications

1. Z. Brzeźniak, F. Hornung, and L. Weis. Uniqueness of martingale solutions for the stochastic nonlinear Schrödinger equation on 3D compact manifolds. Stoch. PDE: Anal. Comp., 10(3):828–857, September 2022. URL https://doi.org/10.1007/s40072-022-00238-w. [preprint] [bibtex]

2. F. Hornung. The stochastic nonlinear Schrödinger equation in unbounded domains and non-compact manifolds. Nonlinear Differ. Equ. Appl., 27:40, July 2020. URL https://doi.org/10.1007/s00030-020-00642-y. [preprint] [bibtex]

3. Z. Brzeźniak, F. Hornung, and U. Manna. Weak martingale solutions for the stochastic nonlinear Schrödinger equation driven by pure jump noise. Stoch. PDE: Anal. Comp., 8(1):1–53, March 2020. URL https://doi.org/10.1007/s40072-019-00141-x. [preprint] [bibtex]

4. Z. Brzeźniak, F. Hornung, and L. Weis. Martingale solutions for the stochastic nonlinear Schrödinger equation in the energy space. Probab. Theory Related Fields, 174(3-4):1273–1338, August 2019. URL https://doi.org/10.1007/s00440-018-0882-5. [preprint] [bibtex]

5. F. Hornung. The nonlinear stochastic Schrödinger equation via stochastic Strichartz estimates. J. Evol. Equ., 18(3):1085–1114, September 2018. URL https://doi.org/10.1007/s00028-018-0433-7. [preprint] [bibtex]

6. C. Kriegler and L. Weis. Spectral multiplier theorems via $H^\infty$ calculus and $R$-bounds. Math. Z., 289(1-2):405–444, June 2018. URL https://doi.org/10.1007/s00209-017-1957-1. [bibtex]

7. T. Hytönen, J. van Neerven, M. Veraar, and L. Weis. Analysis in Banach spaces. Vol. II, volume 67 of Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics]. Springer, Cham, December 2017. Probabilistic methods and operator theory. [bibtex]

8. P. C. Kunstmann and L. Weis. New criteria for the $H^\infty$-calculus and the Stokes operator on bounded Lipschitz domains. J. Evol. Equ., 17(1):387–409, March 2017. URL https://doi.org/10.1007/s00028-016-0360-4. [preprint] [bibtex]

9. C. Kriegler and L. Weis. Paley–Littlewood decomposition for sectorial operators and interpolation spaces. Math. Nachr., 289(11-12):1488–1525, August 2016. URL https://doi.org/10.1002/mana.201400223. [preprint] [bibtex]

Theses

10. S. Schwarz. Fractional order splitting for semilinear evolution equations. PhD thesis, Karlsruhe Institute of Technology (KIT), October 2019. [bibtex]

11. F. Hornung. Global solutions of the nonlinear Schrödinger equation with multiplicative noise. PhD thesis, Karlsruhe Institute of Technology (KIT), April 2018. [bibtex]