Schippa, Robert: Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations. 2019
Inhalt
- Introduction
- Notation and function spaces
- Schwartz functions and the Fourier transform
- Sobolev spaces and Fourier restriction spaces
- Functions of bounded variation and adaptations for dispersive equations
- Function spaces for frequency dependent time localization
- Modifications for tori
- Control of rough wave interactions via frequency dependent time localization
- Bilinear Strichartz estimates
- Linear Strichartz estimates
- Short-time nonlinear estimates
- Energy estimates
- Proof of local well-posedness via Bona-Smith approximation
- First applications
- New local well-posedness results for higher-dimensional Benjamin-Ono equations
- Introduction to higher-dimensional Benjamin-Ono equations
- Proof of new well-posedness results in Euclidean space
- Linear Strichartz estimates
- Bilinear Strichartz estimates
- Function spaces
- Short-time nonlinear estimates
- Energy estimates
- Periodic solutions to fractional Zakharov-Kuznetsov equations
- New regularity results for dispersive PDE with cubic derivative nonlinearities on the circle
- Quadratic dispersion relations
- Function spaces and Strichartz estimates
- Short-time trilinear estimate
- Energy estimates
- Proof of new regularity results for the modified Benjamin-Ono equation
- Modifications for the derivative nonlinear Schrödinger equation
- Cubic dispersion relation
- Local and global well-posedness for dispersion generalized Benjamin-Ono equations on the circle
- Introduction to dispersion generalized Benjamin-Ono equations
- Function spaces
- Linear and bilinear estimates
- Short-time bilinear estimates
- Energy estimates
- Variable-coefficient decoupling and smoothing estimates for elliptic and hyperbolic phase functions
- Introduction to variable-coefficient oscillatory integral operators
- Variable-coefficient decoupling for hyperbolic phase functions
- Basic reductions
- Rescaling of variable-coefficient phase functions
- Approximation by extension operators
- Conclusion of the proof
- Applications of variable-coefficient decoupling
- Discrete L2-restriction theorem
- Decoupling inequalities imply Strichartz inequalities and smoothing inequalities for variable coefficients
- Lp-smoothing estimates for elliptic phase functions with variable coefficients
- Summary
- Bibliography