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Abstract This paper presents some results concerning the size of magnetic fields that support zero modes for the three-dimensional Dirac equation and related problems for spinor equations. It is a well-known fact that for the Schrödinger equation in three dimensions to have a negative energy bound state, the 3 / 2 {3/2} norm of the potential has to be greater than the Sobolev constant. We prove an analogous result for the existence of zero modes, namely that the 3 / 2 {3/2} norm of the magnetic field has to greater than twice the Sobolev constant. The novel point here is that the spinorial nature of the wave function is crucial. It leads to an improved diamagnetic inequality from which the bound is derived. While the results are probably not sharp, other equations are analyzed where the results are indeed optimal.
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This content will become publicly available on February 1, 2026
On the wave turbulence theory of 2D gravity waves, I: Deterministic energy estimates
Abstract Our goal in this paper is to initiate the rigorous investigation of wave turbulence and derivation of wave kinetic equations (WKEs) for water waves models. This problem has received intense attention in recent years in the context of semilinear models, such as Schrödinger equations or multidimensional KdV‐type equations. However, our situation here is different since the water waves equations are quasilinear and solutions cannot be constructed by iteration of the Duhamel formula due to unavoidable derivative loss. This is the first of two papers in which we design a new strategy to address this issue. We investigate solutions of the gravity water waves system in two dimensions. In the irrotational case, this system can be reduced to an evolution equation on the one‐dimensional interface, which is a large torus of size . Our first main result is a deterministic energy inequality, which provides control of (possibly large) Sobolev norms of solutions for long times, under the condition that a certain ‐type norm is small. This energy inequality is of “quintic” type: if the norm is , then the increment of the high‐order energies is controlled for times of the order , consistent with the approximate quartic integrability of the system. In the second paper in this sequence, we will show how to use this energy estimate and a propagation of randomness argument to prove a probabilistic regularity result up to times of the order , in a suitable scaling regime relating and . For our second main result, we combine the quintic energy inequality with a bootstrap argument using a suitable ‐norm of Strichartz‐type to prove that deterministic solutions with Sobolev data of size are regular for times of the order . In particular, on the real line, solutions exist for times of order . This improves substantially on all the earlier extended lifespan results for 2D gravity water waves with small Sobolev data.
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- Award ID(s):
- 2245228
- PAR ID:
- 10598318
- Publisher / Repository:
- Wiley Periodicals LLC
- Date Published:
- Journal Name:
- Communications on Pure and Applied Mathematics
- Volume:
- 78
- Issue:
- 2
- ISSN:
- 0010-3640
- Page Range / eLocation ID:
- 211 to 322
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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