Invariant Gibbs measures for dispersive partial differential equations

Liang, Rui ORCID: 0009-0000-2092-345X (2024). Invariant Gibbs measures for dispersive partial differential equations. University of Birmingham. Ph.D.

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Abstract

This thesis focuses on the construction of Gibbs measures, their invariance under the flows of Hamiltonian partial differential equations, and their application in understanding macroscopic properties of these partial differential equations. The presentation of these focuses will be carried out through the example of Gibbs measure for the fractional nonlinear Schr\"odinger equations posed on the torus.

In Chapter 2, we consider the Cauchy problem for the cubic nonlinear fractional Schr\"odinger equation (FNLS) on the circle, considering initial data distributed via the Gibbs measure. By reformulating the local theory via the random averaging operator theory from Deng-Nahmod-Yue [34, 35], we construct global strong solutions with the flow property for FNLS on the support of the Gibbs measure in the full dispersive range, thus addressing a question proposed by Sun-Tzvetkov [105]. Additionally, we prove the invariance of the Gibbs measure and the sharpness of the result. This chapter mainly comes from [70].

In Chapter 3, we study the Gibbs measures for the focusing mass-critical fractional nonlinear Schr\"odinger equation on the torus $\mathbb{T}^d=(\mathbb{R}/(2\pi\mathbb{Z}))^d$. We identify the critical nonlinearity and optimal mass threshold for normalisablity and non-normalisability of the Gibbs measures for the fractional nonlinear Schr\"odinger equation on the multi-dimensional torus, which extends the works of Lebowitz-Rose-Speer [68], Bourgain [2], and Oh-Sosoe-Tolomeo [85] on the nonlinear Schr\"odinger equations on one-dimensional torus $\mathbb{T}$. To achieve this purpose, we prove an almost sharp fractional Gagliardo-Nirenberg-Sobolev inequality on the multi-dimensional torus. This chapter is from [69].

Type of Work:

Thesis (Doctorates > Ph.D.)

Award Type:

Doctorates > Ph.D.

Supervisor(s):