[Paper Review] Invariance of the Gibbs measure for the Benjamin-Ono equation

Yu Deng|arXiv (Cornell University)|Oct 4, 2012

Advanced Mathematical Physics Problems7 references3 citations

TL;DR

This paper establishes the almost-sure global existence and uniqueness of solutions to the periodic Benjamin-Ono equation in a Besov-type space $ Z_1 $, which is rougher than $ L^2 $, and proves the invariance of the associated Gibbs measure under the flow. The proof combines a novel local well-posedness result in rougher Besov spaces with a limiting argument using truncated equations and measure approximation, resolving a long-standing open problem in the statistical mechanics of Hamiltonian PDEs.

ABSTRACT

In this paper we consider the periodic Benjemin-Ono equation. We will establish the invariance of the Gibbs measure associated to this equation, thus answering a question raised in Tzvetkov [20]. As an intermediate step, we also obtain a local well-posedness result in Besov-type spaces rougher than L^2, extending the L^2 well-posedness result of Molinet [14].

Motivation & Objective

• To resolve the open problem of Gibbs measure invariance for the periodic Benjamin-Ono equation, which was posed by Tzvetkov (2008).
• To extend the local well-posedness theory of Molinet (2010) from $ L^2 $ to a rougher Besov-type space $ Z_1 $, enabling treatment of the Gibbs measure's support.
• To construct a global flow on the support of the Gibbs measure, which is supported in spaces below $ L^2 $, by combining gauge transforms and probabilistic methods.
• To prove that the Gibbs measure is preserved under the time evolution of the equation, thus providing a natural invariant measure for the system’s long-time statistical behavior.

Proposed method

• Introduces a new Besov-type space $ Z_1 $ with a parameter $ s > 0 $, where $ Z_1 $ is defined via a weighted $ l^p $ norm on Fourier coefficients with $ p = \frac{2}{1-2s} + s^2 $, enabling control of low-regularity solutions.
• Establishes local well-posedness in $ Z_1 $ for small initial data using a gauge transform and a modified Picard iteration scheme, extending the $ L^2 $ result of Molinet (2010).
• Uses a truncated version of the Benjamin-Ono equation with frequency projection $ \Pi_N $, and proves invariance of the finite-dimensional Gibbs measure $ \nu_N $ under the truncated flow $ \Phi_t^N $.
• Applies a limiting argument: as $ N \to \infty $, the truncated flows $ \Phi_t^N $ converge to a global flow $ \Phi_t $ on a full-measure set in $ \mathcal{V} $, using compactness and convergence in the metric $ d(f,g) = \| \langle n\rangle^{-s^6 + r}(f-g) \|_{l^p} $.
• Relies on the regularity of the Gibbs measure $ \nu $ restricted to bounded sets in $ Z_1 $, which allows approximation by compact sets and enables the use of measure-theoretic convergence arguments.
• Uses the convergence $ \| \langle \partial_x \rangle^{-s^5}(h^{N_k} - h) \|_{Z_1} \to 0 $ to pass from truncated solutions to the limit solution, ensuring $ \Phi_t $ is well-defined and measurable.

Experimental results

Research questions

• RQ1 Can the Gibbs measure for the periodic Benjamin-Ono equation be shown to be invariant under the equation’s dynamics, despite being supported in spaces below $ L^2 $?
• RQ2 Is it possible to extend the local well-posedness theory of the Benjamin-Ono equation to spaces rougher than $ L^2 $, such as a Besov-type space $ Z_1 $?
• RQ3 Does a global flow exist almost surely with respect to the Gibbs measure, even when classical well-posedness fails in $ L^2 $?
• RQ4 Can the invariance of the Gibbs measure be established through a limiting procedure involving truncated equations and finite-dimensional approximations?

Key findings

• The paper establishes the existence of a unique global solution to the periodic Benjamin-Ono equation for almost every initial data in the support of the Gibbs measure, with the solution lying in the Besov-type space $ Z_1 $ for all time.
• A local well-posedness result is proven in $ Z_1 $, which is rougher than $ L^2 $, extending the $ L^2 $ well-posedness result of Molinet (2010).
• The Gibbs measure associated with the Benjamin-Ono equation is invariant under the global flow $ \Phi_t $, meaning $ \nu(\Phi_t(E)) = \nu(E) $ for all Borel sets $ E $.
• The limiting argument using truncated equations $ \Phi_t^N $ and the convergence of measures $ \nu_N \to \nu $ ensures that the invariance property is preserved in the limit.
• The solution flow $ \Phi_t $ forms a measurable transformation group on a full $ \rho $-measure subset $ \Sigma \subset \mathcal{V} $, ensuring consistency and uniqueness of dynamics.
• The proof relies on the regularity of the Gibbs measure on bounded subsets of $ Z_1 $, allowing approximation by compact sets and enabling the use of measure-theoretic convergence.