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A method of constructing invariant measures at fixed mass

  • Author(s): Brereton, Justin Thomas
  • Advisor(s): Tataru, Daniel
  • et al.
Abstract

Invariant measures are a useful tool in constructing and analyzing solutions u(t,x) to nonlinear dispersive partial differential equations, especially when a deterministic well-posedness result is not known, and have been studied extensively since the formative work of Bourgain in the 1990s. Due to a wealth of research in recent years, most dispersive PDEs are known to admit an invariant measure with little or no restriction placed on the profile of the initial data u_0 = u(0,x). This prompts one to analyze the ergodicity of such a measure. Unfortunately there has been little progress on decomposition of these measures into invariant measures supported on smaller subsets of the initial data space. Oh and Quastel constructed an invariant measure for each fixed mass and momentum for the NLS equation on T, but there has only been this isolated result.

In this thesis we present a more general method of constructing invariant measures supported on H^(1/2-)(T) ∩ {|u|_{L^2}^2 =m} for a fixed mass m. Typically one constructs an invariant measure by multiplying a Gaussian measure µ by a density function Ѱ(u) to produce a new probability measure ρ with density containing the exponential of the remainder terms of a conservation law of the PDE. Combining Ѱ with the Gaussian base ensures that ρ is proportional to the exponential of the conservation law, making ρ invariant with respect to the PDE flow. One can only define ρ to satisfy the Radon-Nikodym derivative

dρ(u) = |Ѱ(u)|_{L^1(µ)}^{-1} Ѱ(u) dµ(u)

if |Ѱ(u)|_{L^1(µ)} is finite, therefore bounding |Ѱ(u)|_{L^1(µ)} is the main obstacle to constructing the invariant measure.

For each m>0 we will construct a base measure µ_m that is supported on the set of functions of mass m and decompose this measure as a sum

dµ_m = m^{-1} Σ_{k ≥ 0} c_{k,m} dν_m^k

for a sequence {ν_m^k : k ≥ 0} of measures derived from scaling the Fourier coefficients of u. We then compute a specific formula for the integral of a function F with respect to each ν_m^k, allowing us to utilize this measure decomposition to integrate functions with respect to µ_m. In the introduction we will demonstrate that this computation is motivated by an alternative proof of the divergence theorem. We use this method to construct an invariant measure at fixed mass supported on H^{1/2-}(T) for two PDEs: the Benjamin-Ono equation, for which L^2(T) well-posedness is already known, and the derivative non-linear Schrodinger equation, for which well-posedness has only been proven at H^{1/2}(T) level.

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