Concentration of the invariant measures for the periodic Zakharov, KdV, NLS and Gross-Piatevskii equations in 1D and 2D

详细信息    查看全文

• 作者：Gordon Blower ^{g.blower@lancaster.ac.uk" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Gibbs measure ; Logarithmic Sobolev inequality ; Transportation
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：1 June 2016
• 年：2016
• 卷：438
• 期：1
• 页码：240-266
• 全文大小：577 K

文摘

This paper concerns Gibbs measures ν for some nonlinear PDE over the D  -torus T^D${\mathbf{T}}^D$. The Hamiltonian H=∫_T^D‖∇u‖²−∫_T^D|u|^p$H={\int }_{{\mathbf{T}}^D}{\Vert \mathrm{\nabla }u\Vert }^2-{\int }_{{\mathbf{T}}^D}|u{|}^p$ has canonical equations with solutions in Ω_N={u∈L²(T^D):∫|u|²≤N}${\mathrm{\Omega }}_N=\{u\in L^2({\mathbf{T}}^D):\int |u{|}^2\le N\}$; this N   is a parameter in quantum field theory analogous to the number of particles in a classical system. For D=1$D=1$ and 2≤p<6$2\le p<6$, Ω_N${\mathrm{\Omega }}_N$ supports the Gibbs measure ν(du)=Z⁻¹e^−H(u)∏_x∈Tdu(x)$\nu (du)=Z^{-1}{e}^{-H(u)}{\prod }_{x\in \mathbf{T}}du(x)$ which is normalized and formally invariant under the flow generated by the PDE. The paper proves that (Ω_N,‖⋅‖_L²,ν)$({\mathrm{\Omega }}_N,{\Vert \cdot \Vert }_{L^2},\nu )$ is a metric probability space of finite diameter that satisfies the logarithmic Sobolev inequalities for the periodic KdV  , the focussing cubic nonlinear Schrödinger equation and the periodic Zakharov system. For suitable subset of Ω_N${\mathrm{\Omega }}_N$, a logarithmic Sobolev inequality also holds in the critical case p=6$p=6$. For D=2$D=2$, the Gross–Piatevskii equation has H=∫_T²‖∇u‖²−∫_T²(V⁎|u|²)|u|²$H={\int }_{{\mathbf{T}}^2}{\Vert \mathrm{\nabla }u\Vert }^2-{\int }_{{\mathbf{T}}^2}(V⁎|u{|}^2)|u{|}^2$, for a suitable bounded interaction potential V and the Gibbs measure ν   lies on a metric probability space (Ω,‖⋅‖_H^−s,ν)$(\mathrm{\Omega },{\Vert \cdot \Vert }_{H^{-s}},\nu )$ which satisfies LSI  . In the above cases, (Ω,d,ν)$(\mathrm{\Omega },d,\nu )$ is the limit in L²$L^2$ transportation distance of finite-dimensional (Ω_n,‖⋅‖,ν_n)$({\mathrm{\Omega }}_n,\Vert \cdot \Vert ,{\nu }_n)$ given by Fourier sums.