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For the Choquard equation, which is a nonlocal nonlinear Schrödinger type equation,
where 
$$-\Delta u+V_{\mu, \nu} u=(I_\alpha\ast \vert u \vert ^{({N+\alpha})/{N}}){ \vert u \vert }^{{\alpha}/{N}-1}u,\quad {\rm in} \ {\open R}^N, $$
$N\ges 3$, Vμ,ν :ℝN → ℝ is an external potential defined for μ, ν > 0 and x ∈ ℝN by Vμ,ν(x) = 1 − μ/(ν2 + |x|2) and 
$I_\alpha : {\open R}^N \to 0$ is the Riesz potential for α ∈ (0, N), we exhibit two thresholds μν, μν > 0 such that the equation admits a positive ground state solution if and only if μν < μ < μν and no ground state solution exists for μ < μν. Moreover, if μ > max{μν, N2(N − 2)/4(N + 1)}, then equation still admits a sign changing ground state solution provided 
$N \ges 4$ or in dimension N = 3 if in addition 3/2 < α < 3 and 
$\ker (-\Delta + V_{\mu ,\nu }) = \{ 0\} $, namely in the non-resonant case.
The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3D inhomogeneous incompressible Navier–Stokes system in the whole space 
$\mathbb{R}^{3}$. This class of data is based on functions which vary slowly in one direction. The idea is that 2D inhomogeneous Navier–Stokes system with large data is globally well-posed and we construct the 3D approximate solutions by the 2D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2D inhomogeneous Navier–Stokes system. We obtained the same optimal decay estimates as the solutions of 2D homogeneous Navier–Stokes system.
We use Toeplitz operators to evaluate the leading term in the asymptotics of the analytic torsion forms associated with a family of flat vector bundles 
$F_{p}$. For 
$p\in \mathbf{N}$, the flat vector bundle 
$F_{p}$ is the direct image of 
$L^{p}$, where 
$L$ is a holomorphic positive line bundle on the fibres of a flat fibration by compact Kähler manifolds. The leading term of the analytic torsion forms is the integral along the fibre of a locally defined differential form.
