Transport by vector fields with Kolmogorov spectrum
The author studies a model of turbulent transport described by the motion in a Gaussian random velocity field \(V(x,t)\) with Kolmogorov spectrum. Let the following assumptions be fulfilled. \(V(x,t)\), \(x\in R^3\), is a zero-mean Gaussian field, stationary in \(x\) and \(t\), isotropic in \(x\), and Markovian in time. The spectral matrix of \(V\) is given by the formula \(\widehat G(k,t)= \exp(-|t|\Omega(k))M(k)\), where \(\Omega(k)\) is scalar and Lipschitz continuous uniformly on any compact, the matrix \(M(k)\) is symmetric. There exist a constant \(c>0\) and a compact set \(K\subset R^3\), such that \(\int_{K}dM(k)>cI\), where \(I\) is the identity matrix. There exist constants \(c_1,c_2>0\) and \(\alpha,\beta\), such that \(\alpha- \beta+3<0\), \(|M^{ab}|(k)\leq c_1(1+|k|)^{\alpha}\), \(a,b=1,2,3\), \(\Omega(k)\geq c_2(1+|k|)^{\beta}\). Here \(|M^{ab}|(k)\) denotes the variation of \(M^{ab}(k)\). Let us denote by \(D^{m,ab}\) the effective diffusivity of the real-valued Gaussian random field \(V^{m}(x,t)\) with spectral matrix \(\widehat G^{m}(k,t)=\exp(-|t|\Omega(k))M(k)\chi\{|k^{a}|< m, a=1,2,3\}\). Then under the conditions mentioned above there exist constants \(c\) and \(m_0\) such that \(|D^{m,ab}|<c\), \(a,b=1,2,3\), for all \(m>m_0\).
scientific article; zbMATH DE number 1445925