On a certain system of degenerate parabolic equations which arises in hydrodynamics
Some mathematical problems arising in the mathematical study of the evolution of a turbulence burst in a fluid are studied in the paper. Denoting \(e\) the turbulent energy and \(\varepsilon\) the rate of the turbulent energy dissipation, the system of equations in the case of a statistically, horizontally-homogeneous turbulent layer with the zero mean velocity of the flow is \[ e_\theta=\alpha(\partial/\partial x)[eB(e,\varepsilon)e_x]-\varepsilon, \] \[ B=B(e,\varepsilon)\equiv e/\varepsilon,\quad\varepsilon_\theta=\beta(\partial/\partial x)[eB(e,\varepsilon)\varepsilon_x]- \varepsilon\gamma B^{-1}(e,\varepsilon), \] where \(\alpha\), \(\beta\), \(\gamma\) are positive numbers, experimentally determined. As this system could degenerate, depending on the ratio \(e\) to \(\varepsilon\), the author studies another system in which \(B\) is replaced by the value \(B_\theta=e_a/\varepsilon_a\equiv\theta\text{ const.}\) and considers \(\theta\) as another time scale, namely, as the relaxation time for \(e_a\). This system has the form \[ e_t= (\partial/\partial x)(\theta_1ee_x)-\varepsilon,\quad (x,t)\in Q_T, \] \[ \varepsilon_t= (\partial/\partial x)(\theta_1e\varepsilon_x)- \theta^{-1}_1\varepsilon, \] which is studied with boundary conditions \(e(x,0)=e_0(x)\), \(\varepsilon(x,0)=\varepsilon_0(x)\), \(x\in\mathbb{R}\). That system is of degenerate parabolic type with some particularities which separate it from the parabolic nonlinear systems, usually studied in the literature. The author proves the existence of a weak solution to the Cauchy problem, estimates the maximum of the modulus of the derivatives \(e_x\) and \(\varepsilon_x\). Using these estimates, he proves the existence of a generalized Cauchy problem, studies its differential properties and describes the structure of the support of a weak solution.
\(k\)-\(\varepsilon\) model of turbulence
scientific article; zbMATH DE number 917087
scientific article