Local in time solution to Kolmogorov’s two-equation model of turbulence

Abstract

We prove the existence of local in time solution to Kolmogorov’s two-equation model of turbulence in three dimensional domain with periodic boundary conditions. We apply Galerkin method for appropriate truncated problem. Next, we obtain estimates for a limit of approximate solutions to ensure that it satisfies the original problem.

Appendix

The function $\Psi _{t}$ may be defined as follows. We set $f(x)= e^{-1/x}$ for $x>0$ and zero elsewhere. We put $g(x)= x-e^{-1/x}$ for $x<0$ and $g(x)=x$ for $x>0$. Then we set

$$\begin{aligned} \tilde{\eta }(x) = \frac{1}{c} \int _{0}^{x} f(y)f(-y+1)dy, \end{aligned}$$

where $c=\int _{0}^{1} f(y)f(-y+1)dy$. Function $\tilde{\eta }$ is smooth function, which vanishes for negative x and is equal to one for $x>1$. Next, we put

$$\begin{aligned} \eta (x)= \tilde{\eta }(2(x-\frac{1}{4})), h(x)= (1- \eta (x))f(x)+ \eta (x)g(x). \end{aligned}$$

Finally, we define

$$\begin{aligned} \Psi _{t}(x) = \frac{b_{\min }^{t}}{2} + \frac{b_{\min }^{t}}{2} h\left( \frac{2}{b_{\min }^{t}} \left( x- \frac{b_{\min }^{t}}{2} \right) \right) . \end{aligned}$$

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