Global in Time Solution to Kolmogorov’s Two-equation Model of Turbulence with Small Initial Data

Abstract

We prove the existence of global in time solution to Kolmogorov’s two-equation model of turbulence in three dimensional domain with periodic boundary conditions under smallness assumption imposed on initial data.

Appendix

In this subsection we collect the special cases of Gagliardo-Nirenberg inequalities used in the paper (for the original formulation and proof see [5, 6, 12]). Here, the constant c depends only on $\Omega$ and we assume that f is periodic function on $\Omega$ it is sufficiently regular to make the right-hand side finite. Firstly, we recall

$$\begin{aligned} \Vert \nabla f \Vert _{4}^{2} \le c \left\| \nabla f \right\| _{2} \left\| \nabla ^{3}f \right\| _{2}. \end{aligned}$$

(98)

The lower order term (say, $L^{2}$ norm) can be omitted, because $\int _{\Omega }\nabla f dx =0$, $\int _{\Omega }\nabla ^{2}f dx =0$ and from Poincaré inequality for functions with vanishing mean we get

$$\begin{aligned} \Vert \nabla f \Vert _{2}^{2}= \left\| \nabla f \right\| _{2} \left\| \nabla f \right\| _{2} \le C_{1} \left\| \nabla f \right\| _{2} \left\| \nabla ^{2}f \right\| _{2}\le C_{2} \left\| \nabla f \right\| _{2} \left\| \nabla ^{3}f \right\| _{2} , \end{aligned}$$

where $C_{1}$, $C_{2}$ depends only on Poincaré constant for $\Omega$. Next, we have

$$\begin{aligned}&\Vert f \Vert _{3}^{2} \le c \left\| \nabla f \right\| _{2} \left\| f \right\| _{2} , \mathrm { if } \int _{\Omega }fdx=0, \end{aligned}$$

(99)

$$\begin{aligned}&\Vert f \Vert _{6} \le c \left\| \nabla f \right\| _{2} , \mathrm { if } \int _{\Omega }fdx=0, \end{aligned}$$

(100)

$$\begin{aligned}&\Vert \nabla f \Vert _{6}^{2} \le c \left\| \nabla ^{3}f \right\| _{2} \left\| \nabla f \right\| _{2}, \end{aligned}$$

(101)

$$\begin{aligned}&\Vert \nabla f \Vert _{4}^{2} \le c\left\| \nabla ^{3}f \right\| _{2} \left\| {\nabla f}\right\| _{\frac{3}{2}}, \end{aligned}$$

(102)

$$\begin{aligned}&\left\| f \right\| _{\infty } \le c(\left\| \nabla ^{2}f \right\| _{2} + \Vert f \Vert _{1}). \end{aligned}$$

(103)

$$\begin{aligned}&\left\| f \right\| _{\infty } \le c \left\| \nabla ^{2}f \right\| _{2}, \mathrm { if} \int _{\Omega }fdx=0, \end{aligned}$$

(104)

$$\begin{aligned}&\left\| {f}\right\| _\frac{3}{2}\le c \left\| {\nabla f}\right\| ^\frac{1}{2}_\frac{3}{2}\Vert f \Vert _{1}^\frac{1}{2}+ c\Vert f \Vert _{1}, \end{aligned}$$

(105)

where c depends only on $\Omega$.