Well-Posedness and Singularity Formation for the Kolmogorov Two-Equation Model of Turbulence in 1-D

Abstract

We study the Kolomogorov two-equation model of turbulence in one space dimension. Two are the main results of the paper. First of all, we establish a local well-posedness theory in Sobolev spaces even in the case of vanishing mean turbulent kinetic energy. Then, we show that there are smooth solutions which blow up in finite time. To the best of our knowledge, these results are the first establishing the well-posedness of the system for vanishing initial data and the occurence of finite time singularities for the model under study.

Appendix A: Well-Posedness of a Porus Medium Equation

The goal of the present appendix is to discuss the well-posedness of the porum medium type equation (5), which we rewrite here in a slightly more general form.

Let $\theta =\theta (t)$ and $\rho =\rho (t)$ be two positive scalar functions depending only on the time variable $t\in {\mathbb {R}}_+$, with the property that

$$\begin{aligned} \forall \,t\ge 0,\qquad \theta (t)>0 \quad \text {and}\quad \rho (t)>0. \end{aligned}$$

(63)

Consider the following initial-value problem, set on ${\mathbb {R}}_+\times \Omega$, where $\Omega ={\mathbb {T}}$ is the one-dimensional torus as above:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _tk-\theta (t)\,\partial ^2_{x}\left( k^2\right) \,=-\rho (t)\,k \\[1ex] k_{|t=0}=k_0. \end{array} \right. \end{aligned}$$

(64)

Here, again the initial datum $k_0\ge 0$ will be taken a positive function. We will assume $k_0\in H^1(\Omega )$. Our goal is to show that this equation is well-posed in $H^1$ without requiring any special assumption on the support of $k_0$. In particular, the regularity requirement over $\sqrt{k_0}$ can be completely dropped here. This hints at the fact that formulating such an assumption is really specific to the non-linear coupling of the unknowns inside the diffusion operator, namely of u and k in system (4), or of u and $\gamma$ in the toy-model (6).

One may argue that the trick relies on the use of $H^1$ regularity estimates, instead of $H^2$ estimates as we use in our paper. Actually, this is not true: let us explain better this point in the next remark, where for simplicity we deal only with the toy-model (6).

Remark A.1

Observe that, when performing $H^1$ estimates on the toy model (6), one finds a new bad term, simpler but similar to (7), namely

$$\begin{aligned} \int _\Omega \partial _x\gamma \,\partial _xu\,\partial ^2_{x}u\,dx=\frac{1}{2}\,\int _\Omega \partial _x\gamma \,\partial _x\left( (\partial _xu)^2\right) \,dx. \end{aligned}$$

It is easy to see that this term can be controlled only in two ways: either by assuming a control on $\sqrt{\gamma }$, as pursued in this paper, or (after an integration by parts) by looking for, at least, a $L^2$ bound on the second order derivative $\partial ^2_{x}\gamma$. Now, the non-linear coupling of the equation for $\gamma$ forces one to propagate also the $H^2$ norm of u, thus coming back to the problem discussed in (7).

This consideration shows somehow the “necessity” of working with $\sqrt{\gamma }$ instead of $\gamma$ in the toy-model (6), and with $\sqrt{k}$ instead of k for the full Kolmogorov system (4).

Next, let us comment on the propagation of the $H^2$ norm for the prous medium equation (64). Direct computations show that one cannot control the $L^2$ norm of $\partial ^2_{x}k$ uniformly in time, as this quantity satisfies an inequality of the form

$$\begin{aligned} \frac{d}{dt}\left\| \partial _x^2k\right\| ^2_{L^2}+\rho (t)\,\left\| \partial ^2_{x}k\right\| ^2_{L^2}+\theta (t)\int _\Omega k\,\left| \partial _x^3k\right| ^2\,dx\,&\lesssim \, \theta (t)\,\left\| \partial _xk\right\| _{L^\infty }\,\left\| \partial ^2_{x}k\right\| ^{2}_{L^2} \\&\lesssim \,\theta (t)\,\left\| \partial ^2_{x}k\right\| ^{2+\delta }_{L^2}, \end{aligned}$$

for a suitable $\delta >0$, and (if one wants to stick to (5), coming from the Kolmogorov system) one only has $\rho (t)\approx (1+t)^{-1}$, whereas $\theta (t)\approx (1+t)$. From this perspective, one cannot hope for a global propagation of $H^2$ regularity for equation (64), in general. Even worse, paper [9] shows blow-up results of smooth solutions for very similar equations.

After those preliminary remarks, we are now ready to state the main result of this appendix. We stress the fact that (as it will appear clear from our proof) this result is purely one-dimensional.

Theorem A.2

Let the scalar functions $\theta =\theta (t)$ and $\rho =\rho (t)$ satisfy assumption (63). Let $k_0\ge 0$ be a positive scalar function such that $k_0\in H^1(\Omega )$.

Then, there exists a unique global in time (weak) solution k to the initial value problem (64), such that $k\ge 0$ and

$$\begin{aligned} k\in L^\infty \big ({\mathbb {R}}_+;H^1(\Omega )\big )\,\cap \,\bigcap _{s<1}C\big ({\mathbb {R}}_+;H^s(\Omega )\big ), \end{aligned}$$

together with $\sqrt{k}\,\partial _x^2k\in L^2_\textrm{loc}\big ({\mathbb {R}}_+;L^2(\Omega )\big )$.

Proof

First of all, we deal with the proof of existence. We focus here in deriving global in time a priori estimates in $H^1$ for smooth solutions of equation (64). From those estimates, one can prove the existence of a solution at the claimed level of regularity by, for instance, applying a similar approximation procedure as the one described in Sect. 4.1.

Thus, assume to have a smooth solution k of equation (64) on ${\mathbb {R}}_+\times \Omega$. First of all, by noticing that k in particular solves

$$\begin{aligned} \partial _tk-2\,\theta (t)\,\partial _x\big (k\,\partial _xk\big )\,=-\rho (t)\,k, \end{aligned}$$

(65)

and performing an energy estimate for this equation, we get

$$\begin{aligned} \frac{1}{2}\,\frac{d}{dt}\left\| k(t)\right\| _{L^2}^2+\rho (t)\,\left\| k(t)\right\| _{L^2}^2+ 2\,\theta (t)\,\int _\Omega k\,\left| \partial _xk\right| ^2\,dx=0. \end{aligned}$$

This relation immediately implies the following $L^2$ bound:

$$\begin{aligned} \forall \,t>0,\quad \left\| k(t)\right\| _{L^2}^2+ & {} \int ^t_0\rho (\tau )\,\left\| k(\tau )\right\| _{L^2}^2\,d\tau \nonumber \\ {}+ & {} 2\,\int ^t_0\theta (\tau )\,\left\| \sqrt{k(\tau )}\,\partial _xk(\tau )\right\| _{L^2}^2\,d\tau \le \left\| k_0\right\| _{L^2}^2, \end{aligned}$$

(66)

where we have replaced the equality by the inequality to be consistent with what one usually gets for weak solutions.

Next, differentiating equation (65) with respect to space, we find an equation for $\partial _xk$:

$$\begin{aligned} \partial _t\partial _xk-2\,\theta (t)\,\partial _x\big (k\,\partial _x^2k\big )\,=-\rho (t)\,\partial _xk+2\,\theta (t)\,\partial _x\left( \big (\partial _xk\big )^2\right) . \end{aligned}$$

By using the equalities

$$\begin{aligned}&\int _\Omega \partial _x\left( \big (\partial _xk\big )^2\right) \,\partial _xk\,dx=2\,\int _\Omega \partial _xk\,\partial _x^2k\,\partial _xk\,dx \quad \text { and }\quad \\&\int _\Omega \partial _x\left( \big (\partial _xk\big )^2\right) \,\partial _xk\,dx\,=-\int _\Omega \big (\partial _xk\big )^2\,\partial _x^2k\,dx, \end{aligned}$$

which follow respectively by developing the derivative and by integration by parts, we see that the value of the integral on the left is actually zero. Then, an energy estimate for $\partial _xk$ yields

$$\begin{aligned} \frac{1}{2}\,\frac{d}{dt}\left\| \partial _xk(t)\right\| _{L^2}^2+\rho (t)\,\left\| \partial _xk(t)\right\| _{L^2}^2+ 2\,\theta (t)\,\int _\Omega k\,\left| \partial ^2_{x}k\right| ^2\,dx=0. \end{aligned}$$

This relation easily implies a $L^2$ bound for $\partial _xk$: we have

$$\begin{aligned}&\forall \,t>0, \nonumber \\&\left\| \partial _xk(t)\right\| _{L^2}^2+\int ^t_0\rho (\tau )\,\left\| \partial _xk(\tau )\right\| _{L^2}^2\,d\tau + 2\,\int ^t_0\theta (\tau )\,\left\| \sqrt{k(\tau )}\,\partial _x^2k(\tau )\right\| _{L^2}^2\,d\tau \le \left\| \partial _xk_0\right\| _{L^2}^2. \end{aligned}$$

(67)

Summing up (66) and (67), we immediately infer the sought global in time $H^1$ estimate for k. This completes the proof of the a priori estimates.

Let us now focus on the uniqueness of solutions at the level of regularity considered in the statement. Uniqueness will follow from a stability estimate in $L^1$ (in particular, one could formulate a uniqueness statement in a larger functional setting), which we are going to prove by employing similar techniques as the ones used for the classical porus medium equation (see e.g. Chapters 4 and 9 of [23]). The justification that we can indeed perform those computations at our level of regularity is the main new part of our argument.

Thus, suppose to have two solutions $k_{1,2}$ of equation (64), such that, for $j\in \{1,2\}$, one has

$$\begin{aligned} k_j\in L^\infty \big ({\mathbb {R}}_+;H^1(\Omega )\big )\quad \text {and}\quad \sqrt{k_j}\,\partial _x^2k_j\in L^2_{\textrm{loc}}\big ({\mathbb {R}}_+;L^2(\Omega )\big ). \end{aligned}$$

Let us set $\delta k:=\,k_1-k_2$ and, for later use, $\Delta k\,:=\,k_1^2-k_2^2$. We notice that $\delta k$ satisfies the equation

$$\begin{aligned} \partial _t\delta k-\theta (t)\,\partial ^2_{x}\left( \Delta k\right) \,=-\rho (t)\,\delta k. \end{aligned}$$

(68)

Observe that

$$\begin{aligned} \Delta k=k_1^2-k_2^2=\big (k_1+k_2\big )\,\delta k\qquad \text { belongs to }\quad L^\infty \big ({\mathbb {R}}_+;H^1(\Omega )\big ). \end{aligned}$$

From this property and equation (68), we deduce that $\partial _t\delta k\in L^\infty _\textrm{loc}\big ({\mathbb {R}}_+;H^{-1}(\Omega )\big )$. The local-in-time condition comes from the presence of the coefficients $\theta$ and $\rho$, for which we have not formulated any boundedness assumption. However, we need to improve the previous regularity property for the time derivative. For this, we recall that each $k_j$ solves equation (65): by writing

$$\begin{aligned} \partial _x\big (k_j\,\partial _xk_j\big )=\sqrt{k_j}\,\sqrt{k_j}\,\partial ^2_{x}k_j+\big (\partial _xk_j\big )^2, \end{aligned}$$

we see that this quantity belongs to $L^\infty _T(L^1)$ for all $T>0$ fixed. Thus, we get $\partial _tk_j\in L^\infty _T(L^1)$ for all $T>0$, which implies that also $\partial _t\delta k$ belongs to this space.

At this point, we take a scalar function $p\in C^1({\mathbb {R}})$ such that $0\le p\le 1$, together with $p(\sigma )=0$ for $\sigma \le 0$ and $p'(\sigma )>0$ for $\sigma >0$. Thanks to the previous analysis, we can multiply both sides of equation (68) by $p(\Delta k)$ and integrate over the space domain: we find

$$\begin{aligned} \int _\Omega \partial _t\delta k\;p(\Delta k)\,dx\,=-\theta (t)\int _\Omega \left| \partial _x\Delta k\right| ^2\,p'(\Delta k)\,dx- \rho (t)\int _\Omega \delta k\;p(\Delta k)\,dx. \end{aligned}$$

Remark that the sign of $\Delta k$ is the same of $\delta k$, owing to the positivity of each $k_j$. Then, the last term on the right-hand side has negative sign. The same can be said about the first term on the right. Thus, we infer that

$$\begin{aligned} \int _\Omega \partial _t\delta k\;p(\Delta k)\,dx\le 0. \end{aligned}$$

We now consider a sequence of functions $\big (p_n\big )_n$ verifyng the properties above and which converges to the function $\textrm{sign}^+$. We recall that $\textrm{sign}^+$ is defined as

$$\begin{aligned} \textrm{sign}^+(\sigma )=1\quad \text {if }\ \sigma >0,\qquad \textrm{sign}^+(\sigma )=0\quad \text {if }\ \sigma \le 0. \end{aligned}$$

Noticing that $\textrm{sign}^+(\delta k)=\textrm{sign}^+(\Delta k)$ and that $\partial _t\delta k\; \textrm{sign}^+(\delta k)=\partial _t(\delta k)_+$ in our functional framework, from the previous computations we deduce that

$$\begin{aligned} \frac{d}{dt}\int _\Omega (\delta k)_+\,dx\le 0. \end{aligned}$$

This in particular implies that

$$\begin{aligned} \int _\Omega \left( k_1-k_2\right) _+\,dx\le \int _\Omega \left( k_{0,1}-k_{0,2}\right) _+\,dx, \end{aligned}$$

(69)

where, for $j=1,2$, we have denoted by $k_{0,j}$ the initial datum related to the solution $k_j$.

Similar computations for $\widetilde{\delta k}:=\,k_2-k_1=-\delta k$ yield an estimate analogous to (69) for the negative part $\big (k_1-k_2\big )_-$. Then, we finally deduce the following $L^1$ stability estimate:

$$\begin{aligned} \forall \,t\ge 0,\qquad \left\| k_1(t)-k_2(t)\right\| _{L^1}\le \left\| k_{0,1}-k_{0,2}\right\| _{L^1}. \end{aligned}$$

This latter estimate in particular implies the sought uniqueness result whenever $k_{0,1}\equiv k_{0,2}$, thus completing the proof of the theorem. $\square$