# American Institute of Mathematical Sciences

August  2018, 23(6): 2371-2391. doi: 10.3934/dcdsb.2018052

## Transient growth in stochastic Burgers flows

 Department of Mathematics & Statistics, McMaster University, Hamilton, Ontario L8S4K1, Canada

* Corresponding author: Bartosz Protas.

Received  March 2017 Revised  September 2017 Published  February 2018

This study considers the problem of the extreme behavior exhibited by solutions to Burgers equation subject to stochastic forcing. More specifically, we are interested in the maximum growth achieved by the "enstrophy" (the Sobolev $H^1$ seminorm of the solution) as a function of the initial enstrophy $\mathcal{E}_0$, in particular, whether in the stochastic setting this growth is different than in the deterministic case considered by Ayala & Protas (2011). This problem is motivated by questions about the effect of noise on the possible singularity formation in hydrodynamic models. The main quantities of interest in the stochastic problem are the expected value of the enstrophy and the enstrophy of the expected value of the solution. The stochastic Burgers equation is solved numerically with a Monte Carlo sampling approach. By studying solutions obtained for a range of optimal initial data and different noise magnitudes, we reveal different solution behaviors and it is demonstrated that the two quantities always bracket the enstrophy of the deterministic solution. The key finding is that the expected values of the enstrophy exhibit the same power-law dependence on the initial enstrophy $\mathcal{E}_0$ as reported in the deterministic case. This indicates that the stochastic excitation does not increase the extreme enstrophy growth beyond what is already observed in the deterministic case.

Citation: Diogo Poças, Bartosz Protas. Transient growth in stochastic Burgers flows. Discrete & Continuous Dynamical Systems - B, 2018, 23 (6) : 2371-2391. doi: 10.3934/dcdsb.2018052
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##### References:
(a) Space-time evolution of the solution $u(t,x)$ and (b) history of the enstrophy ${{\mathcal{E}}(u(t))}$ in a solution of the deterministic Burgers equation with an extreme initial condition $\tilde{g}_{\mathcal{E}_0,T}$. In figure (a) the level sets of $u(t,x)$ are plotted with increments of 0.1.
Errors in the numerical approximations of $\mathcal{E}(\mathbb{E}[u(T)])$ (blue lines and circles) and $\mathbb{E}[\mathcal{E}(u(T))]$ (green lines and squares) as functions of (a) the spatial discretization parameter $M$ with $N = 20,000$ and $S = 1000$ fixed, (b) the temporal discretization parameter $N$ with $M = 1024$ and $S = 1000$ fixed and (c) the sampling discretization parameter $S$ with $M = 1024$ and $N = 20,000$ fixed. The initial data used was $\tilde{g}_{\mathcal{E}_0,T}$ with $\mathcal{E}_0 = 10$ and $T = 1$, and the errors are evaluated with respect to the reference solutions computed with $M = 1024$, $N = 20,000$ and $S = 1000$. The dashed black lines correspond to the power laws (a) $CM^{-2}$, $CM^{-3}$, and $CM^{-4}$, (b) $CN^{-1}$, and (c) $CS^{-1/2}$ with suitably adjusted constants $C$.
Optimal initial conditions $\tilde{g}_{\mathcal{E}_0,T}(x)$ for $\mathcal{E}_0 = 10$ and $T$ ranging from $10^{-3}$ to $1$ [5] (arrows indicate the directions of increase of $T$).
Dependence of (a) the enstrophy $\mathcal{E}(T)$ at a final time $T$ and (b) the maximum enstrophy $\max_{t \in [0,T]} \mathcal{E}(t)$ on the initial enstrophy $\mathcal{E}_0$ for the optimal initial data $\tilde{g}_{\mathcal{E}_0,T}$ with $T$ in the range from $10^{-3}$ to $1$. Arrows indicate the direction of increasing $T$ and the dashed lines correspond to the power law $C \, \mathcal{E}_0^{3/2}$.
[Small noise case: $\sigma^2 = 10^{-2}$] (a) Sample stochastic solution ${ u(t,x)}$ as a function of space and time (the level sets are plotted with the increments of $0.1$), (b) evolution of enstrophy of two sample stochastic solutions $\mathcal{E}({ u(t;\omega_s)})$, $s = 1,2$, (green dash-dotted lines), the enstrophy of the deterministic solution $\mathcal{E}(t)$ (black solid line), the expected value of the enstrophy $\mathbb{E}[\mathcal{E}({ u(t)})]$ (blue dashed line) and the enstrophy of the expected value of the solution $\mathcal{E}(\mathbb{E}[{ u(t)}])$ (red dotted line). The initial data used was $\tilde{g}_{\mathcal{E}_0,T}$ with $\mathcal{E}_0 = 10$ and $T = 1$. The inset in figure (b) shows details of the evolution during the subinterval $[0.35,0.65]$.
[Large noise case: $\sigma^2 = 1$] (see previous figure for details).
The expected value of the enstrophy $\mathbb{E}[\mathcal{E}({ u(t)})]$ (dashed lines), the enstrophy of the expected value of the solution $\mathcal{E}(\mathbb{E}[{ u(t)}])$ (dotted lines) and the enstrophy $\mathcal{E}(t)$ of the deterministic solution (thick solid line) as functions of time for the initial condition $\tilde{g}_{\mathcal{E}_0,T}$ with $\mathcal{E}_0 = 10$, $T = 1$ and different noise levels $\sigma^2$ in the range from $10^{-2}$ to $1$ (the direction of increase of $\sigma^2$ is indicated by arrows).
Normalized PDFs of the maximum enstrophy values $\max_{t \ge 0} {\mathcal{E}}(u(t,\omega))$ for the cases with the initial condition $\tilde{g}_{\mathcal{E}_0,T}$ with $T = 1$ and (a) $\mathcal{E}_0 = 10$, (b) $\mathcal{E}_0 = 10^3$. The noise levels $\sigma^2$ are equal to $10^{-2}$ (green lines and crosses), $10^{-1}$ (blue lines and squares) and $1$ (red lines and circles). To obtain these plots, $S = 10^5$ samples were collected in each case and sorted into $30$ equispaced bins. The solid lines correspond to the standard Gaussian distributions.
(a) The values at $T = 1$ and (b) the maximum values attained in $[0,T]$ of the expected value of the enstrophy $\mathbb{E}[\mathcal{E}({ u(t)})]$ (dashed lines), the enstrophy of the expected value of the solution $\mathcal{E}(\mathbb{E}[{ u(t)}])$ (dotted lines) and the enstrophy $\mathcal{E}(t)$ of the deterministic solution (thick solid line) as functions of the initial enstrophy $\mathcal{E}_0$ for the initial condition $\tilde{g}_{\mathcal{E}_0,T}$ with $\mathcal{E}_0 = 10$, $T = 1$ and different noise levels $\sigma^2$ in the range from $10^{-2}$ to $1$ (the direction of increase of $\sigma^2$ is indicated by arrows)
[Small noise case: $\sigma^2 = 10^{-2}$] Dependence of (a) the enstrophy of the expected value of the solution $\mathcal{E}(\mathbb{E}[{ u(T)}])$ and (b) the expected value of the enstrophy $\mathbb{E}[\mathcal{E}({ u(T)})]$ on the initial enstrophy $\mathcal{E}_0$ using the initial condition $\tilde{g}_{\mathcal{E}_0,T}$ with $T$ varying from $10^{-3}$ to $1$. In (a) the values of $T$ are marked near the right edge of the plot, whereas in (b) the direction of increasing $T$ is indicated with an arrow. The dashed lines correspond to the power law $C \, \mathcal{E}_0^{3/2}$.
[Large noise case: $\sigma^2 = 1$] (see previous figure for details).
Dependence of (a) the maximum enstrophy of the expected value of the solution $\max_{t\in[0,T]} \mathcal{E}(\mathbb{E}[{ u(t)}])$ and (b) the maximum expected value of the enstrophy $\max_{t\in[0,T]} \mathbb{E}[\mathcal{E}({ u(t)})]$ on the initial enstrophy $\mathcal{E}_0$ using the initial conditions $\tilde{g}_{\mathcal{E}_0,T}$ and with noise magnitudes proportional to $\mathcal{E}_0$, cf. (14), with $C_{\sigma}$ in the range from $10^{-3}$ to $10^{-1}$ (arrow indicate the direction of increase of $C_{\sigma}$). The parameter $T$ is chosen to maximize $\max_{t\in[0,T]} \mathcal{E}(\mathbb{E}[{ u(t)}])$ in (a) and $\max_{t\in[0,T]} \mathbb{E}[\mathcal{E}({ u(t)})]$ in (b). The thick black solid line corresponds to the quantity $\max_{t \in [0,T]} \mathcal{E}(t)$ obtained in the deterministic case, whereas the thin black solid line in (a) represents the power law $\mathcal{E}_0^1$.
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