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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D. Blomker and A. Jentzen, Galerkin approximations for the stochastic Burgers equation, SIAM Journal on Numerical Analysis, 51 (2013), 694-715. doi: 10.1137/110845756.

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[29]

M.Hairer and K.Matetski, Optimal rate of convergence for stochastic Burgers-type equations, Stoch.Partial Differ.Equ.Anal.Comput., 4 (2016), 402-437. doi: 10.1007/s40072-015-0067-5.

[30]

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[31]

J. Kim and T. Bewley, A linear systems approach to flow control, Ann. Rev. Fluid Mech., 39 (2007), 383-417.

[32]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.

[33]

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S.Kuksin and A.Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge University Press, 2012.

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L. Lu and C. R. Doering, Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations, Indiana University Mathematics Journal, 57 (2008), 2693-2727. doi: 10.1512/iumj.2008.57.3716.

[38]

B.Meerson, E.Katzav and A.Vilenkin, Large deviations of surface height in the Kardar-Parisi-Zhang equation Phys.Rev.Lett., 116 (2016), 070601, 5pp. doi: 10.1103/PhysRevLett.116.070601.

[39]

D. Pelinovsky, Enstrophy growth in the viscous Burgers equation, Dynamics of Partial Differential Equations, 9 (2012), 305-340. doi: 10.4310/DPDE.2012.v9.n4.a2.

[40]

D. Pelinovsky, Sharp bounds on enstrophy growth in the viscous Burgers equation, Proceedings of Royal Society A, 468 (2012), 3636-3648. doi: 10.1098/rspa.2012.0200.

[41]

S.Pope, Turbulent Flows, 1st edition, Cambridge University Press, Cambridge, UK, 2000.

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G. D. PratoA. Debussche and R. Temam, Stochastic Burgers' equation, Nonlinear Differential Equations and Applications, 1 (1994), 289-402. doi: 10.1007/BF01194987.

[43]

A.Ruszczyński, Nonlinear Optimization, Princeton University Press, 2006.

[44]

Z.-S. SheE. Aurell and U. Frisch, The inviscid Burgers equation with initial data of brownian type, Communications in Mathematical Physics, 148 (1992), 623-641. doi: 10.1007/BF02096551.

[45]

Y. G. Sinai, Statistics of shocks in solutions of inviscid Burgers equation, Communications in Mathematical Physics, 148 (1992), 601-621. doi: 10.1007/BF02096550.

[46]

O. ZikanovA. Thess and R. Grauer, Statistics of turbulence in a generalized random-force-driven Burgers equation, Physics of Fluids, 9 (1997), 1362-1367. doi: 10.1063/1.869250.

show all references

References:
[1]

R.A.Adams and J.F.Fournier, Sobolev Spaces, Elsevier/Academic Press, Amsterdam, 2003.

[2]

A. Alabert and I. Gyöngy, On numerical approximation of stochastic Burgers' equation, in From Stochastic Calculus to Mathematical Finance, Springer(15), 1 (2006).

[3]

S. Albeverio and O. Rozanova, The non-viscous Burgers equation associated with random position in coordinate space: A threshold for blow up behaviour, Mathematical Models and Methods in Applied Sciences, 19 (2009), 749-767. doi: 10.1142/S0218202509003607.

[4]

S. Albeverio and O. Rozanova, Suppression of unbounded gradients in an SDE associated with the Burgers equation, Proceedings of the American Mathematical Society, 138 (2010), 241-251. doi: 10.1090/S0002-9939-09-10020-5.

[5]

D. Ayala and B. Protas, On maximum enstrophy growth in a hydrodynamic system, Physica D, 240 (2011), 1553-1563. doi: 10.1016/j.physd.2011.07.003.

[6]

D. Ayala and B. Protas, Maximum palinstrophy growth in 2D incompressible flows, Journal of Fluid Mechanics, 742 (2014), 340-367. doi: 10.1017/jfm.2013.685.

[7]

D.Ayala and B.Protas, Vortices, maximum growth and the problem of finite-time singularity formation, Fluid Dynamics Research, 46 (2014), 031404, 14 pp.

[8]

D. Ayala and B. Protas, Extreme vortex states and the growth of enstrophy in 3D incompressible flows, Journal of Fluid Mechanics, 818 (2017), 772-806. doi: 10.1017/jfm.2017.136.

[9]

E. BalkovskyG. FalkovichI. Kolokolov and V. Lebedev, Intermittency of burgers' turbulence, Phys. Rev. Lett., 78 (1997), 1452-1455. doi: 10.1103/PhysRevLett.78.1452.

[10]

J. Bec and K. Khanin, Burgers turbulence, Physics Reports, 447 (2007), 1-66. doi: 10.1016/j.physrep.2007.04.002.

[11]

L. BertiniN. Cancrini and G. Jona-Lasinio, The stochastic Burgers equation, Communications in Mathematical Physics, 165 (1994), 211-232. doi: 10.1007/BF02099769.

[12]

D. Blomker and A. Jentzen, Galerkin approximations for the stochastic Burgers equation, SIAM Journal on Numerical Analysis, 51 (2013), 694-715. doi: 10.1137/110845756.

[13]

A. Boritchev, Decaying turbulence in the generalised Burgers equation, Archive for Rational Mechanics and Analysis, 214 (2014), 331-357. doi: 10.1007/s00205-014-0766-5.

[14]

C.Canuto, A.Quarteroni, Y.Hussaini and T.A.Zang, Spectral Methods, Scientific Computation, Springer, 2006.

[15]

A. Chekhlov and V. Yakhot, Kolmogorov turbulence in a random-force-driven Burgers equation, Phys. Rev. E, 52 (1995), R2739-R2742.

[16]

A. Chekhlov and V. Yakhot, Kolmogorov turbulence in a random-force-driven Burgers equation: Anomalous scaling and probability density functions, Phys. Rev. E, 52 (1995), 5681-5684. doi: 10.1103/PhysRevE.52.5681.

[17]

P.A.Davidson, Turbulence.An Introduction for Scientists and Engineers, Oxford University Press, 2004.

[18]

A. Debussche and L. D. Menza, Numerical simulation of focusing stochastic nonlinear Schrödinger equations, Physica D, 162 (2002), 131-154. doi: 10.1016/S0167-2789(01)00379-7.

[19]

C. R. Doering, The 3D Navier-Stokes problem, Annual Review of Fluid Mechanics, 41 (2009), 109-128.

[20]

W. EK. KhaninA. Mazel and Y. Sinai, Invariant measures for burgers equation with stochastic forcing, Annals of Mathematics, 151 (2000), 877-960. doi: 10.2307/121126.

[21]

C.L.Fefferman, Existence and smoothness of the Navier-Stokes equation, http://www.claymath.org/sites/default/files/navierstokes.pdf, 2000, Clay Millennium Prize Problem Description.

[22]

F.Flandoli, Random Perturbation of PDEs and Fluid Dynamic Models, Lecture Notes in Mathematics, Springer, 2015.

[23]

F. Flandoli, A stochastic view over the open problem of well-posedness for the 3D Navier-Stokes equations, Stochastic Analysis: A Series of Lectures, Progr. Probab., Birkhäuser/Springer, Basel, 68 (2015), 221-246.

[24]

J. D. GibbonM. Bustamante and R. M. Kerr, The three-dimensional Euler equations: Singular or non-singular?, Nonlinearity, 21 (2008), 123-129. doi: 10.1088/0951-7715/21/8/T02.

[25]

T. Gotoh and R. H. Kraichnan, Statistics of decaying Burgers turbulence, Physics of Fluids A, 5 (1993), 445-457. doi: 10.1063/1.858868.

[26]

T.Grafke, R.Grauer and T.Schäfer, The instanton method and its numerical implementation in fluid mechanics, Journal of Physics A: Mathematical and Theoretical, 48 (2015), 333001, 39 pp.

[27]

I. Gyöngy, Existence and uniqueness results for semilinear stochastic partial differential equations, Stochastic Processes and their Applications, 73 (1998), 271-299. doi: 10.1016/S0304-4149(97)00103-8.

[28]

I. Gyöngy and D. Nualart, On the stochastic Burgers' equation in the real line, The Annals of Probability, 27 (1999), 782-802. doi: 10.1214/aop/1022677386.

[29]

M.Hairer and K.Matetski, Optimal rate of convergence for stochastic Burgers-type equations, Stoch.Partial Differ.Equ.Anal.Comput., 4 (2016), 402-437. doi: 10.1007/s40072-015-0067-5.

[30]

M. Hairer, Solving the KPZ equation, Annals of Mathematics, 178 (2014), 559-664. doi: 10.4007/annals.2013.178.2.4.

[31]

J. Kim and T. Bewley, A linear systems approach to flow control, Ann. Rev. Fluid Mech., 39 (2007), 383-417.

[32]

A. KiselevF. Nazarov and A. Volberg, Global well-posedness for the critical 2D dissipative quasi-geostrophic equation, Invent. Math., 167 (2007), 445-453. doi: 10.1007/s00222-006-0020-3.

[33]

C. Klein and J.-C. Saut, A numerical approach to blow-up issues for dispersive perturbations of Burgers' equation, Physica D, 295/296 (2015), 46-65. doi: 10.1016/j.physd.2014.12.004.

[34]

H.Kreiss and J.Lorenz, Initial-Boundary Value Problems and the Navier-Stokes Equations, vol.47 of Classics in Applied Mathematics, SIAM, 2004.

[35]

S.Kuksin and A.Shirikyan, Mathematics of Two-Dimensional Turbulence, Cambridge University Press, 2012.

[36]

G.J.Lord, C.E.Powell and T.Shardlow, An Introduction to Computational Stochastic PDEs, Cambridge University Press, 2014.

[37]

L. Lu and C. R. Doering, Limits on enstrophy growth for solutions of the three-dimensional Navier-Stokes equations, Indiana University Mathematics Journal, 57 (2008), 2693-2727. doi: 10.1512/iumj.2008.57.3716.

[38]

B.Meerson, E.Katzav and A.Vilenkin, Large deviations of surface height in the Kardar-Parisi-Zhang equation Phys.Rev.Lett., 116 (2016), 070601, 5pp. doi: 10.1103/PhysRevLett.116.070601.

[39]

D. Pelinovsky, Enstrophy growth in the viscous Burgers equation, Dynamics of Partial Differential Equations, 9 (2012), 305-340. doi: 10.4310/DPDE.2012.v9.n4.a2.

[40]

D. Pelinovsky, Sharp bounds on enstrophy growth in the viscous Burgers equation, Proceedings of Royal Society A, 468 (2012), 3636-3648. doi: 10.1098/rspa.2012.0200.

[41]

S.Pope, Turbulent Flows, 1st edition, Cambridge University Press, Cambridge, UK, 2000.

[42]

G. D. PratoA. Debussche and R. Temam, Stochastic Burgers' equation, Nonlinear Differential Equations and Applications, 1 (1994), 289-402. doi: 10.1007/BF01194987.

[43]

A.Ruszczyński, Nonlinear Optimization, Princeton University Press, 2006.

[44]

Z.-S. SheE. Aurell and U. Frisch, The inviscid Burgers equation with initial data of brownian type, Communications in Mathematical Physics, 148 (1992), 623-641. doi: 10.1007/BF02096551.

[45]

Y. G. Sinai, Statistics of shocks in solutions of inviscid Burgers equation, Communications in Mathematical Physics, 148 (1992), 601-621. doi: 10.1007/BF02096550.

[46]

O. ZikanovA. Thess and R. Grauer, Statistics of turbulence in a generalized random-force-driven Burgers equation, Physics of Fluids, 9 (1997), 1362-1367. doi: 10.1063/1.869250.

Figure 1.  (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.
Figure 2.  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 $C$.
Figure 3.  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 $T$).
Figure 4.  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 $C \, \mathcal{E}_0^{3/2}$.
Figure 5.  [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 $[0.35,0.65]$.
Figure 6.  [Large noise case: $\sigma^2 = 1$] (see previous figure for details).
Figure 7.  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).
Figure 8.  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.
Figure 9.  (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)
Figure 10.  [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 $C \, \mathcal{E}_0^{3/2}$.
Figure 11.  [Large noise case: $\sigma^2 = 1$] (see previous figure for details).
Figure 12.  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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