Estimates for Solutions of a Low-Viscosity Kick-Forced Generalised Burgers Equation

We consider a non-homogeneous generalised Burgers equation: $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = \eta^{\omega},\quad t \in \R,\ x \in S^1.$$ Here, \nu is small and positive, f is strongly convex and satisfies a growth assumption, while \eta^{\omega} is a space-smooth random "kicked" forcing term. For any solution $u$ of this equation, we consider the quasi-stationary regime, corresponding to t>=2. After taking the ensemble average, we obtain upper estimates as well as time-averaged lower estimates for a class of Sobolev norms of $u$. These estimates are of the form C \nu^{-\beta} with the same values of $\beta$ for bounds from above and from below. They depend on \eta and f, but do not depend on the time t or the initial condition

Multidimensional potential Burgers turbulence

We consider the multidimensional generalised stochastic Burgers equation in the space-periodic setting: $\partial \mathbf{u}/\partial t+$ $(\nabla f(\mathbf{u}) \cdot \nabla)$ $\mathbf{u} -\nu \Delta \mathbf{u}=$ $\nabla \eta,\quad t \geq 0,\ \mathbf{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d,$ under the assumption that $\mathbf{u}$ is a gradient. Here $f$ is strongly convex and satisfies a growth condition, $\nu$ is small and positive, while $\eta$ is a random forcing term, smooth in space and white in time. For solutions $\mathbf{u}$ of this equation, we study Sobolev norms of $\mathbf{u}$ averaged in time and in ensemble: each of these norms behaves as a given negative power of $\nu$. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. These quantities behave as a power of the norm of the relevant parameter, which is respectively the separation $\mathbf{l}$ in the physical space and the wavenumber $\mathbf{k}$ in the Fourier space. Our bounds do not depend on the initial condition and hold uniformly in $\nu$. We generalise the results obtained for the one-dimensional case in \cite{BorW}, confirming the physical predictions in \cite{BK07,GMN10}. Note that the form of the estimates does not depend on the dimension: the powers of $\nu, |\mathbf{k}|, \mathbf{l}$ are the same in the one- and the multi-dimensional setting.Comment: arXiv admin note: substantial text overlap with arXiv:1201.556

Decaying Turbulence in Generalised Burgers Equation

We consider the generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1,$$ where $f$ is strongly convex and $\nu$ is small and positive. We obtain sharp estimates for Sobolev norms of $u$ (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for small-scale quantities which characterise the decaying Burgers turbulence, i.e. the dissipation length scale, the structure functions and the energy spectrum. The proof uses a quantitative version of an argument by Aurell, Frisch, Lutsko and Vergassola \cite{AFLV92}. Note that we are dealing with \textit{decaying}, as opposed to stationary turbulence. Thus, our estimates are not uniform in time. However, they hold on a time interval $[T_1, T_2]$, where $T_1$ and $T_2$ depend only on $f$ and the initial condition, and do not depend on the viscosity. These results give a rigorous explanation of the one-dimensional Burgers turbulence in the spirit of Kolmogorov's 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form $k^{-2}$. These results remain valid in the inviscid limit.Comment: arXiv admin note: substantial text overlap with arXiv:1201.5567, arXiv:1107.486

On hyperbolicity of minimizers for 1D random Lagrangian systems

We prove hyperbolicity of global minimizers for random Lagrangian systems in dimension 1. The proof considerably simplifies a related result in [2]. The conditions for hyperbolicity are almost optimal: they are essentially the same as conditions for uniqueness of a global minimizer in [3]

Sharp Estimates for Turbulence in White-Forced Generalised Burgers Equation

We consider the non-homogeneous generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} -\nu \frac{\partial^2 u}{\partial x^2} = \eta,\ t \geq 0,\ x \in S^1.$$ ∂ u ∂ t + f ′ ( u ) ∂ u ∂ x - ν ∂ 2 u ∂ x 2 = η , t ≥ 0 , x ∈ S 1 . Here f is strongly convex and satisfies a growth condition, ν is small and positive, while η is a random forcing term, smooth in space and white in time. For any solution u of this equation we consider the quasi-stationary regime, corresponding to $${t \geq T_1}$$ t ≥ T 1 , where T 1 depends only on f and on the distribution of η. We obtain sharp upper and lower bounds for Sobolev norms of u averaged in time and in ensemble. These results yield sharp upper and lower bounds for natural analogues of quantities characterising the hydrodynamical turbulence. All our bounds do not depend on the initial condition or on t for $${t \geq T_1}$$ t ≥ T 1 , and hold uniformly in ν. Estimates similar to some of our results have been obtained by Aurell, Frisch, Lutsko and Vergassola on a physical level of rigour; we use an argument from their articl

Decaying Turbulence in the Generalised Burgers Equation

We consider the generalised Burgers equation $$\frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2} = 0, \,\, t \geqq 0, \,\, x \in S^1,$$ ∂ u ∂ t + f ′ ( u ) ∂ u ∂ x - ν ∂ 2 u ∂ x 2 = 0 , t ≧ 0 , x ∈ S 1 , where f is strongly convex and ν is small and positive. We obtain sharp estimates for Sobolev norms of u (upper and lower bounds differ only by a multiplicative constant). Then, we obtain sharp estimates for the dissipation length scale and the small-scale quantities which characterise the decaying Burgers turbulence, i.e., the structure functions and the energy spectrum. The proof uses a quantitative version of an argument by Aurell etal. (J Fluid Mech 238:467-486, 1992). Note that we are dealing with decaying, as opposed to stationary turbulence. Thus, our estimates are not uniform in time. However, they hold on a time interval [T 1, T 2], where T 1 and T 2 depend only on f and the initial condition, and do not depend on the viscosity. These results allow us to obtain a rigorous theory of the one-dimensional Burgers turbulence in the spirit of Kolmogorov's 1941 theory. In particular, we obtain two results which hold in the inertial range. On one hand, we explain the bifractal behaviour of the moments of increments, or structure functions. On the other hand, we obtain an energy spectrum of the form k −2. These results remain valid in the inviscid limit

Turbulence for the generalised Burgers equation

In this survey, we review the results on turbulence for the generalised Burgers equation on the circle: u_t+f'(u)u_x=\nu u_{xx}+\eta,\ x \in S^1=\R/\Z, obtained by A.Biryuk and the author in \cite{Bir01,BorK,BorW,BorD}. Here, f is smooth and strongly convex, whereas the constant 0<\nu << 1 corresponds to a viscosity coefficient. We will consider both the case \eta=0 and the case when \eta is a random force which is smooth in x and irregular (kick or white noise) in t. In both cases, sharp bounds for Sobolev norms of u averaged in time and in ensemble of the type C \nu^{-\delta}, \delta>=0, with the same value of \delta for upper and lower bounds, are obtained. These results yield sharp bounds for small-scale quantities characterising turbulence, confirming the physical predictions \cite{BK07}.Comment: arXiv admin note: substantial text overlap with arXiv:1201.5567, arXiv:1107.4866, arXiv:1208.524

Decaying turbulence for the fractional subcritical Burgers equation

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Note on Decaying Turbulence in a Generalised Burgers Equation

We consider a generalised Burgers equation \frac{\partial u}{\partial t} + f'(u)\frac{\partial u}{\partial x} - \nu \frac{\partial^2 u}{\partial x^2}=0,\ t \geq 0,\ x \in S^1, where $f$ is strongly convex and $\nu$ is small and positive. Under mild assumptions on the initial condition, we obtain sharp estimates for small-scale quantities. In particular, upper and lower bounds only differ by a multiplicative constant. The quantities which we estimate are the dissipation length scale and averages for structure functions and the energy spectrum for solutions $u$, which characterise the "Burgulence". Our proof uses a quantitative version of arguments contained in Aurell, Frisch, Lutsko & Vergassola 1992. Our estimates remain true for the inviscid Burgers equation