Ian M Davies, Aubrey Truman, Huaizhong Zhao
Stochastic heat and Burgers equations and their singularities II - Analytical Properties and Limiting Distributions
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ABSTRACT. \noindent We study the inviscid limit, $\mu\to 0$, of the stochastic viscous Burgers equation, for the
velocity field $v^{\mu}(x,t)$, $t>0$, $x\in\mathbb R^d$,
$$
\frac{\partial{v^{\mu}}}{\partial{t}} + (v^{\mu}\cdot\nabla)v^{\mu} = -\nabla
c(x,t) -\epsilon\nabla k(x,t) \dot W_{t} +
\frac{\mu^{2}}{2}\Delta v^{\mu}, \text{for small $\epsilon$,}
$$
with $v^{\mu}(x,0) \equiv \nabla S_{0}(x)$ for some given $S_{0}$, $\dot W_{t}$
representing White Noise. Here we use the Hopf-Cole transformation,
$v^{\mu} = -\mu^{2}\nabla\ln u^{\mu}$, where $u^{\mu}$ satisfies the stochastic heat
equation of Stratonovich type and the Feynmac-Kac Truman-Zhao formula for $u^{\mu}$, where
$$
d u^{\mu}_{t}(x) =\left[\frac{\mu^{2}}{2} \Delta u^{\mu}_{t}(x)
+\mu^{-2}c(x,t)u^{\mu}_{t}(x)\right]\,dt +
\epsilon\mu^{-2}k(x,t)u^{\mu}_{t}(x)\circ dW_{t},
$$
with $u^{\mu}_{0}(x) = T_{0}(x) \exp\left( -S_{0}(x)/\mu^{2} \right)$, $S_{0}$ as before
and $T_{0}$ a smooth positive function.
\medskip
\noindent In an earlier paper, Davies, Truman and Zhao [10], an exact
solution of the stochastic viscous Burgers equation was used to
show how the formal ``blow-up'' of the Burgers velocity field
occurs on {\em random shockwaves} for the $v^{\mu=0}$ solution of Burgers
equation coinciding with the caustics of a corresponding Hamiltonian system with classical flow map $\Phi$.
Moreover, the $u^{\mu=0}$ solution of the stochastic heat
equation has its {\em wavefront} determined by the behaviour of the Hamilton
principal function of the corresponding stochastic mechanics. This led in particular to the level surface of the minimizing Hamilton - Jacobi function developing cusps at points corresponding to points of intersection of the corresponding pre-level surface with the pre-caustic, ``pre-'' denoting the preimage under $\Phi$ determined algebraically. These results were primarily of a geometrical nature.
\medskip
\noindent In this paper we consider small $\epsilon$ and derive
the shape of the random shockwave for the inviscid limit of the stochastic Burgers velocity field
and also give the equation determining the random wavefront for the
stochastic heat equation both correct to first order in $\epsilon$.
\medskip
\noindent In the case $c(x,t)= \frac12x^{T}\Omega^{2}x$, $\nabla
k(x,t)=-a(t)$, we obtain the
exact random shockwave and prove that its shape is
unchanged by the addition of noise, it merely being displaced by a random
brownian vector $N(t)$. By exploiting the Jacobi fields for this problem we obtain the large time limit of the distribution of the Burgers fluid velocity for noises which have infinite time averages, such as almost periodic ones. Here resonance with the underlying $\epsilon = 0$ classical problem has an important effect. Imitating these results for the case of a periodic underlying classical problem perturbed by small noise, arming ourselves with some detailed estimates for Greens functions enables us to make generalisations.
\medskip
\noindent In the stochastic case we have also the possibility of ``infinitely rapid'' changes in the number of cusps on the minimizing level surface of the Hamilton - Jacobi function. This will engender stochastic turbulence
in the Burgers velocity field and, due to its stochasticity, may be of an
``intermittent'' nature. There is no analogue of this in the deterministic case.