Stochastic heat and Burgers equations and their singularities. II. Analytical properties and limiting distributions

Abstract

We study the inviscid limit, μ→0, of the stochastic viscous Burgers equation, for the velocity field vμ(x,t), t>0, x∊Rd, (∂vμ∕∂t)+(vμ·∇)vμ=−∇c(x,t)−ϵ∇k(x,t)Ẇt+(μ2∕2)Δvμ, for small ϵ, with vμ(x,0)≡∇S0(x) for some given S0, Ẇt representing white noise. Here we use the Hopf–Cole transformation, vμ=−μ2∇lnuμ, where uμ satisfies the stochastic heat equation of Stratonovich-type and the Feynmac–Kac Truman–Zhao formula for uμ, where dutμ(x)=[(μ2∕2)Δutμ(x)+μ−2c(x,t)utμ(x)]dt+ϵμ−2k(x,t)utμ(x)∘dWt, with u0μ(x)=T0(x)exp(−S0(x)∕μ2), S0 as before and T0 a smooth positive function. In an earlier paper, Davies, Truman, and Zhao [J. Math. Phys. 43, 3293 (2002)], 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 random shockwaves for the vμ=0 solution of Burgers equation coinciding with the caustics of a corresponding Hamiltonian system with classical flow map Φ. Moreover, the uμ=0 solution of the stochastic heat equation has its wavefront determined by the behavior 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 prelevel surface with the precaustic, “pre” denoting the preimage under Φ determined algebraically. These results were primarily of a geometrical nature. In this paper we consider small ϵ 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 ϵ. In the case c(x,t)=12xTΩ2x, ∇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 ϵ=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 Green’s functions enables us to make generalizations. 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 analog of this in the deterministic case.