Abstract
In this work, we examine the solution properties of the Burgers’ equation with stochastic transport. First, we prove results on the formation of shocks in the stochastic equation and then obtain a stochastic Rankine–Hugoniot condition that the shocks satisfy. Next, we establish the local existence and uniqueness of smooth solutions in the inviscid case and construct a blow-up criterion. Finally, in the viscous case, we prove global existence and uniqueness of smooth solutions.
Highlights
We prove the well-posedness of a stochastic Burgers’ equation of the form ∞du(t, x) + u(t, x)∂xu(t, x) dt + ξk(x)∂xu(t, x) ◦ dWtk = ν∂xxu(t, x) dt, k=1 (1.1)where x ∈ T or R, ν ≥ 0 is constant, {Wtk}k∈N is a countable set of independent Brownian motions, {ξk(·)}k∈N is a countable set of prescribed functions depending only on the spatial variable, and ◦ means that the stochastic integral is interpreted in the Stratonovich sense
Where x ∈ T or R, ν ≥ 0 is constant, {Wtk}k∈N is a countable set of independent Brownian motions, {ξk(·)}k∈N is a countable set of prescribed functions depending only on the spatial variable, and ◦ means that the stochastic integral is interpreted in the Stratonovich sense
Flandoli et al [20] demonstrated that the linear transport equation ∂tu + b(t, x)∇u = 0, which is ill-posed if b is sufficiently irregular, can recover existence and uniqueness of L∞ solutions that is strong in the probabilistic sense, by the addition of a “simple” transport noise, du + b(t, x)∇u dt + ∇u ◦ dWt = 0, (1.2)
Summary
Where x ∈ T or R, ν ≥ 0 is constant, {Wtk}k∈N is a countable set of independent Brownian motions, {ξk(·)}k∈N is a countable set of prescribed functions depending only on the spatial variable, and ◦ means that the stochastic integral is interpreted in the Stratonovich sense. If the set {ξk(·)}k∈N forms a basis of some separable Hilbert space H (for example L2(T)), the process dW :=. H, generalising the notion of a standard Wiener process to infinite dimensions. The multiplicative noise in (1.1) makes the transport velocity stochastic, which allows the Burgers’ equation to retain the form of a transport equation. ∂tu + u ∂xu = 0, where u(t, x) := u(t, x) + Wis a stochastic vector field with noise Wthat is smooth in space and rough in time. Compared with the wellstudied Burgers’ equation with additive noise, where the noise appears as an
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