Initial-boundary value problem for stochastic transport equations
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Initial-boundary value problem for stochastic transport equations Wladimir Neves1 · Christian Olivera2 Received: 3 May 2019 / Revised: 12 August 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper concerns the Dirichlet initial-boundary value problem for stochastic transport equations with non-regular coefficients. First, the existence and uniqueness of the strong stochastic traces is proved. The existence of weak solutions relies on the strong stochastic traces, and also on the passage from the Stratonovich into Itô’s formulation for bounded domains. Moreover, the uniqueness is established without the divergence of the drift vector field bounded from below. Keywords Stochastic partial differential equations · Transport equation · Well-posedness · Initial-boundary value problem
1 Introduction A great deal of attention has recently been given to the study of stochastic partial differential equations. We are interested in random description of physical problems, where the probabilistic term appears as a perturbation of the velocity vector field. In this direction, it was Ogawa [24] who initiated the analysis of wave propagation in random media. In this article we establish global existence and uniqueness of solutions for the stochastic linear transport equations (SLTE for short) in bounded domains. Namely, we consider the following initial-boundary value problem: Given a standard Brownian motion Bt = (Bt1 , ..., Btd ) in Rd , find u(t, x) ∈ R, satisfying
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Wladimir Neves [email protected] Christian Olivera [email protected]
1
Instituto de Matemática, Universidade Federal do Rio de Janeiro, C.P. 68530, Cidade Universitária, 21945-970 Rio de Janeiro, Brazil
2
Departamento de Matemática, Universidade Estadual de Campinas, 13.081-970 Campinas, SP, Brazil
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Stoch PDE: Anal Comp
⎧ ⎨ ∂ u(t, x, ω) + b(t, x) + σ d Bt (ω) · ∇u(t, x, ω) = 0, t dt ⎩ u|t=0 = u 0 , u|T = u b ,
(1.1)
with (t, x) ∈ UT := [0, T ] × U , where T > 0 is any fixed real number, U is an open and bounded domain of Rd (d ∈ N), ω ∈ is an element of the probability space (, P, F), and the stochastic integration is taken in the Stratonovich sense. The parameter σ = 1 most of the time, and equals zero when we talk about (1.1) in the deterministic case. Moreover, we denote by the C 2 -boundary of U , with the outside normal field to U at r ∈ denoted by n(r ), and define T := (0, T ) × . Here, we assume that the initial and boundary data respectively u 0 , u b are measurable and bounded functions with respect to the usual measures, that is, Lebesgue (denoted by d x, or dξ , etc.) and Hausdorff (denoted by Hd−1 (r ) or dr ) tensor dt. The vector field b : (0, T ) × Rd → Rd , called drift, satisfies the following conditions: For any q > 2 and some non-negative functions α, γ ∈ L 1loc (R), b ∈ L q ((0, T ); BVloc (Rd ; Rd )), divb ∈ L 1loc ((0, T ) × Rd ),
(1.2)
|b(t, x)| ≤ α(t),
(1.3)
divb(t, x) ≤ γ (t).
Let us remark that, we assume q > 2 in order to use the machinery developed for
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