The paper deals with linear passive scalar transport-diffusion equation subject to a velocity field which is white noise in time and is mainly active at small scales in space. The purpose is investigating the enhancement of dissipation and decay given by the small-scale transport. We modify and improve estimates from the previous work by Flandoli et al. (J. Differ. Equations 394, 237–277, 2024), in order to investigate a different regime, namely small molecular diffusion and small noise intensity – corresponding to small turbulent kinetic energy. The noise specification is carefully tuned to Kolmogorov theory of turbulent fluids.
The purpose of this paper is to investigate the problem of autocorrelated moving boundaries for a one-dimensional heat equation, where the moving boundary is a fractional Brownian motion B^H with Hurst parameter H >1/2 and the dependent Volterra-type Dirichlet boundary term ξ is verified to lie in the exponential Orlicz space 𝒮_Ψ^(v)(Ω× (0,T]) with v∈ (0,1] . This problem is interpreted as a single-layer potential involving stochastic heat kernel G( t-s,x-B^H(s)) = 1/√(4π (t-s))exp( -( x-B^H(s)) ^2/4(t-s)) , with 0z ) , z> E [ sup _(s,t)∈𝒬 B^H(t)-B^H(s)/(t-s)^1/2+γ] , where 𝒬={(s,t): 0≤ s ≤ t≤ T } and γ∈ [0,H-1/2) . By potential theoretic arguments, we establish the unique weak solution to the fractional Brownian moving boundary problem in L^p(Ω ; C_(v)((0,T];C_b(-∞ ,B^H(· )))) with p>1 and v∈ (0,1] .
We present an example of a linear partial differential equation whose Cauchy problem becomes well-posed when perturbed by noise. Specifically, we make clear how a suitable multiplicative Stratonovich perturbation of Brownian type renders a weakly hyperbolic operator with double involutive characteristics well-posed in the C^∞-category, while its deterministic counterpart is only well-posed in the Gevrey s classes with 1 ≤ s <2 .
We consider time-dependent singular stochastic partial differential equations on the three-dimensional torus. These equations are only well-posed after one adds renormalization terms. In order to construct a well-defined notion of solution, one should put the equation in a more general setting. In this article, we consider the paradigm of paracontrolled distributions, and get concentration results around a stable deterministic equilibrium for solutions of non-autonomous generalizations of the (Phi 34) model. Specifically, we obtain Gaussian-type tail bounds.
