Transport noise restores uniqueness and prevents blow-up in geometric transport equations

In this work, we demonstrate well-posedness and regularisation by noise results for a class of geometric transport equations that contains, among others, the linear transport and continuity equations. This class is known as linear advection of $k$-forms. In particular, we prove global existence and uniqueness of $L^p$-solutions to the stochastic equation, driven by a spatially $\alpha$-H\"older drift $b$, uniformly bounded in time, with an integrability condition on the distributional derivative of $b$, and sufficiently regular diffusion vector fields. Furthermore, we prove that all our solutions are continuous if the initial datum is continuous. Finally, we show that our class of equations without noise admits infinitely many $L^p$-solutions and is hence ill-posed. Moreover, the deterministic solutions can be discontinuous in both time and space independently of the regularity of the initial datum. We also demonstrate that for certain initial data of class $C^\infty_{0},$ the deterministic $L^p$-solutions blow up instantaneously in the space $L^{\infty}_{loc}$. In order to establish our results, we employ characteristics-based techniques that exploit the geometric structure of our equations

The Burgers' equation with stochastic transport: shock formation, local and global existence of smooth solutions

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

Stability, well-posedness and blow-up criterion for the incompressible slice model

In atmospheric science, slice models are frequently used to study the behaviour of weather, and specifically the formation of atmospheric fronts, whose prediction is fundamental in meteorology. In 2013, Cotter and Holm introduced a new slice model, which they formulated using Hamilton's variational principle, modified for this purpose. In this paper, we show the local existence and uniqueness of strong solutions of the related ISM (Incompressible Slice Model). The ISM is a modified version of the Cotter–Holm Slice Model (CHSM) that we have obtained by adapting the Lagrangian function in Hamilton's principle for CHSM to the Euler–Boussinesq Eady incompressible case. Besides proving local existence and uniqueness, in this paper we also construct a blow-up criterion for the ISM, and study Arnold's stability around a restricted class of equilibrium solutions. These results establish the potential applicability of the ISM equations in physically meaningful situations.The first author has been partially supported by the grant MTM2017-83496-P from the Spanish Ministry of Economy and Competitiveness and through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0554). The second author has been supported by the Mathematics of Planet Earth Centre of Doctoral Training, Spai

On the well-posedness of stochastic boussinesq equations with transport noise

Open access at https://arxiv.org/pdf/1807.09493.pdf.The Boussinesq equations play a fundamental role in meteorology. Among other aspects, they aim to model the process of frontogenesis and describe large-scale atmospheric and oceanic flows. In this work, we establish the existence and uniqueness of maximal strong solutions of the stochastic Boussinesq equations with transport noise in Sobolev spaces and construct a blow-up criterion. For this, in particular, we derive some general estimates, which turn out to be crucial for showing the well-posedness of a broader range of stochastic partial differential equations.The first author has been partially supported by the Grant MTM2017-83496-P from the Spanish Ministry of Economy and Competitiveness and through the Severo Ochoa Programme for Centres of Excellence in R&D (SEV-2015-0554). The second author has been supported by the Mathematics of Planet Earth Centre of Doctoral Training (MPE CDT) and Grantham Research Institute on Climate Change and the Environment, London School of Economics and Political Science