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

Alonso-Orán, Diego; León, Aythami Bethencourt de; Takao, So

doi:10.48550/arxiv.1808.07821

Cited by 2 publications

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“…The SALT equations in this form have been studied extensively, for example, in wave-current interactions [Hol19a], uncertainty prediction [GBH19], solution properties of stochastic fluid dynamics [CFH19,AOBdLT18], and turbulent cascades [Hol19b], even when the spatial correlations are nonstationary [GBH18].…”

Section: Salt Equationsmentioning

confidence: 99%

Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler–Boussinesq Equation with Transport Noise

et al. 2020

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The prediction of climate change and its impact on extreme weather events is one of the great societal and intellectual challenges of our time. The first part of the problem is to make the distinction between weather and climate. The second part is to understand the dynamics of the fluctuations of the physical variables. The third part is to predict how the variances of the fluctuations are affected by statistical correlations in their fluctuating dynamics. This paper investigates a framework called LA SALT which can meet all three parts of the challenge for the problem of climate change. As a tractable example of this framework, we consider the Euler-Boussinesq (EB) equations for an incompressible stratified fluid flowing under gravity in a vertical plane with no other external forcing. All three parts of the problem are solved for this case. In fact, for this problem, the framework also delivers global well-posedness of the dynamics of the physical variables and closed dynamical equations for the moments of their fluctuations. Thus, in a well-posed mathematical setting, the framework developed in this paper shows that the mean field dynamics combines with an intricate array of correlations in the fluctuation dynamics to drive the evolution of the mean statistics. The results of the framework for 2D EB model analysis define its climate, as well as climate change, weather dynamics, and change of weather statistics, all in the context of a model system of SPDEs with unique global strong solutions.September 4, 2019Aims of the present paper. In this paper, we derive a stochastic version of the two-dimensional Euler-Boussinesq fluid system which is non-local in probability space, rather than in physical space, in the sense that the expected velocity is assumed to replace the drift velocity in the transport operator for the stochastic fluid flow. This stochastic fluid model is derived by exploiting a novel idea introduced in [DH19], of applying Lagrangian-averaging (LA) in probability space to the fluid equations governed by stochastic advection by Lie transport (SALT) which were introduced in [Hol15]. We follow the LA SALT approach to achieve three results of interest in climate modelling based on the Kelvin circulation theorem for stochastic transport of the Kelvin loop. The three results address the three components of the climate change problem discussed at the outset. First, it answers Lorenz's question about determinism in the affirmative. Namely, by replacing the drift velocity of the stochastic vector field by its expected value, one finds that the expected fluid motion becomes deterministic. This first step leads to the second result of interest in climate change modelling. Namely, it reduces the dynamical equations for the fluctuations to a linear stochastic transport problem with a deterministic drift velocity. Such problems are well-posed. We prove here that the LA SALT version of the 2D EB problem in a vertical plane possesses global strong solutions. The third result addresses the dynamics of the varianc...

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Section: Salt Equationsmentioning

confidence: 99%

Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler–Boussinesq Equation with Transport Noise

et al. 2020

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“…The SALT Burgers equations (without the expectation on the drift velocity) were studied in [47] and it was shown that shocks form almost surely. On the other hand, for LA SALT solutions stay regular.…”

Section: La Salt Burgersmentioning

confidence: 99%

Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids

2020

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We formulate a class of stochastic partial differential equations based on Kelvin's circulation theorem for ideal fluids. In these models, the velocity field is randomly transported by white-noise vector fields, as well as by its own average over realizations of this noise. We call these systems the Lagrangian averaged stochastic advection by Lie transport (LA SALT) equations. These equations are nonlinear and nonlocal, in both physical and probability space. Without taking this average, the equations recover the Stochastic Advection by Lie Transport (SALT) fluid equations introduced by Holm [1]. Remarkably, the introduction of the non-locality in probability space in the form of advecting the velocity by its own mean gives rise to a closed equation for the expectation field which comprises Navier-Stokes equations with Lie-Laplacian 'dissipation'. As such, this form of non-locality provides a regularization mechanism. The formalism we develop is closely connected to the stochastic Weber velocity framework of Constantin and Iyer [2] in the case when the noise correlates are taken to be the constant basis vectors in R 3 and, thus, the Lie-Laplacian reduces to the usual Laplacian. We extend this class of equations to allow for advected quantities to be present and affect the flow through exchange of kinetic and potential energies. The statistics of the solutions for the LA SALT fluid equations are found to be changing dynamically due to an array of intricate correlations among the physical variables. The statistical properties of the LA SALT physical variables propagate as local evolutionary equations which when spatially integrated become dynamical equations for the variances of the fluctuations. Essentially, the LA SALT theory is a nonequilibrium stochastic linear response theory for fluctuations in SALT fluids with advected quantities.

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

Cited by 2 publications

References 0 publications

Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler–Boussinesq Equation with Transport Noise

Modelling the Climate and Weather of a 2D Lagrangian-Averaged Euler–Boussinesq Equation with Transport Noise

Lagrangian Averaged Stochastic Advection by Lie Transport for Fluids

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