A simple proof of uniqueness of the particle trajectories for solutions of the Navier–Stokes equations

Dashti, Masoumeh; Robinson, James C.

doi:10.1088/0951-7715/22/4/003

Cited by 12 publications

(8 citation statements)

References 15 publications

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“…We now give bounds on the decay of the H s norms when u 0 ∈ H . Note that the following gives an alternative proof of the decay rate of weak solutions obtained by Giga and Mirakawa [29], which was obtained using the semigroup approach of Kato and Fujita [45], and which is used in the proof of uniqueness of particle trajectories for 2D weak solutions due to Dashti and Robinson [18].…”

Section: +mentioning

confidence: 99%

See 1 more Smart Citation

Bayesian inverse problems for functions and applications to fluid mechanics

Cotter

Dashti²,

Robinson³

et al. 2009

Inverse Problems

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224

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In this paper we establish a mathematical framework for a range of inverse problems for functions, given a finite set of noisy observations. The problems are hence underdetermined and are often ill-posed. We study these problems from the viewpoint of Bayesian statistics, with the resulting posterior probability measure being defined on a space of functions. We develop an abstract framework for such problems which facilitates application of an infinite-dimensional version of Bayes theorem, leads to a well-posedness result for the posterior measure (continuity in a suitable probability metric with respect to changes in data), and also leads to a theory for the existence of maximizing the posterior probability (MAP) estimators for such Bayesian inverse problems on function space. A central idea underlying these results is that continuity properties and bounds on the forward model guide the choice of the prior measure for the inverse problem, leading to the desired results on well-posedness and MAP estimators; the PDE analysis and probability theory required are thus clearly dileneated, allowing a straightforward derivation of results. We show that the abstract theory applies to some concrete applications of interest by studying problems arising from data assimilation in fluid mechanics. The objective is to make inference about the underlying velocity field, on the basis of either Eulerian or Lagrangian observations. We study problems without model error, in which case the inference is on the initial condition, and problems with model error in which case the inference is on the initial condition and on the driving noise process or, equivalently, on the entire time-dependent velocity field. In order to undertake a relatively uncluttered mathematical analysis we consider the two-dimensional Navier-Stokes equation on a torus. The case of Eulerian observationsdirect observations of the velocity field itself-is then a model for weather forecasting. The case of Lagrangian observations-observations of passive tracers advected by the flow-is then a model for data arising in oceanography. The methodology which we describe herein may be applied to many other inverse problems in which it is of interest to find, given observations, an infinitedimensional object, such as the initial condition for a PDE. A similar approach might be adopted, for example, to determine an appropriate mathematical

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Section: +mentioning

confidence: 99%

“…[12,18] For u 0 ∈ H and f ∈ L 2 (0, T ; H ), let u ∈ L 2 (0, T ; V )∩L ∞ (0, T ; H ) be a weak solution of the two dimensional Navier-Stokes equation(3.1-3.3), extended by periodicity to a function u : R 2 × (0, T ] → R 2 . Then the ordinary differential equation (4.1) has a unique solution z ∈ C(R + , R 2 ).…”

mentioning

confidence: 99%

Bayesian inverse problems for functions and applications to fluid mechanics

Cotter

Dashti²,

Robinson³

et al. 2009

Inverse Problems

Self Cite

150

224

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show abstract

“…The following result shows that the tracer equations (4.4) have a solution, under mild regularity assumptions on the initial data. An analogous result is proved in [6] for the case where the velocity field is governed by the Navier-Stokes equation and the proof may be easily extended to the case of the Stokes equations. We assume throughout that ψ is sufficiently regular that this theorem applies.…”

Section: Lagrangian Data Assimilationmentioning

confidence: 65%

Approximation of Bayesian Inverse Problems for PDEs

Cotter¹,

Dashti²,

Stuart³

2010

SIAM J. Numer. Anal.

113

111

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Abstract. Inverse problems are often ill posed, with solutions that depend sensitively on data. In any numerical approach to the solution of such problems, regularization of some form is needed to counteract the resulting instability. This paper is based on an approach to regularization, employing a Bayesian formulation of the problem, which leads to a notion of well posedness for inverse problems, at the level of probability measures. The stability which results from this well posedness may be used as the basis for quantifying the approximation, in finite dimensional spaces, of inverse problems for functions. This paper contains a theory which utilizes this stability property to estimate the distance between the true and approximate posterior distributions, in the Hellinger metric, in terms of error estimates for approximation of the underlying forward problem. This is potentially useful as it allows for the transfer of estimates from the numerical analysis of forward problems into estimates for the solution of the related inverse problem. It is noteworthy that, when the prior is a Gaussian random field model, controlling differences in the Hellinger metric leads to control on the differences between expected values of polynomially bounded functions and operators, including the mean and covariance operator. The ideas are applied to some non-Gaussian inverse problems where the goal is determination of the initial condition for the Stokes or Navier-Stokes equation from Lagrangian and Eulerian observations, respectively.

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“…Proof of Proposition 1.1. In the light of the above observations the proof follows as in the 2D Navier-Stokes equations, see [10]. We test the equations by −t ∆u m and we have…”

Section: The Space-periodic Casementioning

confidence: 96%