A Criterion for Uniqueness of Lagrangian Trajectories for Weak Solutions of the 3D Navier-Stokes Equations

Robinson, James C.; Sadowski, Witold

doi:10.1007/s00220-009-0819-z

Cited by 24 publications

(44 citation statements)

References 21 publications

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“…It is clear from (78) that the single leading-order principal null direction splits into two whenever σ = 0. This dependence on the shear is reminiscent of-although different from-Robinson's theorem [50,35], which non-perturbatively relates shearfree null geodesic congruences to null electromagnetic fields (i.e., fields which admit only one principal null direction). This theorem implies in particular that if σ = 0, there does not exist an exact Maxwell field whose principal null congruence is tangent to k a .…”

Section: Directions Associated With the Fieldmentioning

confidence: 99%

“…Relations between n = 1 amplitudes are much less clear when comparing scalar and electromagnetic quantities. Unfortunately, even if the n = 0 amplitudes are related via (50), no simple result appears to follow at the following order. This may be seen by noting from (42) that A 1 a is determined by a transport equation whose right-hand side involves…”

Section: Higher-order Amplitudesmentioning

confidence: 99%

“…The majority of this paper assumes that the polarization tensors e a and e ab defined by (50) and (112) are normalized such that e ·ē = 1. This is always possible if at least one of the e ± in (54) or (115) is nonzero.…”

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confidence: 99%

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Gravitational lensing beyond geometric optics: I. Formalism and observables

Harte

2019

Gen Relativ Gravit

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The laws of geometric optics and their corrections are derived for scalar, electromagnetic, and gravitational waves propagating in generic curved spacetimes. Local peeling-type results are obtained, where different components of high-frequency fields are shown to scale with different powers of their frequencies. Additionally, finite-frequency corrections are identified for a number of conservation laws and observables. Among these observables are a field's energy and momentum densities, as well as several candidates for its corrected "propagation directions."

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Section: Directions Associated With the Fieldmentioning

confidence: 99%

Section: Higher-order Amplitudesmentioning

confidence: 99%

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Gravitational lensing beyond geometric optics: I. Formalism and observables

Harte

2019

Gen Relativ Gravit

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“…The null dust is interpreted as a coherent zero rest mass field propagating at the speed of light in the null direction k a . Naturally, a null dust can be realized by propagating electromagnetic [13] or gravitational [11,14] waves. Here we consider the analogous problem for a massless scalar field.…”

Section: Introductionmentioning

confidence: 99%

Scalar field as a null dust

Faraoni

Côté

2019

Eur. Phys. J. C

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We show that a canonical, minimally coupled scalar field which is non-self interacting and massless is equivalent to a null dust fluid (whether it is a test or a gravitating field), in a spacetime region in which its gradient is null. Under similar conditions, the gravitating and nonminimally coupled Brans-Dicke-like scalar of scalar-tensor gravity, instead, cannot be represented as a null dust unless its gradient is also a Killing vector field.

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“…condition on a weak solution u that can lead to the uniqueness of the solution of (2) is the subject of the recent work by Robinson & Sadowski (2008).…”

mentioning

confidence: 99%

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

Dashti

Robinson

2009

Nonlinearity

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Abstract. We give a simple proof of the uniqueness of fluid particle trajectories corresponding to: 1) the solution of the two-dimensional Navier Stokes equations with an initial condition that is only square integrable, and 2) the local strong solution of the three-dimensional equations with an H 1/2 -regular initial condition i.e. with the minimal Sobolev regularity known to guarantee uniqueness. This result was proved by Chemin & Lerner (J Diff Eq 121 (1995) 314-328) using the Littlewood-Paley theory for the flow in the whole space R d , d ≥ 2. We first show that the solutions of theWe then prove, using standard energy methods, that the solution of the Navier-Stokes equations with initial condition in H (d/2)−1 satisfies these conditions. This proof is also valid for the more physically relevant case of bounded domains.Submitted to: Nonlinearity A simple proof of uniqueness of trajectories 2

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A Criterion for Uniqueness of Lagrangian Trajectories for Weak Solutions of the 3D Navier-Stokes Equations

Cited by 24 publications

References 21 publications

Gravitational lensing beyond geometric optics: I. Formalism and observables

Gravitational lensing beyond geometric optics: I. Formalism and observables

Scalar field as a null dust

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

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