Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit

Drivas, Theodore D.; Nguyen, Huy Quang

doi:10.1007/s00332-018-9500-z

Cited by 49 publications

(47 citation statements)

References 47 publications

(87 reference statements)

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“…Numerical evidence that the dissipation rate is independent of ν for large Re is for example given in Sreenivasan (1998), Kaneda et al (2003) and Orlandi et al (2012), and experimental evidence for example in Pearson, Krogstad & van de Water (2002) and Dubrulle (2019). Under certain regularity or smoothness assumptions (existence of a strong L 3 limit; we refer to Duchon & Robert (2000), Drivas & Nguyen (2019) and Drivas & Eyink (2019) for a precise discussion), weak Euler solutions are the ν → 0 limit of Leray-Hopf weak solutions u ν of the Navier-Stokes equations, so that the dissipation rate in the inviscid limit equals the viscous dissipation rate in the limit ν → 0:…”

Section: Energy Dissipation Anomalymentioning

confidence: 99%

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor–Green problem

et al. 2021

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The well-known energy dissipation anomaly in the inviscid limit, related to velocity singularities according to Onsager, still needs to be demonstrated by numerical experiments. The present work contributes to this topic through high-resolution numerical simulations of the inviscid three-dimensional Taylor–Green vortex problem using a novel high-order discontinuous Galerkin discretisation approach for the incompressible Euler equations. The main methodological ingredient is the use of a discretisation scheme with inbuilt dissipation mechanisms, as opposed to discretely energy-conserving schemes, which – by construction – rule out the occurrence of anomalous dissipation. We investigate effective spatial resolution up to $8192^3$ (defined based on the $2{\rm \pi}$ -periodic box) and make the interesting phenomenological observation that the kinetic energy evolution does not tend towards exact energy conservation for increasing spatial resolution of the numerical scheme, but that the sequence of discrete solutions seemingly converges to a solution with non-zero kinetic energy dissipation rate. Taking the fine-resolution simulation as a reference, we measure grid-convergence with a relative $L^2$ -error of $0.27\,\%$ for the temporal evolution of the kinetic energy and $3.52\,\%$ for the kinetic energy dissipation rate against the dissipative fine-resolution simulation. The present work raises the question of whether such results can be seen as a numerical confirmation of the famous energy dissipation anomaly. Due to the relation between anomalous energy dissipation and the occurrence of singularities for the incompressible Euler equations according to Onsager's conjecture, we elaborate on an indirect approach for the identification of finite-time singularities that relies on energy arguments.

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Section: Energy Dissipation Anomalymentioning

confidence: 99%

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor–Green problem

et al. 2021

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“…We emphasize that to date, even the question of whether the weak L 2 t L 2 x inviscid limit holds (against test functions compactly supported in the interior of the domain), remains open. Conditional results have been established recently in terms of interior structure functions [9,11], or in terms of interior vorticity concentration measures [8].…”

Section: Introductionmentioning

confidence: 99%

The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary

Kukavica

Vicol

Wang

2020

Arch Rational Mech Anal

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We address the inviscid limit for the Navier-Stokes equations in a half space, with initial datum that is analytic only close to the boundary of the domain, and has finite Sobolev regularity in the complement. We prove that for such data the solution of the Navier-Stokes equations converges in the vanishing viscosity limit to the solution of the Euler equation, on a constant time interval.April 12, 2019 1 for the solution u of (1.1)-(1.4). Hereū denotes the real-analytic solution of the Euler equations, and u P is the real-analytic solution of the Prandtl boundary layer equations. We refer the reader to [1,10,17,28,35,36,37,39,47,51,52] for the well-posedness theory for the Prandtl equations, to [14,26,18,38] for the identification of ill-posed regimes, and to [19,20,21,22,23] for recent works which show the invalidity of the Prandtl expansion at the level of Sobolev regularity. In [52, 53] Sammartino-Caflisch carefully analyze the error terms in the expansion (1.6), and show that they remain O( √ ν) for an O(1) time interval, by appealing to real-analyticity and an abstract Cauchy-Kowalevski theorem. This strategy has been proven successful for treating the case of a channel [40,34] and the exterior of a disk [5]. Subsequently, in a remarkable work [44], Maekawa proved that the inviscid limit also holds for initial datum whose associated vorticity is Sobolev smooth and is supported at an O(1) distance away from the boundary of the domain. The main new device in [44] is the use of the vorticity boundary condition in the case of the half space [2, 43], using which one may actually establish the validity of the expansion (1.6). Using conormal Sobolev spaces, the authors of [55] have obtained an energy based proof for the Caflisch-Sammartino result, while in [12, 13] it is shown that Maekawa's result can also be proven solely using energy methods, in 2D and 3D respectively. More recently, Nguyen-Nguyen have found in [50] a very elegant proof of the Sammartino-Caflisch result, which for the first time completely avoids the usage of Prandtl boundary layer correctors. Instead, Nguyen-Nguyen appeal to the boundary vorticity formulation, precise bounds for the associated Green's function, and an analysis in boundary-layer weighted spaces. In this paper we use a number of estimates from [50], chief among which are the ones for the Green's function for the Stokes system (see Lemma 3.4 below). Lastly, we mention that in a recent remarkable result [16], Gerard-Varet-Maekawa-Masmoudi establish the stability in a Gevrey topology in x and a Sobolev topology in y, of Euler+Prandtl shear flows (cf. (1.6)), when the Prandtl shear flow is both monotonic and concave. It is worth noting that in all the above cases the Prandtl expansion (1.6) is valid, and thus the Kato-criterion (1.5) holds. However, in general there is a large discrepancy between the question of the vanishing viscosity limit in the energy norm, and the problem of the validity of the Prandtl expansion. It is not clear to which degree these two problems are related.

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“…Recently, the condition on weak-convergence at a.e. time t was removed in [31] in favor of assuming a structure function bound within a more precise "inertial range". Also, as pointed out in [18], Remark 3.4, this condition may be removed by assuming a bound on the space-time structure function defined by…”

Section: Introductionmentioning

confidence: 99%

An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes

Drivas

Eyink

2019

Nonlinearity

Self Cite

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We study in the inviscid limit the global energy dissipation of Leray solutions of incompressible Navier-Stokes on the torus T d , assuming that the solutions have norms for Besov space B σ,∞ 3 (T d ), σ ∈ (0, 1], that are bounded in the L 3 -sense in time, uniformly in viscosity. We establish an upper bound on energy dissipation of the form O(ν (3σ−1)/(σ+1) ), vanishing as ν → 0 if σ > 1/3. A consequence is that Onsagertype "quasi-singularities" are required in the Leray solutions, even if the total energy dissipation vanishes in the limit ν → 0, as long as it does so sufficiently slowly. We also give two sufficient conditions which guarantee the existence of limiting weak Euler solutions u which satisfy a local energy balance with possible anomalous dissipation due to inertial-range energy cascade in the Leray solutions. For σ ∈ (1/3, 1) the anomalous dissipation vanishes and the weak Euler solutions may be spatially "rough" but conserve energy.

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Remarks on the Emergence of Weak Euler Solutions in the Vanishing Viscosity Limit

Cited by 49 publications

References 47 publications

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor–Green problem

Numerical evidence of anomalous energy dissipation in incompressible Euler flows: towards grid-converged results for the inviscid Taylor–Green problem

The Inviscid Limit for the Navier–Stokes Equations with Data Analytic Only Near the Boundary

An Onsager singularity theorem for Leray solutions of incompressible Navier–Stokes

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