Asymptotic self-similar blow up profile for 3-D Euler via physics-informed neural networks

Wang, Yongji; Lai, Ching-Yao; Gómez-Serrano, Javier; Buckmaster, Tristan

doi:10.48550/arxiv.2201.06780

Cited by 7 publications

(7 citation statements)

References 25 publications

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“…Besides being very commonly used for applications in oceanography and atmospheric sciences [26], the 2d Boussinesq system (B) attracted the interest of the mathematical community since several decades and whether finite-time blow-up of solutions emanated from finite-energy, smooth initial data can take place is an outstanding open question (see [27]). In the presence of boundary, several impressive contributions have been very recently made in this direction, among which Chen & Hou [8] showed the blow-up of self similar solutions by means of computer assisted proofs and physics-informed neural networks have been used to construct approximated self-similar blow-up profiles in [28]. The finite-time blow-up for smooth data of [8] is proved for the 3d axisymmetric Euler equations as well.…”

Section: Introduction and Resultsmentioning

confidence: 99%

Strong ill-posedness in $W^{1, \infty}$ of the 2d stably stratified Boussinesq equations and application to the 3d axisymmetric Euler Equations

Bianchini¹,

Hientzsch²,

Iandoli³

2023

Preprint

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We prove the strong ill-posedness in the sense of Hadamard of the two-dimensional Boussinesq equations in W 1,8 pR 2 q without boundary, extending to the case of systems the method that Shikh Khalil & Elgindi arXiv:2207.04556v1 developed for scalar equations. We provide a large class of initial data with velocity and density of small W 1,8 pR 2 q size, for which the horizontal density gradient has a strong L 8 pR 2 q-norm inflation in infinitesimal time, while the vorticity and the vertical density gradient remain bounded. Furthermore, exploiting the three dimensional version of Elgindi's decomposition of the Biot-Savart law, we apply the method to the three-dimensional axisymmetric Euler equations with swirl and away from the vertical axis, showing that a large class of initial data with velocity field uniformly bounded in W 1,8 pR 2 q provides a solution whose swirl component has a strong W 1,8 pR 2 q-norm inflation in infinitesimal time, while the potential vorticity remains bounded at least for small times. Finally, the L 8 -norm inflation of the swirl (and the vorticity field) is quantified from below by an explicit lower bound which depends on time, the size of the data and it is valid for small times.

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Section: Introduction and Resultsmentioning

confidence: 99%

Strong ill-posedness in $W^{1, \infty}$ of the 2d stably stratified Boussinesq equations and application to the 3d axisymmetric Euler Equations

Bianchini¹,

Hientzsch²,

Iandoli³

2023

Preprint

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show abstract

“…The authors of [32] also suggested the existence of approximate self-similar profiles in the in the original Hou-Luo scenario. More recently, Wang, Lai, Gómez-Serrano, and Buckmaster [211] proposed a new method to find self-similar profiles in the Hou-Luo scenario using neural networks .…”

Section: Axi-symmetric Solutions On the Cylinder: The Hou-luo Scenariomentioning

confidence: 99%

Singularity formation in the incompressible Euler equation in finite and infinite time

Drivas¹,

Elgindi²

2022

Preprint

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“…PINNs leverage the universal function approximation property of neural networks 41 and optimization algorithms developed in deep learning to numerically search for solutions to both forward and inverse problems 8,9 . Since their inception, PINNs have been successfully implemented to solve a wide range of problems in fluid dynamics 9,[42][43][44][45][46] .…”

Section: Inferring Ice Viscosity Via Pinnsmentioning

confidence: 99%

Discovering the rheology of Antarctic Ice Shelves via physics-informed deep learning

Wang

Lai

Cowen-Breen

2022

Preprint

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Ice shelves are floating extensions of grounded ice and play a crucial role in slowing ice discharge into the ocean. Ice flows in response to stresses according to the flow law, which is essential for predicting the mass loss of ice sheet. Laboratory experiments have shown that polycrystalline ice obeys Glen’s flow law, a power-law relation between the stress and strain rate that has been applied to ice-sheet models for decades. However, it remains unclear how processes at ice-shelf scales impact the flow law, i.e. rheology, of glacial ice. Here, we reveal the rheology of glacial ice in Antarctic Ice Shelves leveraging the availability of remote-sensing data and physics-informed deep learning. We find that the rheology of ice shelves differs substantially between the compression and extension zones. In the compression zone near the grounding line the rheology of ice closely obeys power laws with exponents in the range 1 < n < 3, consistent with prior laboratory experiments. In the extension zone, which comprises most of the total ice-shelf area, the rheology deviates from laboratory findings and does not follow a clear trend. Our result highlights a need to examine the impact of ice-shelf scale processes on glacial rheology.

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Asymptotic self-similar blow up profile for 3-D Euler via physics-informed neural networks

Cited by 7 publications

References 25 publications

Strong ill-posedness in $W^{1, \infty}$ of the 2d stably stratified Boussinesq equations and application to the 3d axisymmetric Euler Equations

Strong ill-posedness in $W^{1, \infty}$ of the 2d stably stratified Boussinesq equations and application to the 3d axisymmetric Euler Equations

Singularity formation in the incompressible Euler equation in finite and infinite time

Discovering the rheology of Antarctic Ice Shelves via physics-informed deep learning

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