Collapse Versus Blow-Up and Global Existence in the Generalized Constantin–Lax–Majda Equation

Lushnikov, Pavel M.; Silantyev, Denis A.; Siegel, Michael

doi:10.1007/s00332-021-09737-x

Cited by 12 publications

(21 citation statements)

References 35 publications

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“…The right panel shows the curve of λ inferred via PINNs for different values of a within the range of r´1, 1s . The curve collapses well with the result from [23], indicating the accuracy of PINN inversion. in [10] for small, positive a, leading to finite time blowup.…”

Section: The Generalized De Gregorio Equationsupporting

confidence: 64%

“…r x X q 3 P q z P 6 e i S l e 0 c w A y s r 1 9 w n a A k < / l a t e x i t > ( = 1) < l a t e x i t s h a 1 _ b a s e 6 4 = " 3 f 0 3 z H Z u t J P w H m q q G d k e u E f 6 Besides inferring λ for a given value of a, PINN can also directly infer a for a given value of λ. The black line indicates the solution inferred by imposing λ " ´1, which gives a " 0.6887, which shows a good agreement with the result from [23]. The inset shows the comparison of the PINN prediction with the exact solution for a " 0, which shows a perfect agreement.…”

Section: The Generalized De Gregorio Equationsupporting

confidence: 55%

“…We will assume that both Ω and U are odd. Figure 5 demonstrates that the PINN can accurately reproduce the findings of [23], including the prediction that self-similar collapse forms for a below the critical value a c « 0.6887. In should be noted that while the PINN can take advantage of a continuation argument in a in order to reduce computation time, the PINN does not strictly require a continuation argument in order to find the self-similar solutions corresponding to a " ˘1.…”

Section: The Generalized De Gregorio Equationmentioning

confidence: 72%

“…The paper [23], represents the most thorough numerical study of equation (2.1) to date. In [23], self-similar solutions were found in the whole range a P r´1, 1s and beyond. We use their reported parameters as benchmarks for our results.…”

Section: The Generalized De Gregorio Equationmentioning

confidence: 99%

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

Wang¹,

Lai²,

Gómez-Serrano³

et al. 2022

Preprint

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We develop a new numerical framework, employing physics-informed neural networks, to find a smooth self-similar solution for the Boussinesq equations. The solution in addition corresponds to an asymptotic self-similar profile for the 3-dimensional Euler equations in the presence of a cylindrical boundary. In particular, the solution represents a precise description of the Luo-Hou blow-up scenario [G. Luo, T. Hou, Proc. Natl. Acad. Sci. 111(36): 12968-12973, 2014] for 3-dimensional Euler. To the best of the authors' knowledge, the solution is the first truly multi-dimensional smooth backwards selfsimilar profile found for an equation from fluid mechanics. The new numerical framework is shown to be both robust and readily adaptable to other equations. SignificanceThis paper sheds light on the question of finite time blow-up for the 2-dimensional Boussinesq and the 3-dimensional Euler equations with boundary, questions of fundamental importance to the field of mathematical fluid mechanics. Using a novel numerical method employing physics-informed neural networks, we construct smooth backwards self-similar solutions for the Boussinesq equations. The solutions themselves could potentially form the basis of a future computer-assisted proof of blow-up for the 2-dimensional Boussinesq and the 3-dimensional Euler equations with boundary.

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Section: The Generalized De Gregorio Equationsupporting

confidence: 64%

Section: The Generalized De Gregorio Equationsupporting

confidence: 55%

Section: The Generalized De Gregorio Equationmentioning

confidence: 72%

Section: The Generalized De Gregorio Equationmentioning

confidence: 99%

See 2 more Smart Citations

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

Wang¹,

Lai²,

Gómez-Serrano³

et al. 2022

Preprint

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“…For C α solutions with |aα| small enough, advection is not strong enough to inhibit vortex stretching, and blow-ups are also constructed in the above-cited papers. A numerical investigation of the behaviour of the solution with different values of a can be found in Lushnikov-Silantyev-Siegel [15].…”

Section: The Generalized De Gregorio Modelmentioning

confidence: 99%

Exactly self-similar blow-up of the generalized De Gregorio equation

Zheng

2023

Nonlinearity

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We study exactly self-similar blow-up profiles for the generalized De Gregorio model for the three-dimensional Euler equation: w t + a u w x = u x w , u x = H w . We show that for any α ∈ ( 0 , 1 ) such that | a α | is sufficiently small, there is an exactly self-similar C α solution that blows up in finite time. This simultaneously improves on the result in Elgindi and Jeong (2020 Arch. Ration. Mech. Anal. 235 1763–817) by removing the restriction 1 / α ∈ Z and Chen et al (2021 Commun. Pure Appl. Math. 74 1282–350), which only deals with asymptotically self-similar blow-ups.

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Self-Similar Finite-Time Blowups with Smooth Profiles of the Generalized Constantin–Lax–Majda Model

Huang,

Qin,

Wang

et al. 2024

Arch Rational Mech Anal

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Collapse Versus Blow-Up and Global Existence in the Generalized Constantin–Lax–Majda Equation

Cited by 12 publications

References 35 publications

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

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

Exactly self-similar blow-up of the generalized De Gregorio equation

Self-Similar Finite-Time Blowups with Smooth Profiles of the Generalized Constantin–Lax–Majda Model

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