Vorticity and Incompressible Flow. Cambridge Texts in Applied Mathematics

By A. Majda, A. Bertozzi, A. Ogawa. 2002

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Blow-up in multidimensional aggregation equations with mildly singular interaction kernels (2009, methodology): “In previous works [5, 6, 23], the local-in-time well-posedness of solutions with pointy interaction potential K1 has been shown for initial data in Sobolev spaces [6, 23] and for integrable and bounded initial data [5] inspired by ideas from 2D-Euler equations in fluid mechanics [28, 39].”
Finite time blowup for an averaged three-dimensional Navier-Stokes equation (2014, background): “, [32]), the inviscid Euler equations ∂tu = B(u, u) conserve helicity ∫ R3 u · (curlu).”
The role of self-similarity in singularities of partial differential equations (2008, background): “For a critical discussion of this work, see [81, 147].”
On universal approximation and error bounds for Fourier Neural Operators (2021, background): “It is well known that in two space dimensions d = 2, the time interval for existence of solutions [0, T ] can be extended to any finite T as long as u0 ∈ Hr, whereas the corresponding finite-time well-posedness result for three space-dimensions is an outstanding open problem (Majda and Bertozzi, 200”
Uniform Regularity and Vanishing Viscosity Limit for the Free Surface Navier–Stokes Equations (2012, background): “Proof of Theorem 1.2: inviscid limit 78 14.”