Vorticity and incompressible flow

By Andrew J. Majda, Andrea L. Bertozzi. 2001

2,215 citations recorded by the source corpus.

View Paper: https://www.semanticscholar.org/paper/eab904237edc503cae81f00173a2132a22c7e185

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Citations in the corpus

The three-dimensional Euler equations : Where do we stand? (2008, methodology): “An alternative proof using methods of harmonic analysis by Bertozzi and Constantin [17] can also be seen in [6].”
The role of self-similarity in singularities of partial differential equations (2008, background): “This leads to a geometric evolution equation for the filament (see [100], chapter 7, and references therein) that can be transformed, via Hasimoto transformation, into the cubic Nonlinear-Schrödinger in 1D.”
GLOBAL WELL-POSEDNESS OF THE VISCOUS BOUSSINESQ EQUATIONS (2004, methodology): “4 on page 98 of [13]) ∑”
Euler equations for incompressible ideal fluids (2007, background): “We will be presenting only the essence of the essential arguments and not the full details of the proofs (see, for example, [7] or [8] for the details).”
On the Two‐Dimensional Muskat Problem with Monotone Large Initial Data (2016, background): “The interface between the two fluids is a “vortex sheet” [8, 18], since the normal velocity is continuous while the tangential velocity is discontinuous.”
Finite-Time Singularities of an Aggregation Equation in $${\mathbb {R}^n}$$ with Fractional Dissipation (2009, methodology): “By applying the Hodge decomposition theorem [29], one can write G = G +G ) := ∇N +∇P, where N and P are scalar functions.”
On the Finite Time Blowup of the De Gregorio Model for the 3D Euler Equations (2019, background): “The fundamental question regarding the global regularity of the 3D Euler and Navier-Stokes equations for general smooth initial data with finite energy remains open, and it is generally viewed as one of the most important open questions in mathematical fluid mechanics; see the surveys [10, 18, 19, 2”
On the Regularity Conditions for the Navier-Stokes and Related Equations (2005, background): “Leray ([24]) there are many comprehensive literatures on the existence theory of the weak solution of the (NS) equations (say, [23, 31, 26]).”
A Liouville Theorem for the Axially-symmetric Navier-Stokes Equations (2010, unknown): “The radial, angular (or swirl) and axial components v , v and v of the velocity field are governed by (see, for instance, [20]) ⎧⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩ ∂tv r + b · ∇vr − (v θ )2 r + ∂rp = ( − 1 r2 ) v , ∂tv θ + b · ∇vθ + v v r = ( − 1 r2 ) v , ∂tv z + b · ∇vz + ∂zp = v, b = ver + vez, ∇ · b = ∂rv + v r r + ”
On the Euler equations of incompressible fluids (2007, background): “There are books ([6], [28], [33], [105], [107]) and expository articles ([10], [42], [80]) on the subject, too numerous to all be listed here.”
Regularity criterion via two components of vorticity on weak solutions to the Navier-Stokes equations in R 3 (2005, background): “Since the proof is standard, we omit it (see, for example, [18]).”
Finite-Time Blow-up of Solutions of an Aggregation Equation in Rn (2007, methodology): “By virtue of the fact that u ∈ CW ([0, T ];H s(Rn)), it suffices to show that the norm ‖u(t)‖s is a continuous function of time [22, 25].”
Criticality of the Axially Symmetric Navier-Stokes Equations (2015, methodology): “Following [17, 7, 5], we also study the equations for J and Ω:”
Well-posedness of the transport equation by stochastic perturbation (2008, background): “Following [16], this equation is (formally) equivalent to the stochastic ordinary equation depending on a parameter a ∈ R(2) dX t = [∫”