Nonlinear maximum principles for dissipative linear nonlocal operators and applications

By Peter Constantin, Vlad Vicol. 2011

352 citations recorded by the source corpus.

We obtain a family of nonlinear maximum principles for linear dissipative nonlocal operators, that are general, robust, and versatile. We use these nonlinear bounds to provide transparent proofs of global regularity for critical SQG and critical d-dimensional Burgers equations. In addition we give applications of the nonlinear maximum principle to the global regularity of a slightly dissipative anti-symmetric perturbation of 2D incompressible Euler equations and generalized fractional dissipative 2D Boussinesq equations.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/29bd4f8490fb5f2b1ed87a1e3a348190d3c54124

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In Applied Mathematics

In Computational Mechanics

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Citations in the corpus

Eulerian dynamics with a commutator forcing III. Fractional diffusion of order 0 (2017, methodology): “To establish the stated exponential decay of |ux(·, t)| we first prepare with the following refinement of the nonlinear maximum principle, [15] extending [2, Lemma 3.”
Endpoint Sobolev Theory for the Muskat Equation (2020, methodology): “In the critical case α = 1, the global in time well-posedness has been proved by Kiselev-Nazarov-Volberg [42], Caffarelli-Vasseur [14] and Constantin-Vicol [30] (see also [41,48,52,57]).”
Local and global existence of smooth solutions for the stochastic Euler equations with multiplicative noise (2011, methodology): “ is f1 x,y=x and 2( =y), the remaining two terms prevent one from obtaining a bound on kwkL∞ using classical methods, since Calderón-Zygmund operators are not bounded on L∞. Recently it was proven in [CV11] that if one adds an arbitrary amount of dissipation, in the form of a positive power of −∆, o”
On the global regularity for the supercritical SQG equation (2014, methodology): “Following the ideas of [6,7], we now consider the second term in the right-hand side of (3.”
Nonuniqueness of Weak Solutions to the SQG Equation (2016, background): “The critical case D 1 is a quasilinear problem, and the global regularity of smooth solutions for large initial data was established in [12, 63], with different proofs given in [30, 31, 61].”
Critical SQG in Bounded Domains (2016, methodology): “1 1−α Cα(Ω) (d(x))1−α +c f2(x) d(x) (17) for large f= χ∇θ(see (61)). This is a super-cubic bound, and makes the gradient equation look subcritical. Similar bounds were obtained in the whole space in ([10]). Proving the bounds (16) and (17) requires a different approach and new ideas because of the a”
On a transport equation with nonlocal drift (2014, methodology): “We note that the method developed in [41] for general linear drift-diﬀusion equations also yields this result (the methods developed in [3,14,15,31] for global regularity of the critically dissipative SQG equation only appear to work in the case of positive initial datum where the L 2 norm of the…”
On the Loss of Continuity for Super-Critical Drift-Diffusion Equations (2012, result): “Other proofs of this result were given in [25] and [14].”
Finite time singularity for the modified SQG patch equation (2016, background): “(See also the subsequent works [34], [12], [11] for alternative proofs.”