Estimates and regularity results for the DiPerna-Lions flow

By Gianluca Crippa, Camillo De Lellis. 2008

342 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Mechanics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0d35aefe5e35a32ae15711121053eeb50c5a07a8

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Citations in the corpus

Well-posedness and large deviation for degenerate SDEs with Sobolev coefficients (2010, methodology): “In the papers [9], [28], and [23], the vector field b needs to be in W 1,q loc (R ) for some q > 1.”
A review of the mean field limits for Vlasov equations (2014, background): “It is also possible to obtain well posedness directly on the flow as in [46].”
Existence, Uniqueness, Stability and Differentiability Properties of the Flow Associated to Weakly Differentiable Vector Fields (2009, background): “Chapter 7 is devoted to the a priori quantitative estimates for Sobolev vector fields [66].”
Positive Solutions of Transport Equations and Classical Nonuniqueness of Characteristic curves (2020, methodology): “5 employ all some suitable Lusin-Lipschitz type estimates for u, an idea pioneered in [5] and [14] and which has proved quite fruitful in different contexts (see for instance [6–8,13,16]).”
Lagrangian flows for vector fields with gradient given by a singular integral (2012, methodology): “The proof of [14] breaks down when p = 1, since the strong estimate (1.”
Constancy of the Dimension for RCD(K,N) Spaces via Regularity of Lagrangian Flows (2018, methodology): “To fix this issue, we adopt a slightly different approach (borrowed from [28]).”
Suppression of Chemotactic Explosion by Mixing (2015, background): “Lemma Let ρ(x , t) ≥ 0 be smooth local solution to (3) set on Td , d = 2 or 3.”
Lipschitz Bounds and Nonautonomous Integrals (2020, methodology): “We also mention that, over the last years, Sobolev coefficients have ben systematically considered as a replacement of usual Lipschitz ones to find optimal conditions in several other fields of analysis and PDE; see for instance [7, 25].”
Exponential self-similar mixing and loss of regularity for continuity equations (2014, background): “These results show the sharpness of the lower bounds in [6, 13, 9] (point (b) above), for the full range 1 ≤ p ≤ ∞.”
Well-posedness of Multidimensional Diffusion Processes with Weakly Differentiable Coefficients (2015, methodology): “Crippa-DeLellis’ technique [14] was extended to SDE’s in [35, 28]): of course, novelties and improvements appear in these developments, but to the author’s knowledge that they address different aspects (such as strong solutions, equations with jumps, quantitative estimates, etc.”
Differential equations with singular fields (2010, background): “1 is a more precise version of the well known bound (used in [18] in particular) |b(x)− b(y)| ≤ C |x− y| (M |db|(x) +M |db|(y)), (3.”
Global existence of weak solutions for compressible Navier--Stokes equations: Thermodynamically unstable pressure and anisotropic viscous stress tensor (2015, background): “De Lellis in [25] at the PDE level, instead of the ODE level.”