Studying Nonlinear pde by Geometry in Matrix Space

By B. Kirchheim, Stefan Müller, Vladimír Sverák. 2003

127 citations recorded by the source corpus.

Subfields: Computational Theory and Mathematics, Control and Systems Engineering, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/690c929f7c94ed7a9593c71838be36b8b598baac

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In Computational Theory and Mathematics

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In Geometry and Topology

Citations in the corpus

The Euler equations as a differential inclusion (2007, methodology): “Tartar for the analysis of oscillations in linear partial differential systems coupled with nonlinear pointwise constraints [7, 15, 24, 25].”
Dislocation Microstructures and the Effective Behavior of Single Crystals (2005, unknown): “[38, 30] and references therein).”
Lack of Uniqueness for Weak Solutions of the Incompressible Porous Media Equation (2009, methodology): “This is an obstruction for the available versions of convex integration, the ones based on Baire category [5, 10, 15] and direct constructions [9, 11, 14].”
On Admissibility Criteria for Weak Solutions of the Euler Equations (2007, background): “In a recent paper [7], we have shown how the general framework of convex integration [5,12,17,23] combined with Tartar’s programme on oscillation phenomena in conservation laws [24] (see also [13] for an overview) can be applied to (1).”
Regularity of minima: An invitation to the dark side of the calculus of variations (2006, unknown): “The example of Müller & Šverák is completely different from the ones considered in Section 3, and is based on a delicate construction resting on a the use of the so-called Tartar’s “T4-configuration”, and of Gromov’s convex integration theory [168]; for more on the issue see also their paper with Ki”
The $h$-principle and the equations of fluid dynamics (2011, background): “Šverak the existence theory for differential inclusions (see also [17, 18, 27, 45, 46]) has a close relation to the h-principle.”
Mixing solutions for the Muskat problem (2016, background): “to the (lack of) regularity of elliptic systems [29,30,34].”
Convex integration and phenomenologies in turbulence (2019, methodology): “1 Convex integration schemes for the Euler equations Inspired by the work [151, 110], and building on the plane-wave analysis introduced by Tartar [189, 190, 55], De Lellis and Székelyhidi Jr.”
Conformal representation of surfaces, and Plateau's problem for Cartan functionals (2005, background): “Sver´ ak [ 29 ] shows that there are smooth, strongly convex functions f(p) such that divfp(rX) = 0 has weak solutions X 2 Lip(B,R2) which are notC1 in any open subset ofB.”
Dissipative continuous Euler flows (2012, background): “52 of [15]), but as far as we know, there have been no such examples in the literature so far.”