The $h$-principle and the equations of fluid dynamics

By Camillo De Lellis, L. Sz'ekelyhidi. 2011

204 citations recorded by the source corpus.

In this note we survey some recent results for the Euler equations in compressible and incompressible fluid dynamics. The main point of all these theorems is the surprising fact that a suitable variant of Gromov's $h$-principle holds in several cases.

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Citations in the corpus

Enhanced Dissipation and Inviscid Damping in the Inviscid Limit of the Navier–Stokes Equations Near the Two Dimensional Couette Flow (2014, background): “Also, the recent constructions of non-unique energy dissipative solutions of the three dimensional Euler equations [17,22,23,35] are not known to have an analogue in the presence of dissipation [5], with or without boundaries.”
Nonuniqueness of Weak Solutions to the SQG Equation (2016, methodology): “3 It was suggested in [43, 58, 79] that the structure of the SQG nonlinearity is nonamenable to convex integration methods because the multiplier relating u to is an odd function of frequency.”
Dissipative Euler Flows with Onsager‐Critical Spatial Regularity (2014, result): “Furthermore, in [9, 10] the failure of energy conservation (as well as the non-uniqueness) were identified as a weak h-principle - we refer to the survey [11] for more details.”
Dissipative Euler Flows and Onsager's Conjecture (2012, methodology): “Since then these techniques have been used successfully in other equations of fluid dynamics: we refer the interested reader to the survey article [14].”
Convex integration and phenomenologies in turbulence (2019, background): “[47, 190, 50, 51].”
Anomalous dissipation for 1/5-Hölder Euler flows (2015, background): “The 3× 3 symmetric traceless tensor R̊q is related to the so-called Reynolds stress, a quantity which arises naturally when considering highly oscillatory solutions of the Euler equations or, equivalently, when taking weak limits of solutions of (1) (we refer the reader to [10] for a thorough discus”
h-Principles for the Incompressible Euler Equations (2012, background): “Further developments in fluid dynamics inspired by these works appeared subsequently in [5,6,26,28–30] and are surveyed in the note [11].”
On Nonperiodic Euler Flows with Hölder Regularity (2014, methodology): “It is a basic principle of convex integration that approximate solutions used in the construction turn out to be weak limits of solutions to the partial differential equation (or inclusion) that is being solved (see, for example, Appendix A below, or [8,13,23]).”
On solvability and ill-posedness of the compressible Euler system subject to stochastic forces (2017, methodology): “These increments are obtained by an adaptation of the deterministic method of convex integration developed by De Lellis and Székelyhidi [9].”
Dissipative continuous Euler flows (2012, background): “All these aspects (and further developments for some PDEs in fluid dynamics inspired by our work) are surveyed in the note [8].”