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京东 11.11 红包
Phillip Isett:Continuous Incompressible Euler Flows的Local Dissipation of Energy
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https://www.youtube.com/watch?v=Pz1pf77quj0 美国普林斯顿高等研究院 Analysis Seminar Topic: Local Dissipation of Energy for Continuous Incompressible Euler Flows Speaker: Phillip Isett Affiliation: University of Texas at Austin, California Institute of Technology Date: June 14, 2021 I will discuss the construction of continuous solutions to the incompressible Euler equations that exhibit local dissipation of energy and the surrounding motivations. A significant open question, which represents a strong form of the Onsager conjecture, is whether such solutions exist that locally dissipate energy while having the maximal possible regularity of being 1/3-Hölder continuous. Philip Isett——《Nonuniqueness and existence of continuous, globally dissipative Euler flows》: https://doi.org/10.48550/arXiv.1710.11186 We show that Hölder continuous incompressible Euler flows that satisfy the local energy inequality ("globally dissipative" solutions) exhibit nonuniqueness and contain examples that strictly dissipate kinetic energy. The collection of such solutions emanating from a fixed initial data may have positive Hausdorff dimension in the energy space even if the local energy equality is imposed, and the set of initial data giving rise to such an infinite family of solutions is C^0 dense in the space of continuous, divergence free vector fields on the torus T^3. The construction of these solutions involves a new and explicit convex integration approach mirroring Kraichnan's LDIA theory of turbulent energy cascades that overcomes the limitations of previous schemes, which had been restricted to bounded measurable solutions or to continuous solutions that dissipate total kinetic energy.
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