scholarly journals Convex integration and phenomenologies in turbulence

2020 ◽  
Vol 6 (1) ◽  
pp. 173-263 ◽  
Author(s):  
Tristan Buckmaster ◽  
Vlad Vicol
Keyword(s):  
Convex Integration

Convex integration solutions for the geometrically nonlinear two-well problem with higher Sobolev regularity

2020 ◽  
Vol 30 (03) ◽  
pp. 611-651
Author(s):  
Francesco Della Porta ◽  
Angkana Rüland
Keyword(s):  
Dirichlet Problem ◽  
Boundary Conditions ◽  
Geometrically Nonlinear ◽  
Selection Mechanism ◽  
Matrix Space ◽  
Sobolev Regularity ◽  
Convex Integration ◽  
Nonlinear Matrix ◽  
Deformation Gradients ◽  
Higher Sobolev Regularity

In this paper, we discuss higher Sobolev regularity of convex integration solutions for the geometrically nonlinear two-well problem. More precisely, we construct solutions to the differential inclusion [Formula: see text] subject to suitable affine boundary conditions for [Formula: see text] with [Formula: see text] such that the associated deformation gradients [Formula: see text] enjoy higher Sobolev regularity. This provides the first result in the modelling of phase transformations in shape-memory alloys where [Formula: see text], and where the energy minimisers constructed by convex integration satisfy higher Sobolev regularity. We show that in spite of additional difficulties arising from the treatment of the nonlinear matrix space geometry, it is possible to deal with the geometrically nonlinear two-well problem within the framework outlined in [A. Rüland, C. Zillinger and B. Zwicknagl, Higher Sobolev regularity of convex integration solutions in elasticity: The Dirichlet problem with affine data in int[Formula: see text], SIAM J. Math. Anal. 50 (2018) 3791–3841]. Physically, our investigation of convex integration solutions at higher Sobolev regularity is motivated by viewing regularity as a possible selection mechanism of microstructures.

scholarly journals Sobolev homeomorphisms with gradients of low rank via laminates

2018 ◽  
Vol 11 (2) ◽  
pp. 111-138 ◽  
Author(s):  
Daniel Faraco ◽  
Carlos Mora-Corral ◽  
Marcos Oliva
Keyword(s):  
Convex Function ◽  
Low Rank ◽  
Hölder Continuous ◽  
Convex Integration ◽  
Open Set

AbstractLet {\Omega\subset\mathbb{R}^{n}} be a bounded open set. Given {2\leq m\leq n}, we construct a convex function {u\colon\Omega\to\mathbb{R}} whose gradient {f=\nabla u} is a Hölder continuous homeomorphism, f is the identity on {\partial\Omega}, the derivative Df has rank {m-1} a.e. in Ω and Df is in the weak {L^{m}} space {L^{m,w}}. The proof is based on convex integration and staircase laminates.

scholarly journals Uniqueness of rarefaction waves in multidimensional compressible Euler system

2015 ◽  
Vol 12 (03) ◽  
pp. 489-499 ◽  
Author(s):  
Eduard Feireisl ◽  
Ondřej Kreml
Keyword(s):  
Shock Wave ◽  
Rarefaction Wave ◽  
Riemann Problem ◽  
Euler System ◽  
Entropy Solutions ◽  
Infinitely Many Solutions ◽  
Rarefaction Waves ◽  
Convex Integration ◽  
Wave Solutions ◽  
Compressible Euler

We show that 1D rarefaction wave solutions are unique in the class of bounded entropy solutions to the multidimensional compressible Euler system. Such a result may be viewed as a counterpart of the recent examples of non-uniqueness of the shock wave solutions to the Riemann problem, where infinitely many solutions are constructed by the method of convex integration.

The Euler-Reynolds System

2017 ◽  
Author(s):  
Philip Isett
Keyword(s):  
Weak Solutions ◽  
Low Frequency ◽  
Reynolds Equations ◽  
Convex Integration ◽  
System P ◽  
Weak Limits ◽  
Euler Flows

This chapter provides a background on the Euler-Reynolds system, starting with some of the underlying philosophy behind the argument. It describes low frequency parts and ensemble averages of Euler flows and shows that the average of any family of solutions to Euler will be a solution of the Euler-Reynolds equations. It explains how the most relevant type of averaging to convex integration arises during the operation of taking weak limits, which can be regarded as an averaging process. The chapter proceeds by focusing on weak limits of Euler flows and the hierarchy of frequencies, concluding with a discussion of the method of convex integration and the h-principle for weak limits. The method inherently proves that weak solutions to Euler may fail to be solutions.

Implementation of Convex Integration

2021 ◽  
pp. 91-143
Author(s):  
Simon Markfelder
Keyword(s):  
Convex Integration
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