Critical regularity for conservation of energy: the 2D case Milton C. Lopes Filho Instituto de Matemática, Universidade Federal do Rio de Janeiro MathFlows, Porquerolles, September 2015 MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 1 / 25
Collaborators: Alexey Cheskidov (Univ. Illinois, Chicago) Helena J. Nussenzveig Lopes (IM-UFRJ) Roman Shvydkoy (Univ. Illinois, Chicago) MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 2 / 25
Plan of the Talk Kolmogorov 41 1 Onsager conjecture 2 Convex integration counter-examples 3 Regularity threshold 4 Energy flux 5 Littlewood-Paley counterexample 6 Vanishing viscosity solutions 7 Conclusions 8 MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 3 / 25
Kolmogorov 41 Statistical theory of stationary, homogeneous turbulence. Turbulence is described by a family of random variables { u ν = u ν ( x ) } ν> 0 , parametrized by the adimensional viscosity ν > 0, subject to the following hypothesis: H1 Displacements u ν ( x + v ) − u ν ( x ) are statistically independent of x and of the direction of v ; H2 The flow description is self-similar , i.e. there exists an exponent δ > 0 such that, for any v in space and λ > 0, u ν ( x + λ v ) − u ν ( x ) = λ δ ( u ν ( x + v ) − u ν ( x )) . H3 There is a positive dissipation rate ǫ 0 , independent of ν , i.e., in a control volume Q , � ν |∇ u ν | 2 dx = ǫ 0 . lim | Q | ν → 0 Q MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 4 / 25
Kolmogorov 41 With these three hypothesis, one can derive all traditional features of homogeneous turbulence theory, including the fact that δ = 1 / 3, and the energy cascade, with the well-known Kolmogorov’s − 4 / 5 law. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25
Kolmogorov 41 With these three hypothesis, one can derive all traditional features of homogeneous turbulence theory, including the fact that δ = 1 / 3, and the energy cascade, with the well-known Kolmogorov’s − 4 / 5 law. No connection with the Navier-Stokes system is assumed, although one would like that, for each ν fixed, u ν ( x ) be some sort of statistical solution of the Navier-Stokes system, for the sake of consistency with first physical principles. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25
Kolmogorov 41 With these three hypothesis, one can derive all traditional features of homogeneous turbulence theory, including the fact that δ = 1 / 3, and the energy cascade, with the well-known Kolmogorov’s − 4 / 5 law. No connection with the Navier-Stokes system is assumed, although one would like that, for each ν fixed, u ν ( x ) be some sort of statistical solution of the Navier-Stokes system, for the sake of consistency with first physical principles. There are discrepancies between Komogorov 41 and experimental data, which has given rise to alternative theories. Most often criticized is self-similarity, which gives rise to multifractal theories of turbulence. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25
Kolmogorov 41 With these three hypothesis, one can derive all traditional features of homogeneous turbulence theory, including the fact that δ = 1 / 3, and the energy cascade, with the well-known Kolmogorov’s − 4 / 5 law. No connection with the Navier-Stokes system is assumed, although one would like that, for each ν fixed, u ν ( x ) be some sort of statistical solution of the Navier-Stokes system, for the sake of consistency with first physical principles. There are discrepancies between Komogorov 41 and experimental data, which has given rise to alternative theories. Most often criticized is self-similarity, which gives rise to multifractal theories of turbulence. The existence of a nonvanishing dissipation rate is considered fundamental, in general, to turbulence theories, as well as being well-established experimentally. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25
Kolmogorov 41 With these three hypothesis, one can derive all traditional features of homogeneous turbulence theory, including the fact that δ = 1 / 3, and the energy cascade, with the well-known Kolmogorov’s − 4 / 5 law. No connection with the Navier-Stokes system is assumed, although one would like that, for each ν fixed, u ν ( x ) be some sort of statistical solution of the Navier-Stokes system, for the sake of consistency with first physical principles. There are discrepancies between Komogorov 41 and experimental data, which has given rise to alternative theories. Most often criticized is self-similarity, which gives rise to multifractal theories of turbulence. The existence of a nonvanishing dissipation rate is considered fundamental, in general, to turbulence theories, as well as being well-established experimentally. See the 1995 book by U. Frisch for details. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 5 / 25
Onsager Conjecture Assuming that the realizations of the velocities u ν are solutions of the Navier-Stokes equations, the limit ν → 0, after passing to subsequences as needed, gives rise to a weak solution of the Euler system, which should dissipate energy at the residual rate ǫ 0 , by some mysterious means. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 6 / 25
Onsager Conjecture Assuming that the realizations of the velocities u ν are solutions of the Navier-Stokes equations, the limit ν → 0, after passing to subsequences as needed, gives rise to a weak solution of the Euler system, which should dissipate energy at the residual rate ǫ 0 , by some mysterious means. In 1949, Lars Onsager asked which would be the regularity, or more precisely, the irregularity of this ideal flow, in order to make inviscid dissipation possible. By a simple dimensional argument, Onsager concluded that something with less than 1 / 3 of a spatial derivative would be required to support inviscid dissipation. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 6 / 25
Onsager Conjecture Assuming that the realizations of the velocities u ν are solutions of the Navier-Stokes equations, the limit ν → 0, after passing to subsequences as needed, gives rise to a weak solution of the Euler system, which should dissipate energy at the residual rate ǫ 0 , by some mysterious means. In 1949, Lars Onsager asked which would be the regularity, or more precisely, the irregularity of this ideal flow, in order to make inviscid dissipation possible. By a simple dimensional argument, Onsager concluded that something with less than 1 / 3 of a spatial derivative would be required to support inviscid dissipation. His prediction became known as the "Onsager Conjecture". MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 6 / 25
Onsager Conjecture Assuming that the realizations of the velocities u ν are solutions of the Navier-Stokes equations, the limit ν → 0, after passing to subsequences as needed, gives rise to a weak solution of the Euler system, which should dissipate energy at the residual rate ǫ 0 , by some mysterious means. In 1949, Lars Onsager asked which would be the regularity, or more precisely, the irregularity of this ideal flow, in order to make inviscid dissipation possible. By a simple dimensional argument, Onsager concluded that something with less than 1 / 3 of a spatial derivative would be required to support inviscid dissipation. His prediction became known as the "Onsager Conjecture". Recent mathematical work on the Onsager conjecture has followed two fronts: proofs that Euler solutions with 1 / 3 of a derivative are conservative, and examples, mostly constructed by variations on the convex integration scheme, of Euler solutions with less than 1 / 3 of a derivative, exhibiting inviscid dissipation. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 6 / 25
Convex integration Examples of weak solutions of the Euler equations which exhibit inviscid dissipation. Scheffer 93, Shnirelman 95 and De Lellis, Szekelyhidi 2009 - not 1 inviscid dissipation, but nonuniqueness and time-varying energy. MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 7 / 25
Convex integration Examples of weak solutions of the Euler equations which exhibit inviscid dissipation. Scheffer 93, Shnirelman 95 and De Lellis, Szekelyhidi 2009 - not 1 inviscid dissipation, but nonuniqueness and time-varying energy. x , De Lellis, Szekelyhidi 2013, C 1 / 10 − ǫ Shnirelman 2000, L ∞ t L 2 , 2 x , t Isset 2014 C 1 / 5 − ǫ , Buckmaster, De Lellis, Szekelyhidi 2014, x , t t C 1 / 3 − ǫ L 1 . These do not adapt to 2D. x MIlton C. Lopes Filho (IM-UFRJ) Critical regularity for energy conservation September, 2015 7 / 25
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