382 research outputs found
Holder Continuous Solutions of Active Scalar Equations
We consider active scalar equations , where is a divergence-free velocity field, and
is a Fourier multiplier operator with symbol . We prove that when is
not an odd function of frequency, there are nontrivial, compactly supported
solutions weak solutions, with H\"older regularity . In fact,
every integral conserving scalar field can be approximated in by
such solutions, and these weak solutions may be obtained from arbitrary initial
data. We also show that when the multiplier is odd, weak limits of
solutions are solutions, so that the -principle for odd active scalars may
not be expected.Comment: 61 page
A heat flow approach to Onsager's conjecture for the Euler equations on manifolds
We give a simple proof of Onsager's conjecture concerning energy conservation
for weak solutions to the Euler equations on any compact Riemannian manifold,
extending the results of Constantin-E-Titi and
Cheskidov-Constantin-Friedlander-Shvydkoy in the flat case. When restricted to
or , our approach yields an alternative proof
of the sharp result of the latter authors.
Our method builds on a systematic use of a smoothing operator defined via a
geometric heat flow, which was considered by Milgram-Rosenbloom as a means to
establish the Hodge theorem. In particular, we present a simple and geometric
way to prove the key nonlinear commutator estimate, whose proof previously
relied on a delicate use of convolutions.Comment: 15 pages. Improved exposition, corrected typos. Added a criterion for
energy conservation in terms of the H\"older norm in Theorem 1.
On the Endpoint Regularity in Onsager's Conjecture
Onsager's conjecture states that the conservation of energy may fail for 3D incompressible Euler flows with Hölder regularity below 1/3. This conjecture was recently solved by the author, yet the endpoint case remains an interesting open question with further connections to turbulence theory. In this work, we construct energy non-conserving solutions to the 3D incompressible Euler equations with space-time Hölder regularity converging to the critical exponent at small spatial scales and containing the entire range of exponents [0,1/3).
Our construction improves the author's previous result towards the endpoint case. To obtain this improvement, we introduce a new method for optimizing the regularity that can be achieved by a general convex integration scheme. A crucial point is to avoid power-losses in frequency in the estimates of the iteration. This goal is achieved using localization techniques of [IO16b] to modify the convex integration scheme.
We also prove results on general solutions at the critical regularity that may not conserve energy. These include the fact that singularites of positive space-time Lebesgue measure are necessary for any energy non-conserving solution to exist while having critical regularity of an integrability exponent greater than three
Nonuniqueness and existence of continuous, globally dissipative Euler flows
We show that Hölder continuous, globally dissipative incompressible Euler flows (solutions obeying the local energy inequality) are nonunique and contain examples that strictly dissipate energy. The collection of such solutions emanating from a single 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
H\"older Continuous Euler Flows in Three Dimensions with Compact Support in Time
Building on the recent work of C. De Lellis and L. Sz\'{e}kelyhidi, we
construct global weak solutions to the three-dimensional incompressible Euler
equations which are zero outside of a finite time interval and have velocity in
the H\"{o}lder class . By slightly modifying the
proof, we show that every smooth solution to incompressible Euler on coincides on with some
H\"{o}lder continuous solution that is constant outside . We also propose a conjecture related to our main result that
would imply Onsager's conjecture that there exist energy dissipating solutions
to Euler whose velocity fields have H\"{o}lder exponent .Comment: Minor corrections throughout text and some added detail
A Proof of Onsager's Conjecture
For any α < 1/3, we construct weak solutions to the 3D incompressible Euler equations in the class C_tC_^xα that have nonempty, compact support in time on R Ă T^3 and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for α < 1/3 due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent α = 1/3 marks the threshold for conservation of energy for weak solutions in the class L_t^âC_x^α. The previous best results were solutions in the classC_tC_x^α for α < 1/5, due to [Isett], and in the class L_t^1C_x^α for α < 1/3 due to [Buckmaster, De Lellis, SzĂ©kelyhidi], both based on the method of convex integration developed for the incompressible Euler equations by [De Lellis, SzĂ©kelyhidi]. The present proof combines the method of convex integration and a new âGluing Approximationâ technique. The convex integration part of the proof relies on the âMikado flowsâ introduced by [Daneri, SzĂ©kelyhidi] and the framework of estimates developed in the author's previous work
On the conservation laws and the structure of the nonlinearity for SQG and its generalizations
Using a new definition for the nonlinear term, we prove that all weak
solutions to the SQG equation (and mSQG) conserve the angular momentum. This
result is new for the weak solutions of [Resnick, '95] and rules out the
possibility of anomalous dissipation of angular momentum. We also prove
conservation of the Hamiltonian under conjecturally optimal assumptions,
sharpening a well-known criterion of
[Cheskidov-Constantin-Friedlander-Shvydkoy, '08]. Moreover, we show that our
new estimate for the nonlinearity is optimal and that it characterizes the mSQG
nonlinearity uniquely among active scalar nonlinearities with a scaling
symmetry
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