arXiv subjects

Philip Isett

Publications and source records attributed to Philip Isett.

14 recordsLinked to original sources

arXiv2025

Integral formulas for under/overdetermined differential operators via recovery on curves and the finite-dimensional cokernel condition I: General theory

We introduce a new versatile method for constructing solution operators (i.e., right-inverses up to a finite rank operator) for a wide class of underdetermined PDEs $P u = f$, which are regularizing of optimal order and, more interestingly, whose integral kernels have certain prescribed support properties. By duality, we simultaneously obtain integral representation formulas (i.e., left-inverses up to a finite rank operator) for overdetermined PDEs $P^{\ast} v = g$ with analogous properties, which lead to Poincar\'e- or Korn-type inequalities. Our method applies to operators such as the divergence, linearized scalar curvature, and linearized Einstein constraint operators (which are underdetermined), as well as the gradient, Hessian, trace-free part of the Hessian, Killing, and conformal Killing operators (which are overdetermined). The starting point for our construction is a condition - dubbed the recovery on curves condition (RC) - that leads to Green's functions for $P$ supported on prescribed curves. Then the desired integral solution operators (and, by duality, integral representation formulas) are obtained by taking smooth averages over a suitable family of curves. This procedure generalizes the previous constructions of Bogovskii, Oh-Tataru, and Reshetnyak. We furthermore identify a simple algebraic sufficient condition for (RC), namely, that the principal symbol $p(x, \xi)$ of $P$ is full-rank for all non-zero complex vectors $\xi$ (as opposed to real, as in ellipticity). When the principal symbol has constant coefficients, this is equivalent to (RC) and also to the condition that the formal cokernel of $P$ (without any boundary conditions) is finite-dimensional; for this reason, we call it the finite-dimensional cokernel condition (FC). We give a short proof that all operators above satisfy (FC), and thus (RC). Various applications will be considered in subsequent papers.

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arXiv2025

Intermittency and Dissipation Regularity in Turbulence

We lay down a geometric-analytic framework to capture properties of energy dissipation within weak solutions to the incompressible Euler equations. For solutions with spatial Besov regularity, it is proved that the Duchon-Robert distribution has optimal improved regularity in a negative Besov space and, in the case it is a Radon measure, it is absolutely continuous with respect to a suitable Hausdorff measure. This imposes quantitative constraints on the dimension of the, possibly fractal, dissipative set and the admissible structure functions exponents, relating to the phenomenon of ''intermittency'' in turbulence. As a by-product of the approach, we also recover many known ''Onsager singularity'' type results.

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arXiv2024

A proof of Onsager's Conjecture for the SQG equation

We construct solutions to the SQG equation that fail to conserve the Hamiltonian while having the maximal allowable regularity for this property to hold. This result solves the generalized Onsager conjecture on the threshold regularity for Hamiltonian conservation for SQG.

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arXiv2024

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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arXiv2022

Intermittency and lower dimensional dissipation in incompressible fluids

In the context of incompressible fluids, the observation that turbulent singular structures fail to be space filling is known as ``intermittency'' and it has strong experimental foundations. Consequently, as first pointed out by Landau, real turbulent flows do not satisfy the central assumptions of homogeneity and self-similarity in the K41 theory, and the K41 prediction of structure function exponents $\zeta_p=\frac{p}{3}$ might be inaccurate. In this work we prove that, in the inviscid case, energy dissipation that is lower-dimensional in an appropriate sense implies deviations from the K41 prediction in every $p-$th order structure function for $p>3$. By exploiting a Lagrangian-type Minkowski dimension that is very reminiscent of the Taylor's frozen turbulence hypothesis, our strongest upper bound on $\zeta_p$ coincides with the $\beta-$model proposed by Frisch, Sulem and Nelkin in the late 70s, adding some rigorous analytical foundations to the model. More generally we explore the relationship between dimensionality assumptions on the dissipation support and restrictions on the $p-$th order absolute structure functions. This approach differs from the current mathematical works on intermittency by its focus on geometrical rather than purely analytical assumptions. The proof is based on a new local variant of the celebrated Constantin-E-Titi argument that features the use of a third order commutator estimate, the special double regularity of the pressure, and mollification along the flow of a vector field.

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arXiv2020

A direct approach to nonuniqueness and failure of compactness for the SQG equation

We give an alternative proof of the nonuniqueness of weak solutions to the surface quasigeostrophic equation (SQG) first shown in [Buckmaster-Shkoller-Vicol, '16]. Our approach proceeds directly at the level of the scalar field. Furthermore, we prove that every smooth scalar field with compact support that conserves the integral can be realized as a weak limit of solutions to SQG.

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arXiv2017

Nonuniqueness and existence of continuous, globally dissipative Euler flows

We show that H\"{o}lder 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 ${\mathbb T}^3$. 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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arXiv2017

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 H\"{o}lder 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 H\"{o}lder 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 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 \cite{IOnonpd} to modify the convex integration scheme. We also prove results on general solutions at the critical regularity that may not conserve energy. These include a theorem on intermittency stating roughly that energy dissipating solutions cannot have absolute structure functions satisfying the Kolmogorov-Obukhov scaling for any $p > 3$ if their singular supports have space-time Lebesgue measure zero.

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arXiv2016

A Proof of Onsager's Conjecture

For any $\alpha < 1/3$, we construct weak solutions to the $3D$ incompressible Euler equations in the class $C_tC_x^\alpha$ that have nonempty, compact support in time on ${\mathbb R} \times {\mathbb T}^3$ and therefore fail to conserve the total kinetic energy. This result, together with the proof of energy conservation for $\alpha > 1/3$ due to [Eyink] and [Constantin, E, Titi], solves Onsager's conjecture that the exponent $\alpha = 1/3$ marks the threshold for conservation of energy for weak solutions in the class $L_t^\infty C_x^\alpha$. The previous best results were solutions in the class $C_tC_x^\alpha$ for $\alpha < 1/5$, due to the author, and in the class $L_t^1 C_x^\alpha$ for $\alpha < 1/3$ due to Buckmaster, De Lellis and Sz\'{e}kelyhidi, both based on the method of convex integration developed for the incompressible Euler equations by De Lellis and Sz\'ekelyhidi. The present proof combines the method of convex integration and a new "gluing approximation" technique. The convex integration part of the proof relies on the "Mikado flows" introduced by [Daneri, Sz\'ekelyhidi] and the framework of estimates developed in the author's previous work.

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arXiv2015

On Nonperiodic Euler Flows with Hölder Regularity

In [Isett,13], the first author proposed a strengthening of Onsager's conjecture on the failure of energy conservation for incompressible Euler flows with Hölder regularity not exceeding $1/3$. This stronger form of the conjecture implies that anomalous dissipation will fail for a generic Euler flow with regularity below the Onsager critical space $L_t^\infty B_{3,\infty}^{1/3}$ due to low regularity of the energy profile. In this paper, we establish two theorems that may be viewed as first steps towards establishing the conjectured failure of energy regularity for generic solutions with Hölder exponent less than $1/5$. Our first result shows that any non-negative function with compact support and Hölder regularity $1/2$ can be prescribed as the energy profile of an Euler flow in the class $C^{1/5-ε}_{t,x}$. The exponent $1/2$ is sharp in view of a regularity result of [Isett,13]. The proof employs an improved greedy algorithm scheme that builds upon that in [Buckmaster-De Lellis-Székelyhidi, 13]. Our second result shows that any given smooth Euler flow can be perturbed in $C^{1/5-ε}_{t,x}$ on any pre-compact subset of ${\mathbb R}\times {\mathbb R}^3$ to violate energy conservation. In particular, there exist nonzero $C^{1/5-ε}_{t,x}$ solutions to Euler with compact space-time support, generalizing previous work of the first author [Isett,12] to the nonperiodic setting.

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arXiv2014

Holder Continuous Solutions of Active Scalar Equations

We consider active scalar equations $\partial_t θ+ \nabla \cdot (u \, θ) = 0$, where $u = T[θ]$ is a divergence-free velocity field, and $T$ is a Fourier multiplier operator with symbol $m$. We prove that when $m$ is not an odd function of frequency, there are nontrivial, compactly supported solutions weak solutions, with Hölder regularity $C^{1/9-}_{t,x}$. In fact, every integral conserving scalar field can be approximated in ${\cal D}'$ by such solutions, and these weak solutions may be obtained from arbitrary initial data. We also show that when the multiplier $m$ is odd, weak limits of solutions are solutions, so that the $h$-principle for odd active scalars may not be expected.

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arXiv2014

Hölder Continuous Euler Flows in Three Dimensions with Compact Support in Time

Building on the recent work of C. De Lellis and L. Székelyhidi, 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 Hölder class $C_{t,x}^{1/5 - ε}$. By slightly modifying the proof, we show that every smooth solution to incompressible Euler on $(-2, 2) \times {\mathbb T}^3$ coincides on $(-1, 1) \times {\mathbb T}^3$ with some Hölder continuous solution that is constant outside $(-3/2, 3/2) \times {\mathbb T}^3$. 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 Hölder exponent $1/3 - ε$.

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arXiv2014

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 $\mathbb{T}^{d}$ or $\mathbb{R}^{d}$, 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.

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arXiv2013

Regularity in time along the coarse scale flow for the incompressible Euler equations

One of the most remarkable features of known nonstationary solutions to the incompressible Euler equations is the phenomenon known as the Taylor hypothesis, which predicts that coarse scale averages of the velocity carry the fine scale features of the flow. In this work, we develop a time regularity theory for Euler weak solutions based on quantitative expressions of this hypothesis. We assume only that our velocity field is H\"{o}lder continuous in the spatial variables, which is well-motivated by problems related to turbulence, but precludes the application of Lagrangian methods or local well-posedness theory. Despite the dramatic lack of well-posedness, we obtain a rich theory of regularity in time for solutions, especially concerning advective derivatives. In particular, any Euler flow of class $v \in L_t^\infty C_x^\alpha$ has continuous advective derivatives of any order less than $\frac{\alpha}{1-\alpha}$, and every point has a trajectory passing through it that is $C^r$ for all $r < \frac{1}{1-\alpha}$, and one that is $C^\infty$ if $v$ is $C^1$ or $v \in \bigcap_{\alpha < 1} L_t^\infty C_x^\alpha$ has borderline regularity. In a follow up work, we show that all trajectories are of class $C^{1/(1-\alpha)}$ in time whenever $1/(1-\alpha) \notin {\mathbb Z}$, whether or not the trajectories are unique.

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