publications

For any positive regularity parameter β<\frac12, we construct non-conservative weak solutions of the 3D incompressible Euler equations which lie in H^βuniformly in time. In particular, we construct solutions which have an L^2-based regularity index strictly larger than \frac13, thus deviating from the H^\frac13-regularity corresponding to the Kolmogorov-Obhukov \frac53 power spectrum in the inertial range.

We review recent developments in the field of mathematical fluid dynamics which utilize techniques that go under the umbrella name convex integration. In the hydrodynamical context, these methods produce paradoxical solutions to the fluid equations which defy physical laws. These counterintuitive solutions have a number of properties that resemble predictions made by phenomenological theories of fluid turbulence. The goal of this review is to highlight some of these similarities while maintaining an emphasis on rigorous mathematical statements. We focus our attention on the construction of weak solutions for the incompressible Euler, Navier–Stokes, and magneto-hydrodynamic equations which violate these systems’ physical energy laws.

In this paper we construct unstable shocks in the context of 2D isentropic compressible Euler in azimuthal symmetry. More specifically, we construct initial data that when viewed in self-similar coordinates, converges asymptotically to the unstable C^\frac15 self-similar solution to the Burgers’ equation. Moreover, we show the behavior is stable in C^8 modulo a two dimensional linear subspace. Under the azimuthal symmetry assumption, one cannot impose additional symmetry assumptions in order to isolate the corresponding manifold of initial data leading to stability: rather, we rely on modulation variable techniques in conjunction with a Newton scheme.

Abstract We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high-order energy estimates in Sobolev spaces. © 2022 Wiley Periodicals LLC.

Chapter: The purpose of these lecture notes is to employ a heuristic approach in designing a convex integration scheme that produces non-conservative weak solutions to the Euler equations.

We construct weak solutions to the ideal magneto-hydrodynamic (MHD) equations which have finite total energy, and whose magnetic helicity is not a constant function of time. In view of Taylor’s conjecture, this proves that there exist finite energy weak solutions to ideal MHD which cannot be attained in the infinite conductivity and zero viscosity limit. Our proof is based on a Nash-type convex integration scheme with intermittent building blocks adapted to the geometry of the MHD system.

The purpose of this note is two-fold: A) We give a brief introduction into the problem of rigorously justifying the fundamental equations of wave turbulence theory (the theory of nonequilibrium statistical mechanics of nonlinear waves), and B) we describe a recent work of the authors in which they obtain the so-called wave kinetic equation, predicted in wave turbulence theory, for the nonlinear Schrödinger equation on short but nontrivial time scales.

Consider the cubic nonlinear Schrödinger equation set on a d-dimensional torus, with data whose Fourier coefficients have phases which are uniformly distributed and independent. We show that, on average, the evolution of the moduli of the Fourier coefficients is governed by the so-called wave kinetic equation, predicted in wave turbulence theory, on a nontrivial timescale.

In this review article we discuss a number of recent results concerning \em wild weak solutions of the incompressible Euler and Navier-Stokes equations. These results build on the groundbreaking works of De Lellis and Székelyhidi Jr., who extended Nash’s fundamental ideas on C^1 flexible isometric embeddings, into the realm of fluid dynamics. These techniques, which go under the umbrella name \em convex integration, have fundamental analogies with the phenomenological theories of hydrodynamic turbulence [51,54,55,200]. Mathematical problems arising in turbulence (such as the Onsager conjecture) have not only sparked new interest in convex integration, but certain experimentally observed features of turbulent flows (such as intermittency) have also informed new convex integration constructions. First, we give an elementary construction of nonconservative C^0+_x,t weak solutions of the Euler equations, first proven by De Lellis-Székelyhidi Jr. [52,53].. Second, we present Isett’s [108] recent resolution of the flexible side of the Onsager conjecture. Here, we in fact follow the joint work [21] of De Lellis-Székelyhidi Jr. and the authors of this paper, in which weak solutions of the Euler equations in the regularity class C^\frac 13-_x,t are constructed, attaining any energy profile. Third, we give a concise proof of the authors’ recent result [23], which proves the existence of infinitely many weak solutions of the Navier-Stokes in the regularity class C^0_t L^2+_x ∩C^0_t W^1,1+_x. We conclude the article by mentioning a number of open problems at the intersection of convex integration and hydrodynamic turbulence.

We prove non-uniqueness for a class of weak solutions to the Navier-Stokes equations which have bounded kinetic energy, integrable vorticity, and are smooth outside a fractal set of singular times with Hausdorff dimension strictly less than 1.

Consider the surface quasi-geostrophic equation with random diffusion, white in time. We show global existence and uniqueness in high probability for the associated Cauchy problem satisfying a Gevrey type bound. This article is inspired by a recent work of Glatt-Holtz and Vicol [16].

For initial datum of finite kinetic energy, Leray has proven in 1934 that there exists at least one global in time finite energy weak solution of the 3D Navier-Stokes equations. In this paper we prove that weak solutions of the 3D Navier-Stokes equations are not unique in the class of weak solutions with finite kinetic energy. Moreover, we prove that Hölder continuous dissipative weak solutions of the 3D Euler equations may be obtained as a strong vanishing viscosity limit of a sequence of finite energy weak solutions of the 3D Navier-Stokes equations.

We prove that given any β<1/3, a time interval [0,T], and given any smooth energy profile e : [0,T] \to (0,∞), there exists a weak solution v of the three-dimensional Euler equations such that v ∈C^β([0,T]\times \mathbbT^3), with e(t) = \int_\mathbbT^3 |v(x,t)|^2 dx for all t∈[0,T]. Moreover, we show that a suitable h-principle holds in the regularity class C^\beta_t,x, for any β<1/3. The implication of this is that the dissipative solutions we construct are in a sense typical in the appropriate space of subsolutions as opposed to just isolated examples.

This paper is devoted to the analysis of the continuous resonant (CR) equation, in dimensions greater than 2. This equation arises as the large box (or high frequency) limit of the nonlinear Schrodinger equation on the torus, and was derived in a companion paper by the same authors. We initiate the investigation of the structure of (CR), its local well-posedness, and the existence of stationary waves.

We consider the nonlinear Schrödinger (NLS) equation posed on the box [0,L]^d with periodic boundary conditions. The aim is to describe the long-time dynamics by deriving effective equations for it when L is large and the characteristic size εof the data is small. Such questions arise naturally when studying dispersive equations that are posed on large domains (like water waves in the ocean), and also in theory of statistical physics of dispersive waves, that goes by the name of "wave turbulence". Our main result is deriving a new equation, the continuous resonant (CR) equation, that describes the effective dynamics for large L and small εover very large time-scales. Such time-scales are well beyond the (a) nonlinear time-scale of the equation, and (b) the Euclidean time-scale at which the effective dynamics are given by (NLS) on \mathbb R^d. The proof relies heavily on tools from analytic number theory, such as a relatively modern version of theHardy-Littlewood circle method, which are modified and extended to be applicable in a PDE setting.

We prove that weak solutions of the inviscid SQG equations are not unique, thereby answering Open Problem 11 in the survey arXiv:1111.2700 by De Lellis and Székelyhidi Jr. Moreover, we also show that weak solutions of the dissipative SQG equation are not unique, even if the fractional dissipation is stronger than the square root of the Laplacian.

For any ε>0 we show the existence of continuous periodic weak solutions v of the Euler equations which do not conserve the kinetic energy and belong to the space L^1_t (C_x^\frac13-ε), namely x\mapsto v (x,t) is (\frac13-ε)-Hölder continuous in space at a.e. time t and the integral ∫[v(⋅, t)]_\frac13-ε dt is finite. A well-known open conjecture of L. Onsager claims that such solutions exist even in the class L^\infty_t (C_x^\frac13-ε).

In recent work by Isett (arXiv:1211.4065), and later by Buckmaster, De Lellis, Isett and Székelyhidi Jr. (arXiv:1302.2815), iterative schemes where presented for constructing solutions belonging to the Hölder class C^1/5-ε of the 3D incompressible Euler equations which do not conserve energy. The cited work is partially motivated by a conjecture of Lars Onsager in 1949 relating to the existence of C^1/3-ε solutions to the Euler equations which dissipate energy. In this note we show how the later scheme can be adapted in order to prove the existence of non-trivial Hölder continuous solutions which for almost every time belong to the critical Onsager Hölder regularity C^1/3-ε and have compact temporal support.

Recently the second and fourth authors developed an iterative scheme for obtaining rough solutions of the 3D incompressible Euler equations in Hölder spaces. The motivation comes from Onsager’s conjecture. The construction involves a superposition of weakly interacting perturbed Beltrami flows on infinitely many scales. An obstruction to better regularity arises from the errors in the linear transport of a fast periodic flow by a slow velocity field. In a recent paper the third author has improved upon the methods, introducing some novel ideas on how to deal with this obstruction, thereby reaching a better Hölder exponent — albeit weaker than the one conjectured by Onsager. In this paper we give a shorter proof of this final result, adhering more to the original scheme of the second and fourth authors and introducing some new devices. More precisely we show that for any positive ε, there exist periodic solutions of the 3D incompressible Euler equations that dissipate the total kinetic energy and belong to the Hölder class C^\frac15-ε.

In this paper we will prove the existence of weak solutions to the Korteweg-de Vries initial value problem on the real line with H^-1 initial data; moreover, we will study the problem of orbital and asymptotic H^s stability of solitons for integers s≥-1; finally, we will also prove new a priori H^-1 bounds for solutions to the Korteweg-de Vries equation. The paper will utilise the Miura transformation to link the Korteweg-de Vries equation to the modified Korteweg-de Vries equation.