Convex Integration Research Articles

A new convex integration approach for the compressible Euler equations and failure of the local maximal dissipation criterion

Abstract In this paper we establish a new convex integration approach for the barotropic compressible Euler equations in two space dimensions. In contrast to existing literature, our new method generates not only the momentum for given density, but also the energy and the energy flux. This allows for a simple way to construct admissible solutions, i.e. solutions which satisfy the energy inequality. Moreover using the convex integration method developed in this paper, we show that the local maximal dissipation criterion fails in the following sense: There exist wild solutions which beat the self-similar solution of the one-dimensional Riemann problem extended to two dimensions. Hence the local maximal dissipation criterion rules out the self-similar solution. The convex integration machinery itself is carried out in a very general way. Hence this paper provides a universal framework for convex integration which not even specifies the form of the partial differential equations under consideration. Therefore this general framework is applicable in many different situations.

Convex integration solution of two-dimensional hyperbolic Navier–Stokes equations*

Abstract Hyperbolic Navier–Stokes equations replace the heat operator within the Navier–Stokes equations with a damped wave operator. Due to this second-order temporal derivative term, there exist no known bounded quantities for its solution; consequently, various standard results for the Navier–Stokes equations such as the global existence of a weak solution, that is typically constructed via Galerkin approximation, are absent in the literature. In this manuscript, we employ the technique of convex integration on the two-dimensional hyperbolic Navier–Stokes equations to construct a weak solution with prescribed energy and thereby prove its non-uniqueness. The main difficulty is the second-order temporal derivative term, which is too singular to be estimated as a linear error. One of our novel ideas is to use the time integral of the temporal corrector perturbation of the Navier–Stokes equations as the temporal corrector perturbation for the hyperbolic Navier–Stokes equations.

Remarks on Regularization by Noise, Convex Integration and Spontaneous Stochasticity

AbstractThis note is devoted to a discussion of the potential links and differences between three topics: regularization by noise, convex integration, spontaneous stochasticity. All of them deal with the effect on large scales of a small-scale perturbation of fluid dynamic equations. The effects sometimes have something in common, like convex integration and spontaneous stochasticity, sometimes they look the opposite, as in regularization by noise. We are not aware of rigorous links or precise explanations of the differences, and hope to drive new research with this comparative examination.

Stationary solutions to stochastic 3D Euler equations in Hölder space

We establish the existence of infinitely many global and stationary solutions in C(R,Cϑ) space for some ϑ>0 to the three-dimensional Euler equations driven by an additive stochastic forcing. The result is based on a new stochastic version of the convex integration method, incorporating the stochastic convex integration method developed in Hofmanová et al. (2022) and pathwise estimates to derive uniform moment estimates independent of time.

Non-uniqueness of Admissible Solutions for the 2D Euler Equation with Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} Vortex Data

For any 2<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$2<p<\\infty $$\\end{document} we prove that there exists an initial velocity field v∘∈L2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v^\\circ \\in L^2$$\\end{document} with vorticity ω∘∈L1∩Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega ^\\circ \\in L^1\\cap L^p$$\\end{document} for which there are infinitely many bounded admissible solutions v∈CtL2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$v\\in C_tL^2$$\\end{document} to the 2D Euler equation. This shows sharpness of the weak–strong uniqueness principle, as well as sharpness of Yudovich’s proof of uniqueness in the class of bounded admissible solutions. The initial data are truncated power-law vortices. The construction is based on finding a suitable self-similar subsolution and then applying the convex integration method. In addition, we extend it for 1<p<∞\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$1<p<\\infty $$\\end{document} and show that the energy dissipation rate of the subsolution vanishes at t=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$t=0$$\\end{document} if and only if p≥32\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$p\\ge \ icefrac {3}{2}$$\\end{document}, which is the Onsager critical exponent in terms of Lp\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$L^p$$\\end{document} control on vorticity in 2D.

Nonuniqueness of Weak Solutions to the Dissipative Aw–Rascle Model

We prove nonuniqueness of weak solutions to multi-dimensional generalisation of the Aw-Rascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of admissible weak solutions.

Non-uniqueness of forced active scalar equations with even drift operators

We consider forced active scalar equations with even and homogeneous degree 0 drift operator on Td. Inspired by the non-uniqueness construction for dyadic fluid models (Dai and Friedlander, 2022; Filonov and Khodunov, 2021), by implementing a sum-difference convex integration scheme we obtain non-unique weak solutions for the active scalar equation in space Ct0Cxα with α<12d+1. Without external forcing, Isett and Vicol (2015) constructed non-unique weak solutions for such active scalar equations with spatial regularity Cxα for α<14d+1.

𝐶^{1,1-𝜖} isometric embeddings

In this paper we use the convex integration technique enhanced by an extra iteration originally due to Källén and revisited by Kröner to provide a local h h -principle for isometric embeddings in the class C 1 , 1 − ε C^{1,1-\varepsilon } for n n -dimensional manifolds in codimension 1 2 n ( n + 1 ) \frac {1}{2}n(n+1) .

Onsager's ‘ideal turbulence’ theory

In 1945–1949, Lars Onsager made an exact analysis of the high-Reynolds-number limit for individual turbulent flow realisations modelled by incompressible Navier–Stokes equations, motivated by experimental observations that dissipation of kinetic energy does not vanish. I review here developments spurred by his key idea that such flows are well described by distributional or ‘weak’ solutions of ideal Euler equations. 1/3 Hölder singularities of the velocity field were predicted by Onsager and since observed. His theory describes turbulent energy cascade without probabilistic assumptions and yields a local, deterministic version of the Kolmogorov $4/5$ th law. The approach is closely related to renormalisation group methods in physics and envisages ‘conservation-law anomalies’, as discovered later in quantum field theory. There are also deep connections with large-eddy simulation modelling. More recently, dissipative Euler solutions of the type conjectured by Onsager have been constructed and his $1/3$ Hölder singularity proved to be the sharp threshold for anomalous dissipation. This progress has been achieved by an unexpected connection with work of John Nash on isometric embeddings of low regularity or ‘convex integration’ techniques. The dissipative Euler solutions yielded by this method are wildly non-unique for fixed initial data, suggesting ‘spontaneously stochastic’ behaviour of high-Reynolds-number solutions. I focus in particular on applications to wall-bounded turbulence, leading to novel concepts of spatial cascades of momentum, energy and vorticity to or from the wall as deterministic, space–time local phenomena. This theory thus makes testable predictions and offers new perspectives on large-eddy simulation in the presence of solid walls.

Wild solutions to scalar Euler-Lagrange equations

We study very weak solutions to scalar Euler-Lagrange equations associated with quadratic convex functionals. We investigate whether W 1 , 1 W^{1,1} solutions are necessarily W loc 1 , 2 W^{1,2}_{\operatorname {loc}} , which would make the theories by De Giorgi-Nash and Schauder applicable. We answer this question positively for a suitable class of functionals. This is an extension of Weyl’s classical lemma for the Laplace equation to a wider class of equations under stronger regularity assumptions. Conversely, using convex integration, we show that outside this class of functionals, there exist W 1 , 1 W^{1,1} solutions of locally infinite energy to scalar Euler-Lagrange equations. In addition, we prove an intermediate result which permits the regularity of a W 1 , 1 W^{1,1} solution to be improved to W loc 1 , 2 W^{1,2}_{\operatorname {loc}} under suitable assumptions on the functional and solution.

Non-classical solutions of the $p$-Laplace equation

In this paper we settle Iwaniec and Sbordone’s 1994 conjecture concerning very weak solutions to the p -Laplace equation. Namely, on the one hand we show that distributional solutions of the p -Laplace equation in W^{1,r} for p \neq 2 and r&gt;\max\,\{ 1,p-1\} are classical weak solutions if their weak derivatives belong to certain cones. On the other hand, we construct via convex integration non-energetic distributional solutions if this cone condition is not met, thus disproving Iwaniec and Sbordone’s conjecture in general.

Phase transitions in the fractional three-dimensional Navier–Stokes equations

The fractional Navier–Stokes equations on a periodic domain [0,L]3 differ from their conventional counterpart by the replacement of the −νΔu Laplacian term by νsAsu , where A=−Δ is the Stokes operator and νs=νL2(s−1) is the viscosity parameter. Four critical values of the exponent s⩾0 have been identified where functional properties of solutions of the fractional Navier–Stokes equations change. These values are: s=13 ; s=34 ; s=56 and s=54 . In particular: (i) for s>13 we prove an analogue of one of the Prodi–Serrin regularity criteria; (ii) for s⩾34 we find an equation of local energy balance and; (iii) for s>56 we find an infinite hierarchy of weak solution time averages. The existence of our analogue of the Prodi–Serrin criterion for s>13 suggests the sharpness of the construction using convex integration of Hölder continuous solutions with epochs of regularity in the range 0<s<13 .

Glimm’s method and density of wild data for the Euler system of gas dynamics *

We adapt Glimm’s approximation method to the framework of convex integration to show density of wild data for the (complete) Euler system of gas dynamics. The desired infinite family of entropy admissible solutions emanating from the same initial data is obtained via convex integration of suitable Riemann problems pasted with local smooth solutions. In addition, the wild data belong to BV class.

Adhesion and volume filling in one-dimensional population dynamics under Dirichlet boundary condition

We generalize the one-dimensional population model of Anguige &amp; Schmeiser [1] reflecting the cell-to-cell adhesion and volume filling and classify the resulting equation into the six types. Among these types, we fix one that yields a class of advection-diffusion equations of forward-backward-forward type and prove the existence of infinitely many global-in-time weak solutions to the initial-Dirichlet boundary value problem when the maximum value of an initial population density exceeds a certain threshold. Such solutions are extracted from the method of convex integration by Müller &amp; Šverák [12]; they exhibit fine-scale density mixtures over a finite time interval, then become smooth and identical, and decay exponentially and uniformly to zero as time approaches infinity. TE check: Please check the reference citation in abstract.

On convex integration solutions to the surface quasi-geostrophic equation driven by generic additive noise

On convex integration solutions to the surface quasi-geostrophic equation driven by generic additive noise

Rigidity and flexibility of isometric extensions

In this paper we consider the rigidity and flexibility of C^{1, \theta} isometric extensions. We show that the Hölder exponent \theta_0=\frac12 is critical in the following sense: if u\in C^{1,\theta} is an isometric extension of a smooth isometric embedding of a codimension one submanifold \Sigma and \theta&gt; \frac12 , then the tangential connection agrees with the Levi-Civita connection along \Sigma . On the other hand, for any \theta&lt;\frac12 we can construct C^{1,\theta} isometric extensions via convex integration which violate such property. As a byproduct we get moreover an existence theorem for C^{1, \theta} isometric embeddings, \theta&lt;\frac12 , of compact Riemannian manifolds with C^1 metrics and sharper amount of codimension.

Weak and parabolic solutions of advection–diffusion equations with rough velocity field

We study the Cauchy problem for the advection–diffusion equation ∂tu+div(ub)=Δu\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial _t u + {{\\,\\mathrm{\ extrm{div}}\\,}}(u\\varvec{b}) = \\Delta u$$\\end{document} associated with a merely integrable divergence-free vector field b\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varvec{b}$$\\end{document} defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.

Global existence and non-uniqueness of 3D Euler equations perturbed by transport noise

We construct Hölder continuous, global-in-time probabilistically strong solutions to 3D Euler equations perturbed by Stratonovich transport noise. Kinetic energy of the solutions can be prescribed a priori up to a stopping time, that can be chosen arbitrarily large with high probability. We also prove that there exist infinitely many Hölder continuous initial conditions leading to non-uniqueness of solutions to the Cauchy problem associated with the system. Our construction relies on a flow transformation reducing the SPDE under investigation to a random PDE, and convex integration techniques introduced in the deterministic setting by De Lellis and Székelyhidi, here adapted to consider the stochastic case. In particular, our novel approach allows to construct probabilistically strong solutions on [0,∞)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$[0,\\infty )$$\\end{document} directly.

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 fine scale features of the flow are advected by the mean velocity. In this work, we develop an extensive theory of time regularity for Euler weak solutions in any dimension based on quantitative realizations of this idea. Our work provides the key estimates for showing that the particle trajectories of any Euler flow that is C α C^\alpha in the spatial variables uniformly in time are of class C 1 / ( 1 − α ) C^{1/(1-\alpha )} when 1 / ( 1 − α ) 1/(1-\alpha ) is not an integer, whether or not the trajectories or solutions are unique. In particular, we prove the smoothness of trajectories in borderline spaces such as v ∈ C 1 v \in C^1 or bounded vorticity in any dimension. An essential point is the existence and improved regularity of advective derivatives of high order.

Nonuniqueness in law of stochastic 3D Navier–Stokes equations

We consider the stochastic Navier–Stokes equations in three dimensions and prove that the law of analytically weak solutions is not unique. In particular, we focus on three examples of a stochastic perturbation: an additive, a linear multiplicative and a nonlinear noise of cylindrical type, all driven by a Wiener process. In these settings, we develop a stochastic counterpart of the convex integration method introduced recently by Buckmaster and Vicol. This permits us to construct probabilistically strong and analytically weak solutions defined up to a suitable stopping time. In addition, these solutions fail to satisfy the corresponding energy inequality at a prescribed time with a prescribed probability. Then we introduce a general probabilistic construction used to extend the convex integration solutions beyond the stopping time and in particular to the whole time interval [0,\infty) . Finally, we show that their law is distinct from the law of solutions obtained by Galerkin approximation. In particular, nonuniqueness in law holds on an arbitrary time interval [0,T] , T&gt;0 .