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László Székelyhidi Jr

Publications and source records attributed to László Székelyhidi Jr.

At least 19 recordsLinked to original sources
arXiv2026

Onsager's Conjecture for Ideal Magnetohydrodynamics

For any $0\leγ<1/3$ we construct weak solutions $(v,B,p)$ of the ideal MHD equations with $v,B\in C^γ(\mathbb T^3\times\mathbb R)$, which conserve neither the total energy nor the cross-helicity and have nontrivial magnetic helicity. We also establish anisotropic Hölder bounds with distinct velocity and magnetic exponents and stronger regularity along every magnetic field line. The admissible exponents connect the Goldreich--Sridhar \cite{GoldreichSridhar1995} spatial and parallel pair $1/3, 1/2$ with the $1/4$ spatial scaling of the Iroshnikov--Kraichnan weak-turbulence picture \cite{Iroshnikov1963,Kraichnan1965}. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is regarded as a path of volume-preserving diffeomorphisms; the proof is then based on the interplay between classical convex integration techniques and geometric constructions at the level of the Lie algebra of this Lie group. Our work substantially extends the recent result of Enciso, Peñafiel-Tomás and Peralta-Salas \cite{EnPePe} and can be used to reprove \cite{GiRa} of Giri and Radu without a Newton iteration.

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arXiv2026

$C^{1/5^{-}}$ Convex Integration Solutions of Ideal MHD

For any $0\leq γ< 1/5$, we construct weak solutions $(v, B, p )$ of the Ideal MHD Equations which do not conserve the total kinetic energy, the cross-helicity and lie in $C^γ(\mathbb{T}^3\times\mathbb{R})$. In the spirit of Arnold's formulation of ideal hydrodynamics, a solution is thought of as a path of volume-preserving diffeomorphisms; the proof is then based on the interplay between classical convex integration techniques and geometric constructions at the level of the Lie algebra of this Lie group. Our work substantially extends the recent work of and building on the recent work of Enciso, Peñafiel-Tomás and Peralta-Salas.

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arXiv2024

Contact discontinuities for 2-D isentropic Euler are unique in 1-D but wildly non-unique otherwise

We develop a general framework for studying non-uniqueness of the Riemann problem for the isentropic compressible Euler system in two spatial dimensions, and in this paper we present the most delicate result of our method: non-uniqueness of the contact discontinuity. Our approach is computational, and uses the pressure law as an additional degree of freedom. The stability of the contact discontinuities for this system is a major open problem (see Gui-Qiang Chen and Ya-Guang Wang [Nonlinear partial differential equations, volume 7 of Abel Symposia. Springer, Heidelberg, 2012.]). We find a smooth pressure law $p$, verifying the physically relevant condition $p'>0$, such that for the isentropic compressible Euler system with this pressure law, contact discontinuity initial data is wildly non-unique in the class of bounded, admissible weak solutions. This result resolves the question of uniqueness for contact discontinuity solutions in the compressible regime. Moreover, in the same regularity class in which we have non-uniqueness of the contact discontinuity, i.e. $L^\infty$, with no $BV$ regularity or self-similarity, we show that the classical contact discontinuity solution to the two-dimensional isentropic compressible Euler system is in fact unique in the class of bounded, admissible weak solutions if we restrict to 1-D solutions.

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arXiv2023

On the isometric version of Whitney's strong embedding theorem

We prove a version of Whitney's strong embedding theorem for isometric embeddings within the general setting of the Nash-Kuiper h-principle. More precisely, we show that any $n$-dimensional smooth compact manifold admits infinitely many global isometric embeddings into $2n$-dimensional Euclidean space, of Hölder class $C^{1,θ}$ with $θ<1/3$ for $n=2$ and $θ<(n+2)^{-1}$ for $n\geq3$. The proof is performed by Nash-Kuiper's convex integration construction and applying the gluing technique of the authors on short embeddings with small amplitude.

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arXiv2023

The Baire category method for intermittent convex integration

We use a convex integration construction from \cite{ModenaSattig2020} in a Baire category argument to show that weak solutions to the transport equation with incompressible vector fields with Sobolev regularity are generic in the Baire category sense. Using the construction of \cite{BurczakModenaSzekelyhidi20} we prove an analog statement for the 3D Navier-Stokes equations.

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arXiv2022

Nonexistence of $T_4$ configurations for hyperbolic systems and the Liu entropy condition

We study the constitutive set $\mathcal{K}$ arising from a $2\times 2$ system of conservation laws in one space dimension, endowed with one entropy and entropy-flux pair. The convexity properties of the set $\mathcal{K}$ relate to the well-posedness of the underlying system and the ability to construct solutions via convex integration. Relating to the convexity of $\mathcal{K}$, in the particular case of the $p$-system, Lorent and Peng [Calc. Var. Partial Differential Equations, 59(5):Paper No. 156, 36, 2020] show that $\mathcal{K}$ does not contain $T_4$ configurations. Recently, Johansson and Tione [arXiv e-prints, page arXiv:2208.10979, August 2022] showed that $\mathcal{K}$ does not contain $T_5$ configurations. In this paper, we provide a substantial generalization of these results, based on a careful analysis of the shock curves for a large class of $2\times 2$ systems. In particular, we provide several sets of hypothesis on general systems which can be used to rule out the existence of $T_4$ configurations in the constitutive set $\mathcal{K}$. In particular, our results show the nonexistence of $T_4$ configurations for every well-known $2\times 2$ hyperbolic system of conservation laws which verifies the Liu entropy condition.

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arXiv2021

Magnetic helicity, weak solutions and relaxation of ideal MHD

We revisit the issue of conservation of magnetic helicity and the Woltjer-Taylor relaxation theory in magnetohydrodynamics in the context of weak solutions. We introduce a relaxed system for the ideal MHD system, which decouples the effects of hydrodynamic turbulence such as the appearance of a Reynolds stress term from the magnetic helicity conservation in a manner consistent with observations in plasma turbulence. As by-products we answer two open questions in the field: We show the sharpness of the $L^3$ integrability condition for magnetic helicity conservation and provide turbulent bounded solutions for MHD dissipating energy and cross helicity but with (arbitrary) constant magnetic helicity.

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arXiv2020

Dissipative Euler flows for vortex sheet initial data without distinguished sign

We construct infinitely many admissible weak solutions to the 2D incompressible Euler equations for vortex sheet initial data. Our initial datum has vorticity concentrated on a simple closed curve in a suitable Hölder space and the vorticity may not have a distinguished sign. Our solutions are obtained by means of convex integration; they are smooth outside a "turbulence" zone which grows linearly in time around the vortex sheet. As a byproduct, this approach shows how the growth of the turbulence zone is controlled by the local energy inequality and measures the maximal initial dissipation rate in terms of the vortex sheet strength.

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arXiv2020

A new approach to the Rayleigh-Taylor instability

In this article we consider the inhomogeneous incompressible Euler equations describing two fluids with different constant densities under the influence of gravity as a differential inclusion. By considering the relaxation of the constitutive laws we formulate a general criterion for the existence of infinitely many weak solutions which reflect the turbulent mixing of the two fluids. Our criterion can be verified in the case that initially the fluids are at rest and separated by a flat interface with the heavier one being above the lighter one - the classical configuration giving rise to the Rayleigh-Taylor instability. We construct specific examples when the Atwood number is in the ultra high range, for which the zone in which the mixing occurs grows quadratically in time.

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arXiv2019

Global Nash-Kuiper theorem for compact manifolds

We obtain global extensions of the celebrated Nash-Kuiper theorem for $C^{1,θ}$ isometric immersions of compact manifolds with optimal Hölder exponent. In particular for the Weyl problem of isometrically embedding a convex compact surface in 3-space, we show that the Nash-Kuiper non-rigidity prevails upto exponent $θ<1/5$. This extends previous results on embedding 2-discs as well as higher dimensional analogues.

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arXiv2019

Bounded solutions of ideal MHD with compact support in space-time

We show that in 3-dimensional ideal magnetohydrodynamics there exist infinitely many bounded solutions that are compactly supported in space-time and have non-trivial velocity and magnetic fields. The solutions violate conservation of total energy and cross helicity, but preserve magnetic helicity. For the 2-dimensional case we show that, in contrast, no nontrivial compactly supported solutions exist in the energy space.

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arXiv2019

Very weak solutions to the two dimensional Monge-Ampére equation

In this short note we revisit the convex integration approach to constructing very weak solutions to the 2D Monge-Ampére equation with Hölder-continuous first derivatives of exponent $β<1/5$. Our approach is based on combining the approach of Lewicka-Pakzad \cite{Lewicka:2015gz} with a new diagonalization procedure which avoids the use of conformal coordinates, which was introduced by the second author with De Lellis and Inauen in \cite{DeLellis:2015wm} for the isometric immersion problem.

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arXiv2019

On turbulence and geometry: from Nash to Onsager

This article is a short nontechnical survey of recent progresses in fluid dynamics and differential geometry, relating a conjecture of Lars Onsager to the work of Nash on isometric embeddings.

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arXiv2018

$C^{1,α}$ isometric extensions

In this paper we consider the Cauchy problem for isometric immersions. More precisely, given a smooth isometric immersion of a codimension one submanifold we construct $C^{1,α}$ isometric extensions for any $α<\frac{1}{n(n+1)+1}$ via the method of convex integration.

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arXiv2017

Piecewise constant subsolutions for the Muskat problem

We show the existence of infinitely many admissible weak solutions for the incompressible porous media equations for all Muskat-type initial data with $C^{3,α}$-regularity of the interface in the unstable regime and for all non-horizontal data with $C^{3,α}$-regularity in the stable regime. Our approach involves constructing admissible subsolutions with piecewise constant densities. This allows us to give a rather short proof where it suffices to calculate the velocity and acceleration at time zero - thus emphasizing the instantaneous nature of non-uniqueness due to discontinuities in the initial data.

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