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Vikram Giri

Publications and source records attributed to Vikram Giri.

14 recordsLinked to original sources
arXiv2026

Non-conservation of a generalized helicity in the Euler equations

For a $C^1_{t,x}$ solution $u$ to the incompressible 3D Euler equations, the helicity $H(u(t))=\int_{\mathbb{T}^3} u \cdot \textrm{curl}\, u$ is constant in time. For general low-regularity weak solutions, it is not always clear how to define the helicity, or whether it must be constant in time in the case that there is a clear definition. In this paper, we define a generalized helicity which extends the classical definitions and construct weak solutions of Euler of almost Onsager-critical regularity in $L^3$ with prescribed generalized helicity and kinetic energy.

math.AP↗
arXiv2025

Non-uniqueness of Hölder continuous solutions for Inhomogeneous Incompressible Euler flows

We consider the inhomogeneous (or density dependent) incompressible Euler equations in a three-dimensional periodic domain. We construct density $\varrho$ and velocity $u$ such that, for any $α<1/7$, both of them are $α$-Hölder continuous and $(\varrho, u)$ is a weak solution to the underlying equations. The proof is based on typical convex integration techniques using Mikado flows as building blocks. As a main novelty with respect to the related literature, our result produces a Hölder continuous density.

math.AP↗
arXiv2025

The $L^3$-based strong Onsager theorem

In this work, we prove the $L^3$-based strong Onsager conjecture for the three-dimensional Euler equations. Our main theorem states that there exist weak solutions which dissipate the total kinetic energy, satisfy the local energy inequality, and belong to $C^0_t (W^{\frac 13-, 3} \cap L^{\infty-})$. More precisely, for every $β<\frac 13$, we can construct such solutions in the space $C^0_t ( B^β_{3,\infty} \cap L^{\frac{1}{1-3β}} )$.

math.AP↗
arXiv2025

A Spectral Gap for Spinors on Hyperbolic Surfaces

The purpose of this note is to construct a sequence of spin hyperbolic surfaces $Σ_n$ with genus going to infinity and with a uniform spectral gap for the Dirac operator. Our construction is completely explicit. In particular, the $Σ_n$ can be taken to be a tower of covers, with each $Σ_n$ an arithmetic hyperbolic surface.

math.NT↗
arXiv2025

The structure of Lonely Runner spectra

For each subtorus $T$ of $(\mathbb{R}/\mathbb{Z})^n$, let $D(T)$ denote the (infimal) $L^\infty$-distance from $T$ to the point $(1/2,\ldots, 1/2)$. The $n$-th Lonely Runner spectrum $\mathcal{S}(n)$ is defined to be the set of all values achieved by $D(T)$ as $T$ ranges over the $1$-dimensional subtori of $(\mathbb{R}/\mathbb{Z})^n$ that are not contained in the coordinate hyperplanes. The Lonely Runner Conjecture predicts that $\mathcal{S}(n) \subseteq [0,1/2-1/(n+1)]$. Rather than attack this conjecture, we study the structure of the sets $\mathcal{S}(n)$. The main purpose of this note is to show that the set of accumulation points of $\mathcal{S}(n)$ is precisely $\mathcal{S}(n-1)$.

math.CO↗
arXiv2024

The Onsager conjecture in 2D: a Newton-Nash iteration

For any $γ<1/3$, we construct a nontrivial weak solution $u$ to the two-dimensional, incompressible Euler equations, which has compact support in time and satisfies $u\in C^γ(\mathbb R_t \times \mathbb T^2_x)$. In particular, the constructed solution does not conserve energy and, thus, settles the flexible part of the Onsager conjecture in two dimensions. The proof involves combining the Nash iteration technique with a new linear Newton iteration.

math.AP↗
arXiv2024

An Onsager-type theorem for SQG

We construct non-trivial weak solutions $θ\in C_t^0C_x^{0-}$ to the surface quasi-geostrophic (SQG) equations, which have compact support in time and, thus, violate the conservation of the Hamiltonian. The result is sharp in view of the fact that such a conservation law holds for all weak solutions in the class $C_{t,x}^0 \subset L_{t,x}^3$ (Isett-Vicol, 2015) and resolves the Onsager conjecture for SQG. The construction is achieved by means of a Nash iteration together with the linear decoupling method recently introduced in Giri-Radu (2023).

math.AP↗
arXiv2024

Hyperbolic 3-manifolds with uniform spectral gap for coclosed 1-forms

We study two quantifications of being a homology sphere for hyperbolic 3-manifolds, one geometric and one topological: the spectral gap for the Laplacian on coclosed 1-forms and the size of the first torsion homology group. We first construct a sequence of closed hyperbolic integer homology spheres with volume tending to infinity and a uniform coclosed 1-form spectral gap. This answers a question asked by Lin--Lipnowski. We also find sequences of hyperbolic rational homology spheres with the same properties that geometrically converge to a tame limit manifold. Moreover, we show that any such sequence must have unbounded torsion homology growth. Finally we show that a sequence of closed hyperbolic rational homology 3-spheres with uniformly bounded rank and a uniform coclosed 1-form spectral gap must have torsion homology that grows exponentially in volume.

math.GT↗
arXiv2023

A wavelet-inspired $L^3$-based convex integration framework for the Euler equations

In this work, we develop a wavelet-inspired, $L^3$-based convex integration framework for constructing weak solutions to the three-dimensional incompressible Euler equations. The main innovations include a new multi-scale building block, which we call an intermittent Mikado bundle; a wavelet-inspired inductive set-up which includes assumptions on spatial and temporal support, in addition to $L^p$ and pointwise estimates for Eulerian and Lagrangian derivatives; and sharp decoupling lemmas, inverse divergence estimates, and space-frequency localization technology which is well-adapted to functions satisfying $L^p$ estimates for $p$ other than $1$, $2$, or $\infty$. We develop these tools in the context of the Euler-Reynolds system, enabling us to give both a new proof of the intermittent Onsager theorem (An Intermittent Onsager Theorem, Inventiones Mathematicae, (2023), 233) in this paper, and a proof of the $L^3$-based strong Onsager conjecture in a companion paper (arXiv:2305.18509).

math.AP↗
arXiv2023

Instability and nonuniqueness for the $2d$ Euler equations in vorticity form, after M. Vishik

In this expository work, we present Vishik's theorem on non-unique weak solutions to the two-dimensional Euler equations on the whole space, \[ \partial_t ω+ u \cdot \nabla ω= f \, , \quad u = \frac{1}{2π} \frac{x^\perp}{|x|^2} \ast ω\, , \] with initial vorticity $ω_0 \in L^1 \cap L^p$ and $f \in L^1_t (L^1 \cap L^p)_x$, $p < \infty$. His theorem demonstrates, in particular, the sharpness of the Yudovich class. An important intermediate step is the rigorous construction of an unstable vortex, which is of independent physical and mathematical interest. We follow the strategy of Vishik but allow ourselves certain deviations in the proof and substantial deviations in our presentation, which emphasizes the underlying dynamical point of view.

math.AP↗
arXiv2021

On Non-uniqueness of continuous entropy solutions to the isentropic compressible Euler equations

We consider the Cauchy problem for the isentropic compressible Euler equations in a three-dimensional periodic domain under general pressure laws. For any smooth initial density away from the vacuum, we construct infinitely many entropy solutions with no presence of shock. In particular, the constructed density is smooth and the momentum is $α$-Hölder continuous for $α<1/7$. Also, we provide a continuous entropy solution satisfying the entropy inequality strictly.

math.AP↗
arXiv2021

Non-uniqueness of integral curves for autonomous Hamiltonian vector fields

In this work we prove the existence of an autonomous Hamiltonian vector field in W^{1,r}(T^d;R^d) with r< d-1and d>=4 for which the associated transport equation has non-unique positive solutions. As a consequence of Ambrosio superposition principle, we show that this vector field has non-unique integral curves with a positive Lebesgue measure set of initial data and moreover we show that the Hamiltonian is not constant along these integral curves.

math.AP↗
arXiv2020

Smoothing does not give a selection principle for transport equations with bounded autonomous fields

We give an example of a bounded divergence free autonomous vector field in $\mathbb R^3$ (and of a nonautonomous bounded divergence free vector field in $\mathbb R^2$) and of a bounded initial data for which the Cauchy problem for the corresponding transport equation has $2$ distinct solutions. We then show that both solutions are limits of classical solutions of transport equations for appropriate smoothings of the vector fields and of the initial data.

math.AP↗