arXiv subjects

Matei P. Coiculescu

Publications and source records attributed to Matei P. Coiculescu.

14 recordsLinked to original sources

arXiv2026

Extending the Córdoba-Martínez-Zoroa IPM Blow-Up to Uniformly Space-Time Smooth Forcing

By adapting the techniques used in the IPM blowup result of Córdoba-Martínez-Zoroa with spatially smooth force, we prove finite-time blow-up for the IPM equation on $\mathbb T^2$ with a uniformly spacetime smooth force. In particular, we show there exist a smooth odd initial density, a smooth odd force $F\in C^\infty([0,1]\times\mathbb T^2)$, and a classical solution $ρ$ on $[0,1)$ whose density gradient and spatial velocity gradient diverge in $L^\infty$ as $t\uparrow 1$. Nevertheless, $ρ(t)$ converges in $C^η$ for every $0\leqη<1$.

math.AP↗

arXiv2026

Ill-Posedness of the Euler Equations Linearized around Homogeneous Steady States

Let $L^2_m(\mathbb{R}^2)$ be the space of square-integrable functions on $\mathbb{R}^2$ with $m$-fold rotational symmetry. Let $\overlineω(r,θ) = r^{-α}f(θ)$ be a homogeneous steady state of the two-dimensional incompressible Euler equations with $(m\cdot l)$-fold rotational symmetry. If $f(θ)$ is a constant function we say that $\overlineω$ is a radial power-law vortex. We prove that the incompressible Euler equations in vorticity form, linearized around any homogeneous steady state $\overlineω$ that is not a radial power-law vortex, are ill-posed on $L^2_m(\mathbb{R}^2)$ for any $m\geq 2$, any $l\geq 1$ and $α\in (0,1)$.

math.AP↗

arXiv2026

Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations

We investigate homothetic forward self-similar solutions of the incompressible Navier-Stokes equations: the solutions $\overline{U}$ for which both $\overline{U}$ and $β\overline{U}$ are self-similar profiles for some nontrivial $β$. Homothetic solutions are, in addition, the only solutions for which a singular limit argument can be used to prove non-uniqueness of Leray-Hopf solutions along the lines of the Jia-Šverák program. In three dimensions, and for sufficiently regular initial data, we prove a Liouville theorem that rules out the existence of non-trivial homothetic solutions. In two dimensions, our Liouville theorem proves that the only decaying homothetic solution is the Oseen vortex. In addition, we prove that the Euler operator linearized around the Oseen vortex is stable. On the other hand, we also discover a homothetic solution for which an unstable approximate eigenvalue of the linearized Euler operator exists.

math.AP↗

arXiv2026

Multi-Sink Solutions to the Self-Similar Euler Equations

We construct examples and provide a classification of self-similar solutions to the two-dimensional incompressible Euler equations whose pseudo-velocity fields possess more than one stagnation point. These solutions are also homogeneous steady states of the Euler equations. In contrast, we prove that any homogeneous self-similar solution with bounded vorticity away from the origin necessarily admits only a single stagnation point, located at the origin. The solutions we construct develop velocity cusps along rays from the origin, and this allows for additional stagnation points of the pseudo-velocity field.

math.AP↗

arXiv2026

Stability of the Inviscid Power-Law Vortex

We prove that the power-law vortex $\overlineω(x) = β|x|^{-α}$, which explicitly solves the stationary unforced incompressible Euler equations in $\mathbb{R}^2$ in both physical and self-similar coordinates, is exponentially linearly stable in self-similar coordinates with the natural scaling. This result, which is valid for functions in a weighted $L^2$ space and in the un-weighted $L^2$ space with a mild symmetry condition, answers a question from the monograph by Albritton et al. Moreover, we prove that in physical coordinates the linearization around the power law vortex cannot generate an unstable $C_0$-semigroup.

math.AP↗

arXiv2025

Conditional Liouville theorems for the Navier-Stokes equations

We present a novel approach to the Liouville problem for the stationary Navier-Stokes equations. As an application of our method, we prove conditional Liouville theorems with assumptions on the antiderivative of the velocity that represent substantial improvements on what was heretofore known.

math.AP↗

arXiv2025

Non-Uniqueness of Smooth Solutions of the Navier-Stokes Equations from Critical Data

We consider the Cauchy problem for the incompressible Navier-Stokes equations in dimension three and construct initial data in the critical space $BMO^{-1}$ from which there exist two distinct global solutions, both smooth for all $t>0$. One consequence of this construction is the sharpness of the celebrated small data global well-posedness result of Koch and Tataru. This appears to be the first example of non-uniqueness for the Navier-Stokes equations with data at the critical regularity. The proof is based on a non-uniqueness mechanism proposed by the second author in the context of the dyadic Navier-Stokes equations.

math.AP↗

arXiv2024

Partial Regularity and Blowup for an Averaged Three-Dimensional Navier-Stokes Equation

We prove two results that together strongly suggest that obtaining a positive answer to the Navier-Stokes global regularity question requires more than a refinement of partial regularity theory. First we prove that there exists a class of bilinear operators $\mathfrak{B}$, which contains the Euler bilinear operator $\mathcal{E}(u,v):=\frac{1}{2}\mathbb{P}(u\cdot\nabla v + v\cdot\nabla u)$, such that for any $B\in \mathfrak{B}$, $n\geq3$, $α\in ((n+1)/4, (n+2)/4)$, and smooth solution $u$ of the pseudodifferential equation $\partial_t u +(-Δ)^αu +B(u,u)=0$ on $\mathbb{R}^n\times [0,T)$, we have that $u$ is also smooth at time $T$ away from a closed set of Hausdorff dimension at most $n+2-4α$. Next we prove that, for the Euclidean space $\mathbb{R}^3$, there exists an operator $C(u,v)\in \mathfrak{B}$ that is an averaged version of $\mathcal{E}$, that formally allows the dissipation of energy by the "cancellation identity" $\langle C(u,u), u\rangle =0$, and whose corresponding pseudodifferential equation $\partial_t u +(-Δ)^αu +C(u,u)=0$ admits a solution that blows up in finite time for all $α\in (0,5/4)$.

math.AP↗

arXiv2023

An Application of the Theory of Viscosity Solutions to Higher Order Differential Equations

We directly apply the theory of viscosity solutions to partial differential equations of order greater than two. We prove that there exists a solution in $C^{2,α}(B_R)\cap C(\overline{B_R})$ for the inhomogeneous $\infty$-Bilaplacian equation on a ball $B_R\subset \mathbb{R}^n$: $$Δ_\infty^2 u:=(Δu)^3 |D(Δu)|^2 =f(x)$$ with Navier Boundary conditions ($u=g\in C(\partial B_R), Δu =0 \textrm{ on } \partial B_R$). We also prove that there exists a solution in $C^{1,α}(\mathbb{R}^n)$ for all $α>0$ to the eigenvalue problem on $\mathbb{R}^n$: $$Δ_\infty^2 u =-λu+f(x)$$ whenever $n\geq 3, λ<0,$ and $f(x)$ is continuous, bounded, and supported on an annulus.

math.AP↗

arXiv2022

The Affine Shape of a Figure-Eight under the Curve Shortening Flow

We consider the curve shortening flow applied to a class of figure-eight curves: those with dihedral symmetry, convex lobes, and a monotonicity assumption on the curvature. We prove that when (non-conformal) linear transformations are applied to the solution so as to keep the bounding box the unit square, the renormalized limit converges to a quadrilateral which we call a bowtie. Along the way we prove that suitably chosen arcs of our evolving curves, when suitably rescaled, converge to the Grim Reaper Soliton under the flow. Our Grim Reaper Theorem is an analogue of a theorem of S. Angenent, which is proven in the locally convex case.

math.AP↗

arXiv2021

Some New Results in Geometric Analysis

This thesis presents three results in geometric analysis. We first analyze the curve-shortening flow on figure eight curves in the plane. Afterwards, we examine the point-wise curvature preserving flow on space curves. Lastly, we present an abridgment of our work on a family of three-dimensional Lie groups, which, when equipped with canonical left-invariant metrics, interpolate between Sol and hyperbolic space.

math.DG↗

arXiv2021

An Interpolation from Sol to Hyperbolic Space

We study a one-parameter family of nonisomorphic solvable Lie groups, which, when equipped with canonical left-invariant metrics, $$ds^2=e^{-2z}dx^2+e^{2αz}dy^2+dz^2$$ becomes an interpolation from a model of the Sol geometry to a model of Hyperbolic Space, with a stop at $\mathbb{H}^2\times \mathbb{R}$. These Lie groups are also Bianchi groups of Type VI with orthogonal coordinates. As a continuation of joint work with Richard Schwartz on Sol, we primarily analyze those Lie groups in our interpolation with some positive sectional curvature. Our main result is a characterization of the cut locus at the identity of the group that maximizes scalar curvature.

math.DG↗

arXiv2021

The Spheres of Sol

Let Sol be the three-dimensional solvable Lie group equipped with its standard left-invariant Riemannian metric. We give a precise description of the cut locus of the identity, and a maximal domain in the Lie algebra on which the Riemannian exponential map is a diffeomorphism. As a consequence, we prove that the metric spheres in Sol are topological spheres, and we characterize their singular points almost exactly.

math.DG↗

arXiv2020

Stationary Solutions of the Curvature Preserving Flow on Space Curves

We study a geometric flow on curves, immersed in $\mathbb{R}^3$, that have strictly positive torsion. The evolution equation is given by $$X_{t}=\frac{1}{\sqrtτ} \textbf{B}$$ where $τ$ is the torsion and $\textbf{B}$ is the unit binormal vector. In the case of constant curvature, we find all of the stationary solutions and linearize the PDE for torsion around stationary solutions admitting an explicit formula. Afterwards, we prove the $L^2(\mathbb{R})$ linear stability of the stationary solutions corresponding to helices with constant curvature and constant torsion.

math.DG↗