arXiv subjects

Stan Palasek

Publications and source records attributed to Stan Palasek.

17 recordsLinked to original sources

arXiv2026

On the non-uniqueness of solutions of the axi-symmetric swirl-free Navier-Stokes equations, I

In this paper we construct numerically a new class of unstable self-similar solutions of the incompressible Navier-Stokes equations in $\mathbb{R}^3$. Our solutions are axially symmetric and homogeneous of degree $-1$ at $\infty$, and are unstable in the sense that the linearization around these solutions contains unstable modes. Solutions of this type have been discovered numerically by Guillod and \v{S}ver\'ak and Hou, Wang, and Yang, and have applications to proving non-uniqueness results. The main novelty in this paper is that we discover the existence of such solutions in the space of axially symmetric swirl-free (ASSF) vector fields. These approximate solutions are defined on all of $\mathbb R^3$ and achieve global pointwise residuals of order $10^{-10}$. We discuss the numerical construction of these solutions in detail, as well as their relevance to the problem of non-uniqueness of solutions of the incompressible Navier-Stokes equations in 3D, in the space of ASSF solutions.

math.AP↗

arXiv2026

Finite-time blow-up in an elementary model of the 3D Navier-Stokes equations

We demonstrate finite-time blow-up in a simple, realistic shell model of the 3D Navier-Stokes equations, equipped with "smooth" (i.e., rapidly decaying in frequency) initial data and forcing. Previously studied models either exhibit a turbulent cascade that regularizes the three-dimensional viscous dynamics, or rely on highly artificial interactions not transparently realized in the true Euler nonlinearity. We also treat the inviscid, unforced case and obtain singularity formation just above the energy level. We conclude with a discussion of the prospects for embedding the behavior of the dyadic model into the full Euler and Navier-Stokes equations.

math.AP↗

arXiv2025

Instantaneous Type I blow-up and non-uniqueness of smooth solutions of the Navier-Stokes equations

For any smooth, divergence-free initial data, we construct a solution of the Navier--Stokes equations that exhibits Type~I blow-up of the $L^\infty$ norm at time $T_*>0$, while remaining smooth in space and time on $\mathbb T^d\times([0,T]\setminus\{T_*\})$. An instantaneous injection of energy from infinite wavenumber initiates a bifurcation from the classical solution, producing an infinite family of spatially smooth solutions with the same data and thereby violating uniqueness of the Cauchy problem. A key ingredient is the first known construction of a complete inverse energy cascade realized by a classical Navier--Stokes flow, which transfers energy from infinitely high to low frequencies. The result holds in all dimensions $d\geq2$.

math.AP↗

arXiv2025

Arbitrary norm growth in the 3D Navier-Stokes equations

We construct a family of smooth initial data for the Navier-Stokes equations, bounded in $BMO^{-1}(\mathbb T^3)$, that gives rise to arbitrarily large global solutions. As a consequence, we rule out various hypothetical a priori estimates for strong solutions in terms of a critical norm of the initial data. To our knowledge, this is the first example of unbounded norm growth in the well-posed setting. The solutions exhibit an inverse cascade across an unbounded number of modes, with growth resulting from repeated squaring by the quadratic nonlinearity. This mechanism relies on largeness of the data in $B^{-1}_{\infty,\infty}$ and is fundamentally distinct from instantaneous norm inflation and ill-posedness phenomena.

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

Non-uniqueness in the Leray-Hopf class for a dyadic Navier-Stokes model

The uniqueness of Leray-Hopf solutions to the incompressible Navier-Stokes equations remains a significant open question in fluid mechanics. This paper proposes a potential mechanism for non-uniqueness, illustrated in a natural dyadic shell model. We show that, for the Obukhov model with $d>2$, there exist initial data at the critical regularity that give rise to two distinct Leray-Hopf solutions. These solutions exhibit an approximately discretely self-similar structure, with non-uniqueness resulting from a partial breaking of the scaling symmetry. The fundamental observation is that, in a certain scenario, the dynamics reduce to a sequence of weakly coupled finite-dimensional systems. Moreover, the predominant nonlinear interactions are identical to those arising in convex integration, suggesting the possibility of a similar construction in the full PDE setting.

math.AP↗

arXiv2023

Non-uniqueness up to the Onsager threshold for the forced SQG equation

We establish new non-uniqueness results for the forced inviscid surface quasi-geostrophic equation, via an alternating formulation of convex integration techniques. Our results imply non-uniquenesss in the class of weak solutions with $|\nabla|^{-1}θ\in C_tC_x^α$, for any $α<1$.

math.AP↗

arXiv2023

Convex integration above the Onsager exponent for the forced Euler equations

We establish new non-uniqueness results for the Euler equations with external force on $\mathbb{T}^{d}$ $(d\geq3)$. By introducing a novel alternating convex integration scheme, we construct non-unique, almost-everywhere smooth, Hölder-continuous solutions with regularity $\frac{1}{2}-$, which is notably above the Onsager threshold of $\frac{1}{3}$. The solutions we construct differ significantly in nature from those which arise from the recent unstable vortex construction of Vishik; in particular, our solutions are genuinely $d$-dimensional ($d\geq3$), and give non-uniqueness results for any smooth data. To the best of our knowledge, this is the first instance of a convex integration construction above the Onsager exponent.

math.AP↗

arXiv2022

A minimum critical blowup rate for the high-dimensional Navier-Stokes equations

We prove quantitative regularity and blowup theorems for the incompressible Navier-Stokes equations in $\mathbb R^d$, $d\geq4$ when the solution lies in the critical space $L_t^\infty L_x^d$. Explicit subcritical bounds on the solution are obtained in terms of the critical norm. A consequence is that $\|u(t)\|_{L_x^d(\mathbb R^d)}$ grows at a minimum rate of $(\log\log\log\log(T_*-t)^{-1})^c$ along a sequence of times approaching a hypothetical blowup at $T_*$. These results quantify a theorem of Dong and Du and extend the three-dimensional work of Tao.

math.AP↗

arXiv2022

Epochs of regularity for wild Hölder-continuous solutions of the Hypodissipative Navier-Stokes System

We consider the hypodissipative Navier-Stokes equations on $[0,T]\times\mathbb{T}^{d}$ and seek to construct non-unique, Hölder-continuous solutions with epochs of regularity (smooth almost everywhere outside a small singular set in time), using convex integration techniques. In particular, we give quantitative relationships between the power of the fractional Laplacian, the dimension of the singular set, and the regularity of the solution. In addition, we also generalize the usual vector calculus arguments to higher dimensions with Lagrangian coordinates.

math.AP↗

arXiv2021

Improved quantitative regularity for the Navier-Stokes equations in a scale of critical spaces

We prove a quantitative regularity theorem and blowup criterion for classical solutions of the three-dimensional Navier-Stokes equations satisfying certain critical conditions. The solutions we consider have $\|r^{1-\frac3q}u\|_{L_t^\infty L_x^q}<\infty$ where $r=\sqrt{x_1^2+x_2^2}$ and either $q\in(3,\infty)$, or $u$ is axisymmetric and $q\in(2,3]$. Using the strategy of Tao (2019), we obtain improved subcritical estimates for such solutions depending only on the double exponential of the critical norm. One consequence is a double logarithmic lower bound on the blowup rate. We make use of some tools such as a decomposition of the solution that allows us to use energy methods in these spaces, as well as a Carleman inequality for the heat equation suited for proving quantitative backward uniqueness in cylindrical regions.

math.AP↗

arXiv2016

On the Strategic Allocation of Social Gratification

Members of social networks are given opportunities to bestow positive recognition upon one another by means of constructs such as "likes" and "retweets." Although recipients no doubt experience utility from these actions, one might question why these constructs with no intrinsic value for the sender are exchanged at all. Here we formulate a metric for the prestige of a member of a social network based on his or her place within the network and the rate at which "likes" are exchanged within his or her social circle. Simulation reveals that the 1% most strategically-optimized networks exchange likes at an average rate 23.5% higher than that of their random counterparts. This suggests that purely strategic agents, even with no concern for altruism or the general welfare, experience utility from giving social gratification. Further, we show that prestige-maximization creates a selective pressure for structural features associated with social networks including clustering and the small-world property.

cs.SI↗

arXiv2014

A Reputation-Based Model for Decision-Making in Online Social Networks

The online exchange of social recognition including, for instance, the Facebook "like" appears to produce a scarce allocation without a clear utility function defined for anyone involved. Given the importance attached to such digital commodities by both users and advertisers, it is of interest to study the forces governing their economics. Here we propose a centrality measure akin to eigenvector centrality to describe an individual's perceived importance in an online social network. It is shown in silico that strategically maximizing this prestige metric results in finite nontrivial rates of "like" endowment. Furthermore, it is found that systems of reputation-seeking agents are supported most robustly by networks with the features of real human societies including preferential attachment and the small-world property. We conclude that the incentive system studied here can produce realistic behavior and may therefore provide a framework for a more general model of decision-making in online communities.

cs.SI↗

arXiv2014

Measuring Prestige in Online Social Networks

We study the locally-defined social capital metric of Palasek (2013) for determining individuals' prestige within an online social network. From it we derive an equivalent global measure by considering random walks over the network itself. This result inspires a novel expression quantifying the strategic desirability of a potential social connection. We show in silico that ideal social neighbors tend to satisfy a "big fish in a small pond" criterion and that the distribution of neighbor-desirability throughout a network is governed by anti-homophily.

cs.SI↗

arXiv2014

Non-Cooperativity in Bayesian Social Learning

We describe a Bayesian model for social learning of a random variable in which agents might observe each other over a directed network. The outcomes produced are compared to those from a model in which observations occur randomly over a complete graph. In both cases we observe a nontrivial level of observation which maximizes learning, though individuals have strong incentive to defect from the societal optimum. The implications of such competition over information commons are discussed.

cs.SI↗

arXiv2014

Arbitrage-free exchange rate ensembles over a general trade network

It is assumed that under suitable economic and information-theoretic conditions, market exchange rates are free from arbitrage. Commodity markets in which trades occur over a complete graph are shown to be trivial. We therefore examine the vector space of no-arbitrage exchange rate ensembles over an arbitrary connected undirected graph. Consideration is given for the minimal information for determination of an exchange rate ensemble. We conclude with a topical discussion of exchanges in which our analyses may be relevant, including the emergent but highly-regulated (and therefore not a complete graph) market for digital currencies.

econ.GN↗

arXiv2013

Primordial RNA Replication and Applications in PCR Technology

The emergence of self-replication and information transmission in life's origin remains unexplained despite extensive research on the topic. A hypothesis explaining the transition from a simple organic world to a complex RNA world is offered here based on physical factors in hydrothermal vent systems. An interdisciplinary approach is taken using techniques from thermodynamics, fluid dynamics, oceanography, statistical mechanics, and stochastic processes to examine nucleic acid dynamics and kinetics in a hydrothermal vent from first principles. Analyses are carried out using both analytic and computational methods and confirm the plausibility of a reaction involving the PCR-like assembly of ribonucleotides. The proposal is put into perspective with established theories on the origin of life and more generally the onset of order and information transmission in prebiotic systems. A biomimicry application of this hypothetical process to PCR technology is suggested and its viability is evaluated in a rigorous logical analysis. Optimal temperature curves begin to be established using Monte Carlo simulation, variational calculus, and Fourier analysis. The converse argument is also made but qualitatively, asserting that the success of such a modification to PCR would in turn reconfirm the biological theory.

q-bio.BM↗