One-dimensional model equations for hyperbolic fluid flow

Do, Tam; Hoang, Vu; Radosz, Maria; Xu, Xiaoqian

doi:10.1016/j.na.2016.03.002

Cited by 6 publications

(4 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The study of such nonlocal and nonlinear one-dimensional equations is a wide research area with a large literature. For other similar equations and related results we refer to [4,5,8,12,13,22,14,15,23,25,28,24].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

See 1 more Smart Citation

On a nonlocal differential equation describing roots of polynomials under differentiation

Granero-Belinchón¹

2018

Preprint

View full text Add to dashboard Cite

In this work we study the nonlocal transport equation derived recently by Steinerberger when studying how the distribution of roots of a polynomial behaves under iterated differentation of the function. In particular, we study the well-posedness of the equation, establish some qualitative properties of the solution and give conditions ensuring the global existence of both weak and strong solutions. Finally, we present a link between the equation obtained by Steinerberger and a one-dimensional model of the surface quasi-geostrophic equation used by Chae, Córdoba, Córdoba & Fontelos.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…A similar approach has been used in the study of free boundary problems for incompressible fluids (see [7,20,21] and the references therein). First, we observe that (1) can be equivalently written as (14) B t u `uΛu ´HuB x u u 2 `pHuq 2 " 0 px, tq on S 1 ˆr0, T s.…”

Section: Link Between (1) and (4)mentioning

confidence: 99%

On a nonlocal differential equation describing roots of polynomials under differentiation

Granero-Belinchón¹

2018

Preprint

View full text Add to dashboard Cite

show abstract

“…One of the earliest of these models was proposed by Constantin, Lax and Majda [2], and later inspired other models [6], [3], see also [5], [8] for recent related work. It is fascinating that many natural questions about solutions to these models remain unanswered.…”

Section: Introductionmentioning

confidence: 99%

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

Kiselev

2016

J Nonlinear Sci

Self Cite

View full text Add to dashboard Cite

Abstract. The question of the global regularity vs finite time blow up in solutions of the 3D incompressible Euler equation is a major open problem of modern applied analysis. In this paper, we study a class of one-dimensional models of the axisymmetric hyperbolic boundary blow up scenario for the 3D Euler equation proposed by Hou and Luo [12] based on extensive numerical simulations. These models generalize the 1D Hou-Luo model suggested in [12], for which finite time blow up has been established in [1]. The main new aspects of this work are twofold. First, we establish finite time blow up for a model that is a closer approximation of the three dimensional case than the original Hou-Luo model, in the sense that it contains relevant lower order terms in the Biot-Savart law that have been discarded in [12], [1]. Secondly, we show that the blow up mechanism is quite robust, by considering a broader family of models with the same main term as in the Hou-Luo model. Such blow up stability result may be useful in further work on understanding the 3D hyperbolic blow up scenario.

show abstract

“…One interesting aspect is that the equation has similarities to one-dimensional transport equations that are studied in fluid dynamics, see e.g. [9,10,11,12,14,18,20,38,39,51]. Granero-Belinchon [29] studied an analogue of the equation on the one-dimensional torus.…”

mentioning

confidence: 99%

A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation

O’Rourke¹,

Steinerberger²

2019

Preprint

View full text Add to dashboard Cite

Let pn : C → C be a random complex polynomial whose roots are sampled i.i.d. from a radial distribution u(r)rdr in the complex plane. A natural question is how the distribution of roots evolves under repeated (say n/2−times) differentiation of the polynomial. We derive a mean-field expansion for the evolution of ψ(s) = u(s)sWe discuss some numerical examples suggesting that this particular solution may be stable. We prove that the solution is linearly stable. The linear stability analysis reduces to the classical Hardy integral inequality. Many open problems are discussed.

show abstract

One-dimensional model equations for hyperbolic fluid flow

Abstract: Abstract. In this paper we study the singularity formation for two nonlocal 1D active scalar equations, focusing on the hyperbolic flow scenario. Those 1D equations can be regarded as simplified models of some 2D fluid equations.

Cited by 6 publications

References 13 publications

On a nonlocal differential equation describing roots of polynomials under differentiation

On a nonlocal differential equation describing roots of polynomials under differentiation

Stability of Blowup for a 1D Model of Axisymmetric 3D Euler Equation

A Nonlocal Transport Equation Modeling Complex Roots of Polynomials under Differentiation

Contact Info

Product

Resources

About