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Mimi Dai

Publications and source records attributed to Mimi Dai.

At least 55 records · Page 3Linked to original sources
arXiv2020

Phenomenologies of intermittent Hall MHD turbulence

We introduce the concept of intermittency dimension for the magnetohydrodynamics (MHD) to quantify the intermittency effect. With dependence on the intermittency dimension, we derive phenomenological laws for intermittent MHD turbulence with and without the Hall effect. In particular, scaling laws of dissipation wavenumber, energy spectra and structure functions are predicted. Moreover, we are able to provide estimates for energy spectra and structure functions which are consistent with the predicted scalings.

math.AP↗
arXiv2020

Dyadic models with intermittency dependence for the Hall MHD

We derive dyadic models for the magnetohydrodynamics with Hall effect by including the intermittency dimension as a parameter. For such dyadic models, existence of global weak solutions is established. In addition, local strong solution is obtained; while global strong solution is obtained in the case of high intermittency dimension. Moreover, we show that positive solution with large initial data develops blow-up in finite time provided the intermittency dimension is lower than a threshold.

math.AP↗
arXiv2019

On well-posedness of generalized Hall-magneto-hydrodynamics

We obtain local well-posedness result for the generalized Hall-magneto-hydrodynamics system in Besov spaces ${\dot B^{-(2α_1-γ)}_{\infty, \infty}} \times {\dot B^{-(2α_2-β)}_{\infty,\infty}(\mathbb R^3)}$ with suitable indexes $α_1, α_2, β$ and $γ.$ As a corollary, the generalized electron magneto-hydrodynamics system is globally well-posed in ${\dot B^{-(2α_2-2)}}_{\infty, \infty}(\mathbb R^3)$ for small initial data.

math.AP↗
arXiv2019

On the determining wavenumber for the nonautonomous subcritical SQG equation

A time-dependent determining wavenumber was introduced in [5] to estimate the number of determining modes for the surface quasi-geostrophic (SQG) equation. In this paper we continue this investigation focusing on the subcritical case and study trajectories inside an absorbing set bounded in $L^\infty$. Utilizing this bound we find a time-independent determining wavenumber that improves the estimate obtained in [5]. This classical approach is more direct, but it is contingent on the existence of the $L^\infty$ absorbing set.

math.AP↗
arXiv2019

Kolmogorov's dissipation number and the number of degrees of freedom for the 3D Navier-Stokes equations

Kolmogorov's theory of turbulence predicts that only wavenumbers below some critical value, called Kolmogorov's dissipation number, are essential to describe the evolution of a three-dimensional fluid flow. A determining wavenumber, first introduced by Foias and Prodi for the 2D Navier-Stokes equations, is a mathematical analog of Kolmogorov's number. The purpose of this paper is to prove the existence of a time-dependent determining wavenumber for the 3D Navier-Stokes equations whose time average is bounded by Kolmogorov's dissipation wavenumber for all solutions on the global attractor whose intermittency is not extreme.

math.AP↗
arXiv2019

Determining modes for the surface Quasi-Geostrophic equation

We introduce a determining wavenumber for the surface quasi-geostrophic (SQG) equation defined for each individual trajectory and then study its dependence on the force. While in the subcritical and critical cases this wavenumber has a uniform upper bound, it may blow up when the equation is supercritical. A bound on the determining wavenumber provides determining modes, which in some sense measure the number of degrees of freedom of the flow, or resolution needed to describe a solution to the SQG equation.

math.AP↗
arXiv2019

Long time behavior of solutions to the 3D Hall-magneto-hydrodynamics system with one diffusion

This paper studies the asymptotic behavior of smooth solutions to the generalized Hall-magneto-hydrodynamics system (1.1) with one single diffusion on the whole space $\mathbb R^3$. We establish that, in the inviscid resistive case, the energy $\|b(t)\|_2^2$ vanishes and $\|u(t)\|_2^2$ converges to a constant as time tends to infinity provided the velocity is bounded in $W^{1-α,\frac3α}(\mathbb R^3)$; in the viscous non-resistive case, the energy $\|u(t)\|_2^2$ vanishes and $\|b(t)\|_2^2$ converges to a constant provided the magnetic field is bounded in $W^{1-β,\infty}(\mathbb R^3)$. In summary, one single diffusion, being as weak as $(-Δ)^αb$ or $(-Δ)^βu$ with small enough $α, β$, is sufficient to prevent asymptotic energy oscillations for certain smooth solutions to the system.

math.AP↗
arXiv2018

Local well-posedness for the Hall-MHD system in optimal Sobolev Spaces

We show that the viscous resistive magneto-hydrodynamics system with Hall effect is locally well-posed in $H^s(\mathbb R^n)\times H^{s+1-\varepsilon}(\mathbb R^n)$ with $s>\frac{n}2-1$ and any small enough $\varepsilon>0$ such that $s+1-\varepsilon>\frac{n}2$. This space is to date the largest local well-posedness space in the class of Sobolev spaces for the system. It is also optimal according to the predominant scalings of the two equations in the system.

math.AP↗
arXiv2018

Low modes regularity criterion for a chemotaxis-Navier-Stokes system

In this paper we study the regularity problem of a three dimensional chemotaxis-Navier-Stokes system on a periodic domain. A new regularity criterion in terms of only low modes of the oxygen concentration and the fluid velocity is obtained via a wavenumber splitting approach. The result improves many existing criteria in the literature.

math.AP↗
arXiv2018

Propagation of regularity for the MHD system in optimal Sobolev space

We study the problem of propagation of regularity of solutions to the incompressible viscous non-resistive magneto-hydrodynamics system. According to scaling, the Sobolev space $H^{\frac n2-1}(\mathbb R^n)\times H^{\frac n2}(\mathbb R^n)$ is critical for the system. We show that if a weak solution $(u(t),b(t))$ is in $H^{s}(\mathbb R^n)\times H^{s+1}(\mathbb R^n)$ with $s>\frac n2-1$ at a certain time $t_0$, then it will stay in the space for a short time, provided the initial velocity $u(0)\in H^s(\mathbb R^n)$. In the case that the uniqueness of weak solution in $H^{s}(\mathbb R^n)\times H^{s+1}(\mathbb R^n)$ is known, the assumption of $u(0)\in H^s(\mathbb R^n)$ is not necessary.

math.AP↗
arXiv2017

The existence of a global attractor for the forced critical surface quasi-geostrophic equation in $L^2$

We prove that the critical surface quasi-geostrophic equation driven by a force $f$ possesses a compact global attractor in $L^2(\mathbb T^2)$ provided $f\in L^p(\mathbb T^2)$ for some $p>2$. First, the De Giorgi method is used to obtain uniform $L^\infty$ estimates on viscosity solutions. Even though this does not provide a compact absorbing set, the existence of a compact global attractor follows from the continuity of solutions, which is obtained by estimating the energy flux using the Littlewood-Paley decomposition.

math.AP↗
arXiv2017

Regularity criterion for the 3D Hall-magneto-hydrodynamics

This paper studies the regularity problem for the 3D incompress- ible resistive viscous Hall-magneto-hydrodynamic (Hall-MHD) system. The Kolmogorov 41 phenomenological theory of turbulence predicts that there exists a critical wavenumber above which the high frequency part is dominated by the dissipation term in the fluid equation. Inspired by this idea, we apply an approach of splitting the wavenumber combined with an estimate of the energy flux to obtain a new regularity criterion. The regularity condition presented here is weaker than conditions in the existing criteria (Prodi-Serrin type criteria) for the 3D Hall-MHD system.

math.AP↗
arXiv2017

Stability of Solutions to the Quasi-Geostrophic Equations in $\mathbb R^2$

We consider the stationary Quasi-Geostrophic equation in the whole space $\mathbb R^2$ driven by a force $f$. Under certain smallness assumptions of $f$, we establish the existence of solutions with finite $L^2$ norm. This solution is unique among all solutions with finite energy. The unique solution $Θ$ is also shown to be stable in the sense: any solution of the evolutionary Quasi-Geostrophic equation driven by $f$ and starting with finite energy, will return to $Θ$.

math.AP↗
arXiv2016

Norm-inflation results for the BBM equation

Considered here is the periodic initial-value probem for the regularized long-wave (BBM) equation \[u_t+u_x+uu_x-u_{xxt}=0.\] Adding to previous work in the literature, it is shown here that for any $s < 0$, there is smooth initial data that is small in the $L_2$-based Sobolev spaces $H^s$, but the solution emanating from it becomes arbitrarily large in arbitrarily small time. This so called {\it norm inflation} result has as a consequence the previously determined conclusion that this problem is ill-posed in these negative-norm spaces.

math.AP↗