Searcharxiv⌕ Search

arXiv subjects

Mimi Dai

Publications and source records attributed to Mimi Dai.

68 records · Page 4Linked to original sources
arXiv2016

Regularity problem for the nematic LCD system with Q-tensor in $\mathbb R^3$

We study the regularity problem of a nematic liquid crystal model with local configuration represented by Q-tensor in three dimensions. It was an open question whether the classical Prodi-Serrin condition implies regularity for this model. Applying a wavenumber splitting method, we show that a solution does not blow-up under certain extended Beale-Kato-Majda condition solely imposed on velocity. This regularity criterion automatically implies that the classical Prodi-Serrin or Beale-Kato-Majda condition prevents blow-up of solutions.

math.AP↗
arXiv2016

Regularity criterion and energy conservation for the supercritical Quasi-Geostrophic equation

This paper studies the regularity and energy conservation problems for the 2D supercritical quasi-geostrophic (SQG) equation. We apply an approach of splitting the dissipation wavenumber to obtain a new regularity condition which is weaker than all the Prodi-Serrin type regularity conditions. Moreover, we prove that any weak solution of the supercritical SQG in $L^2(0,T; B^{1/2}_{2,c(\mathbb N)})$ satisfies energy equality.

math.AP↗
arXiv2015

On asymptotic isotropy for a hydrodynamic model of liquid crystals

We study a PDE system describing the motion of liquid crystals by means of the $Q-$tensor description for the crystals coupled with the incompressible Navier-Stokes system. Using the method of Fourier splitting, we show that solutions of the system tend to the isotropic state at the rate $(1 + t)^{-3/2}$ as $t \to \infty$.

math.AP↗
arXiv2015

Ill-posedness of the Navier-Stokes and magneto-hydrodynamics systems

We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed due to the discontinuity of weak solutions in a wide range of spaces. Specifically, we construct initial data which has finite energy and is small in certain spaces, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time $t=0$ in such spaces. An analogous result is also obtained for the Navier-Stokes equation which extends the previous result of ill-posedness in $\dot B^{-1}_{\infty,\infty}$ by Cheskidov and Shvydkoy to spaces that are not necessarily critical. The region of the spaces where the norm inflation occurs almost touches $L^2$.

math.AP↗
arXiv2015

Norm inflation for generalized magneto-hydrodynamic system

We consider the three-dimensional incompressible magneto-hydrodynamic system with fractional powers of the Laplacian. We discover a wide range of spaces where the norm inflation occurs and hence small initial data results are out of reach. The norm inflation occurs not only in scaling invariant (critical) spaces, but also in supercritical, and, surprisingly, subcritical ones.

math.AP↗
arXiv2015

Analysis of a diffuse interface model of multispecies tumor growth

We consider a diffuse interface model for tumor growth recently proposed in [Y. Chen, S.M. Wise, V.B. Shenoy, J.S. Lowengrub, A stable scheme for a nonlinear, multiphase tumor growth model with an elastic membrane, Int. J. Numer. Methods Biomed. Eng., 30 (2014), 726-754]. In this new approach sharp interfaces are replaced by narrow transition layers arising due to adhesive forces among the cell species. Hence, a continuum thermodynamically consistent model is introduced. The resulting PDE system couples four different types of equations: a Cahn-Hilliard type equation for the tumor cells (which include proliferating and dead cells), a Darcy law for the tissue velocity field, whose divergence may be different from 0 and depend on the other variables, a transport equation for the proliferating (viable) tumor cells, and a quasi-static reaction diffusion equation for the nutrient concentration. We establish existence of weak solutions for the PDE system coupled with suitable initial and boundary conditions. In particular, the proliferation function at the boundary is supposed to be nonnegative on the set where the velocity ${\bf u}$ satisfies ${\bf u}\cdotν>0$, where $ν$ is the outer normal to the boundary of the domain. We also study a singular limit as the diffuse interface coefficient tends to zero.

math.AP↗
arXiv2014

Ill-posedness of the incompressible magneto-hydrodynamics system in $\dot{B}_{\infty,\infty}^{-1}$

We demonstrate that the three dimensional incompressible magneto-hydrodynamics (MHD) system is ill-posed in the largest scaling invariant space $\dot{B}_{\infty,\infty}^{-1}$. The construction method of initial data used in this paper is different from the one in a previous work [DQS] of the author. Specifically, we construct initial data which has finite energy and is small enough in $H^s\cap \dot{B}_{\infty,\infty}^{-1}$, with $s<1/2$, such that any Leray-Hopf type of weak solution to the MHD system starting from this initial data is discontinuous at time $t=0$ in the metric of $\dot{B}_{\infty,\infty}^{-1}$.

math.AP↗
arXiv2013

Existence of regular solutions to the full Liquid Crystal System

We study the general Ericksen-Leslie system with non-constant density, which describes the flow of nematic liquid crystal. In particular the model investigated here is associated with Parodi's relation. We prove that: in two dimension, the solutions are globally regular with general data; in three dimension, the solutions are globally regular with small initial data, or for short time with large data. Moreover, a weak-strong type of uniqueness result is obtained.

math.AP↗
arXiv2013

Norm Inflation for Generalized Navier-Stokes Equations

We consider the incompressible Navier-Stokes equation with a fractional power $α\in[1,\infty)$ of the Laplacian in the three dimensional case. We prove the existence of a smooth solution with arbitrarily small in $\dot{B}_{\infty,p}^{-α}$ ($2 0$ in arbitrarily small time. This extends the result of Bourgain and Pavlović for the classical Navier-Stokes equation which utilizes the fact that the energy transfer to low modes increases norms with negative smoothness indexes. It is remarkable that the space $\dot{B}_{\infty,\infty}^{-α}$ is supercritical for $α>1$. Moreover, the norm inflation occurs even in the case $α\geq 5/4$ where the global regularity is known.

math.AP↗
arXiv2012

Asymptotic Behavior of Solutions to the Liquid Crystal System in $H^m(\mathbb{R}^3)$

In this paper we study the large time behavior of regular solutions to a nematic liquid crystal system in Sobolev spaces $H^m(\R^3)$ for $m\geq 0$.We obtain optimal decay rates in $H^m(R^3)$ spaces, in the sense that the rates coincide with the rates of the underlying linear counterpart. The fluid under consideration has constant density and small initial data.

math.AP↗
arXiv2011

Norm inflation for incompressible magneto-hydrodynamic system ]{Norm inflation for incompressible magneto-hydrodynamic system in $\dot{B}_{\infty}^{-1,\infty}$

We demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse. Efforts are made to present a very expository development of the inginious construction of Bourgain and Pavlović for Navier-Stokes equation.

math.AP↗
arXiv2011

Norm inflation for incompressible magneto-hydrodynamic system in $\dot{B}_{\infty}^{-1,\infty}$

We demonstrate that the solutions to the Cauchy problem for the three dimensional incompressible magneto-hydrodynamics (MHD) system can develop diferent types of norm inflations in $\dot{B}_{\infty}^{-1, \infty}$. Particularly the magnetic field can develop norm inflation in short time even when the velocity remains small and vice verse. Efforts are made to present a very expository development of the inginious construction of Bourgain and Pavlović for Navier-Stokes equations.

math.AP↗