Search

In this paper, we consider an aggregation equation with fractional diffusion and large shear flow, which arise from modelling chemotaxis in bacteria. Without the advection, the solution of aggregation equation may blow up in finite time. First, we study the enhanced dissipation of shear flow by resolvent estimate method, where the fractional Laplacian $(-\Delta)^{\alpha/2}$ is considered and $\alpha\in (0,2)$. Next, we show that the enhanced dissipation of shear flow can suppress blow-up of solution to aggregation equation with fractional diffusion and establish global classical solution in the case of $\alpha\geq 3/2$. Here we develop some new technical to overcome the difficult of low regularity for fractional Laplacian., Comment: 33 pages. arXiv admin note: text overlap with arXiv:2308.15287

In this paper, we investigate the transition threshold problem concerning the 2-D Navier-Stokes equations in the context of Couette flow $(y,0)$ at high Reynolds number $Re$ in whole space. By utilizing Green's function estimates for the linearized equations around Couette flow, we initially establish refined dissipation estimates for the linearized Navier-Stokes equations with a precise decay rate $(1+t)^{-1}.$ As an application, we prove that if the initial perturbation of vorticity satisfies$$\|\omega_{0}\|_{H^{1}\cap L^1}\leq c_0\nu^{\frac{3}{4}}$$ for some small constant $c_0$ independent of the viscosity $\nu$, then we can reach the conclusion that the solution remains within $O\left( \nu ^{\frac{3}{4}}\right) $ of the Couette flow., Comment: 20pages

In this paper, we study the 3-D parabolic-parabolic and parabolic-elliptic Keller-Segel models with Couette flow in $\mathbb{R}^3$. We prove that the blow-up phenomenon of solution can be suppressed by enhanced dissipation of large Couette flows. Here we develop Green's function method to describe the enhanced dissipation via a more precise space-time structure and obtain the global existence together with pointwise estimates of the solutions. The result of this paper shows that the enhanced dissipation exists for all frequencies in the case of whole space and it is reason that we obtain global existence for 3-D Keller-Segel models here. It is totally different from the case with the periodic spatial variable $x$ in [2,10]. This paper provides a new methodology to capture dissipation enhancement and also a surprising result which shows a totally new mechanism., Comment: 22 page

In this paper, we consider the Cauchy problem for the three dimensional parabolic-elliptic Keller-Segel equation with a large non-shear incompressible flow. Without advection, there exist solution with arbitrarily mass which blow up in finite time. Firstly, we introduce a three dimensional non-shear incompressible flow and study the enhanced dissipation of such flows by resolvent estimate method. Next, we show that the enhanced dissipation of such flow can suppress blow-up of solution to three dimensional parabolic-elliptic Keller-Segel equation and establish global classical solution with large initial data., Comment: 25 page

Space-time behaviors for 3D compressible bipolar Navier-Stokes-Poisson system (BNSP) with unequal viscosities are given. The space-time estimate of electric field $\nabla\phi$ is the most important thing when deducing generalized Huygens' principle for BNSP since this estimate only can be obtained by $\nabla\phi=\frac{\nabla}{\Delta}(\rho-n)$ from the Poisson equation. Thus, it requires to prove that the space-time estimate of $\rho-n$ only contains diffusion wave. The appearance of these unequal coefficients results that one cannot follow ideas for the special case, where the original system was rewritten as a compressible NS system and a compressible (unipolar) NSP system after a linear combination of unknowns. This linear combination brings special structure for nonlinear terms, and this structure was also used to get desired space-time estimate for $\rho-n$. Moreover, Green's function of the subsystem NSP does not contain Huygens wave is equally important in [36]. However, for the general case, the benefits from this linear combination will not exist any longer. First, we have to directly consider an $8\times8$ Green's matrix of the original system. Second, all of entries in Green's function in low frequency actually contain wave operators. This generally produces the Huygens' wave for each entry in Green's function, as a result, one cannot achieve that the space-time estimate of $\rho-n$ only contains the diffusion wave as usual. We overcome this difficulty by taking more detailed spectral analysis and developing new estimates arising from subtle cancellations in Green's function. Third, due to loss of the special structure of nonlinear terms from the linear combination, we shall develop new nonlinear convolution estimates such that we can ultimately obtain the expected space-time estimate for $\nabla\phi$ and further verify the generalized Huygens' principle., Comment: It has been accepted for publication in SCIENCE CHINA Mathematics

We study the mixing effect for a generalized Keller-Segel system with degenerate dissipation and advection by a weakly mixing. Here the attractive operator has weak singularity, namely, the negative derivative appears in the nonlinear term by singular integral. Without advection, the solution of equation blows up in finite time. We show that the global well-posedness of solution with large advection. Since dissipation term degenerate into the damping, the enhanced dissipation effect of mixing no longer occurs, we prove that the mixing effect can weak the influence of nonlinear term. In this case, the mixing effect is similar with inviscid damping of shear flow. Combining to the mixing effect and damping effect of degenerate dissipation, the global $L^\infty$ estimate of solution is established., Comment: 19 page. arXiv admin note: text overlap with arXiv:2103.04484

Strain engineering is a powerful and widely used strategy for boosting the performance of electronic and optoelectronic devices. Here, we demonstrate an approach to tune the photoelectric properties of Platinum sulfide (PtS$_2$) by using a ferroelectric substrate PMN-PT as the strain generator. It is found that both the drain current and responsivity of the PtS$_2$ photodetector is directly coupled to the electrostriction of PMN-PT, showing a high strain-tuned ratio $10^{3}$, high responsivity up to $6.3\times 10^{3}$ A/W and detectivity of $9.3\times 10^{12}$ Jones. Additionally, a high photogain $\approx 5\times 10^{5}$ is obtained at a gate voltage Vg = 15 V. Our results provide an effective method for manipulating electrical properties and optimizing performance of two dimensional layered (2D) materials based optoelectronic devices., Comment: 15 pages, 4 figures

We introduce the planar helical flows on three dimensional torus and study the dissipation enhancement of such flows. We then use such flows as transport flows to solve the three dimensional advective Kuramoto-Sivashinsky and Keller-Segel equations. The global well-posedness of the Kuramoto-Sivashinsky equation is achieved when the linearized operator does not have growing mode in the direction orthogonal to the flow. The global classical solution of the three dimensional Keller-Segel is ensured for any size of the torus with arbitrarily large initial data.

In this paper, we consider the Cauchy problem for a generalized parabolic-elliptic Keller-Segel equation with fractional dissipation and the additional mixing effect of advection by an incompressible flow. Under suitable mixing condition on the advection, we study well-posedness of solution with large initial data. We establish the global $L^\infty$ estimate of the solution through nonlinear maximum principle, and obtain the global classical solution.

In this paper, we study the large time behavior of solutions to the Cauchy problem for the anisotropic conservation laws in two dimensional space. Without any smallness assumption on the initial data, the decay rates of solutions in $L^2$ space and homogeneous Sobolev space $\dot{H}^\gamma$ are obtained by using the method of time-frequency decomposition and the classical energy method.

The Cauchy problem of the bipolar Navier-Stokes-Poisson system (1.1) in dimension three is considered. We obtain the pointwise estimates of the time-asymptotic shape of the solution, which exhibit generalized Huygens' principle as the Navier-Stokes system. This phenomenon is the the most important difference from the unipolar Navier-Stokes-Poisson system. Due to non-conservative structure of the system (1.1) and interplay of two carriers which counteracts the influence of electric field (a nonlocal term), some new observations are essential for the proof. We make full use of the conservative structure of the system for the total density and total momentum, and the mechanism of the linearized unipolar Navier-Stokes-Poisson system together with the special form of the nonlinear terms in the system for the difference of densities and the difference of momentums. Lastly, as a byproduct, we extend the usual $L^2(\mathbb{R}^3)$-decay rate to $L^p(\mathbb{R}^3)$-decay rate with $p>1$ and also improve former decay rates in part., Comment: 38 pages

Many exotic physics anticipated in topological insulators require a gap to be opened for their topologica surface states by breaking time reversal symmetry. The gap opening has been achieved by doping magnetic impurities, which however inevitably create extra carriers and disorder that undermine the electronic transport. In contrast, the proximity to a ferromagnetic/ferrimagnetic insulator may improve the device quality, thus promises a better way to open the gap while minimizing the side-effects. Here, we grow thin single-crystal Sb1.9Bi0.1Te3 micro flakes on insulating ferrimagnet BaFe12O19 by using the van der Waals epitaxy technique. The micro flakes show a negative magnetoresistance in weak perpendicular fields below 50 K, which can be quenched by increasing temperature. The signature implies the weak localization effect as its origin, which is absent in intrinsic topological insulators, unless a surface state gap is opened. The surface state gap is estimated to be 10 meV by using the theory of the gap-induced weak localization effect. These results indicate that the magnetic proximity effect may open the gap for the topological surface attached to BaM insulating ferrimagnet. This heterostructure may pave the way for the realization of new physical effects as well as the potential applications of spintronics devices.