539 research outputs found
The Optimal Temporal Decay Estimates for the Fractional Power Dissipative Equation in Negative Besov Spaces
In this paper, we first generalize a new energy approach, developed by Y. Guo
and Y. Wang \cite{GW12}, in the framework of homogeneous Besov spaces for
proving the optimal temporal decay rates of solutions to the fractional power
dissipative equation, then we apply this approach to the supercritical and
critical quasi-geostrophic equation and the critical Keller-Segel system. We
show that the negative Besov norm of solutions is preserved along time
evolution, and obtain the optimal temporal decay rates of the spatial
derivatives of solutions by the Fourier splitting approach and the
interpolation techniques.Comment: 23 pages. arXiv admin note: text overlap with arXiv:1508.0202
Well-posedness and Gevrey Analyticity of the Generalized Keller-Segel System in Critical Besov Spaces
In this paper, we study the Cauchy problem for the generalized Keller-Segel
system with the cell diffusion being ruled by fractional diffusion:
\begin{equation*} \begin{cases}
\partial_{t}u+\Lambda^{\alpha}u-\nabla\cdot(u\nabla \psi)=0\quad &\mbox{in}\
\
\mathbb{R}^n\times(0,\infty),
-\Delta \psi=u\quad &\mbox{in}\ \
\mathbb{R}^n\times(0,\infty),
u(x,0)=u_0(x), \ \ &\mbox{in}\ \ \mathbb{R}^n. \end{cases} \end{equation*} In
the case that , we prove local well-posedness for any initial
data and global well-posedness for small initial data in critical Besov spaces
with ,
, and analyticity of solutions for initial data with , . Moreover, the global existence and analyticity of solutions with
small initial data in critical Besov spaces
is also established. In the
limit case that , we prove global well-posedness for small initial
data in critical Besov spaces
with and , and show
analyticity of solutions for small initial data in
with and
, respectively.Comment: 24 page
A Beale-Kato-Majda Blow-up Criterion for a Hydrodynamic System Modeling Vesicle and Fluid Interactions
In this paper, we establish an analog of the Beale-Kato-Majda type criterion
for singularities of smooth solutions of a hydrodynamic system modeling vesicle
and fluid interactions. The result shows that the maximum norm of the vorticity
alone controls the breakdown of smooth solutions.Comment: 16 page
Logarithmical Blow-up Criteria for the Nematic Liquid Crystal Flows
We investigate the blow-up criterion for the local in time classical solution
of the nematic liquid crystal flows in dimension two and three. More precisely,
is the maximal time interval if and only if (i) for ,
{align*}
\int_{0}^{T_{*}}\frac{\|\omega\|_{\dot{B}^{0}_{\infty,\infty}}+\|\nabla
d\|_{\dot{B}^{0}_{\infty,\infty}}^{2}}{\sqrt{1+\text{ln}(e+\|\omega\|_{\dot{B}^{0}_{\infty,\infty}}
+\|\nabla d\|_{\dot{B}^{0}_{\infty,\infty}})}}\text{d}t=\infty, {align*} or
{align*} \int_{0}^{T_{*}}\frac{\|\nabla
u\|_{\dot{B}^{-1}_{\infty,\infty}}^{2}+\|\nabla
d\|_{\dot{B}^{0}_{\infty,\infty}}^{2}}{\sqrt{1+\text{ln}(e+\|\nabla
u\|_{\dot{B}^{-1}_{\infty,\infty}} +\|\nabla
d\|_{\dot{B}^{0}_{\infty,\infty}})}}\text{d}t=\infty; {align*} and (ii) for
, {align*} \int_{0}^{T_{*}}\frac{\|\nabla
d\|_{\dot{B}^{0}_{\infty,\infty}}^{2}}{\sqrt{1+\text{ln}(e +\|\nabla
d\|_{\dot{B}^{0}_{\infty,\infty}})}}\text{d}t=\infty. {align*}Comment: 17 page
Logarithmically Improved Blow-up Criteria for a Phase Field Navier-Stokes Vesicle-Fluid Interaction Model
In this paper, we study a hydrodynamical system modeling the deformation of
vesicle membrane under external incompressible viscous flow fields. The system
is in the Eulerian formulation and is governed by the coupling of the
incompressible Navier-Stokes equations with a phase field equation. In the
three dimensional case, we establish two logarithmically improved blow-up
criteria for local smooth solutions of this system in terms of the vorticity
field only in the homogeneous Besov spaces.Comment: 22 page
Blow-up Criteria for the Three Dimensional Nonlinear Dissipative System Modeling Electro-hydrodynamics
In this paper, we investigate some sufficient conditions for the breakdown of
local smooth solutions to the three dimensional nonlinear nonlocal dissipative
system modeling electro-hydrodynamics. This model is a strongly coupled system
by the well-known incompressible Navier-Stokes equations and the classical
Poisson-Nernst-Planck equations. We show that the maximum of the vorticity
field alone controls the breakdown of smooth solutions, which reveals that the
velocity field plays a more dominant role than the density functions of charged
particles in the blow-up theory of the system. Moreover, some Prodi-Serrin type
blow-up criteria are also established.Comment: 21 page
A regularity criterion for the solution of the nematic liquid crystal flows in terms of -norm
In this paper, we investigate regularity criterion for the solution of the
nematic liquid crystal flows in dimension three and two. We prove the solution
is smooth up to time provided that there exists a positive constant
such that (i) for n=3, |(u,\nabla
d)|_{L^{\infty}(0,T;\dot{B}^{-1}_{\infty,\infty})}\leq \varepsilon_{0}, and
(ii) for , |\nabla d|_{L^{\infty}(0,T;\dot{B}^{-1}_{\infty,\infty})}\leq
\varepsilon_{0}.Comment: 15. arXiv admin note: text overlap with arXiv:1209.562
Existence of Solutions for the Debye-H\"{u}ckel System with Low Regularity Initial Data
In this paper we study existence of solutions for the Cauchy problem of the
Debye-H\"{u}ckel system with low regularity initial data. By using the
Chemin-Lerner time-space estimate for the heat equation, we prove that there
exists a unique local solution if the initial data belongs to the Besov space
for ,
and , and furthermore, if the initial
data is sufficiently small then the solution is global. This result improves
the regularity index of the initial data space in previous results on this
model.Comment: 9 page
Global Existence and Stability for a Hydrodynamic System in the Nematic Liquid Crystal Flows
In this paper we consider a coupled hydrodynamical system which involves the
Navier-Stokes equations for the velocity field and kinematic transport
equations for the molecular orientation field. By applying the Chemin-Lerner's
time-space estimates for the heat equation and the Fourier localization
technique, we prove that when initial data belongs to the critical Besov spaces
with negative-order, there exists a unique local solution, and this solution is
global when initial data is small enough. As a corollary, we obtain existence
of global self-similar solutions. In order to figure out the relation between
the solution obtained here and weak solution of standard sense, we establish a
stability result, which yields in a direct way that all global weak solutions
associated with the same initial data must coincide with the solution obtained
here, namely, weak-strong uniqueness holds.Comment: 18 page
Sparsity Aware Normalized Least Mean p-power Algorithms with Correntropy Induced Metric Penalty
For identifying the non-Gaussian impulsive noise systems, normalized LMP
(NLMP) has been proposed to combat impulsive-inducing instability. However, the
standard algorithm is without considering the inherent sparse structure
distribution of unknown system. To exploit sparsity as well as to mitigate the
impulsive noise, this paper proposes a sparse NLMP algorithm, i.e., Correntropy
Induced Metric (CIM) constraint based NLMP (CIMNLMP). Based on the first
proposed algorithm, moreover, we propose an improved CIM constraint variable
regularized NLMP(CIMVRNLMP) algorithm by utilizing variable regularized
parameter(VRP) selection method which can further adjust convergence speed and
steady-state error. Numerical simulations are given to confirm the proposed
algorithms.Comment: 5 pages, 4 figures, submitted for DSP201
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