Compressible Navier-Stokes equations with vacuum state in one dimension
Daoyuan Fang Ting Zhang
Communications on Pure & Applied Analysis 2004, 3(4): 675-694 doi: 10.3934/cpaa.2004.3.675
In this paper, we consider the one-dimensional compressible Navier-Stokes equations for isentropic flow connecting to vacuum state with a continuous density when viscosity coefficient depends on the density. Precisely, the viscosity coefficient $\mu$ is proportional to $\rho^\theta$ and $0<\theta<1/2$, where $\rho$ is the density. The global existence of weak solutions is proved.
keywords: vacuum global existence of weak solutions Navier-Stokes equations
Regularity criterion for 3D Navier-Stokes equations in Besov spaces
Daoyuan Fang Chenyin Qian
Communications on Pure & Applied Analysis 2014, 13(2): 585-603 doi: 10.3934/cpaa.2014.13.585
Several regularity criterions of Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. The results show that a weak solution $u$ becomes regular if the gradient of velocity component $\nabla_{h}{u}$ (or $ \nabla{u_3}$) satisfies the additional conditions in the class of $L^{q}(0,T; \dot{B}_{p,r}^{s}(\mathbb{R}^{3}))$, where $\nabla_{h}=(\partial_{x_{1}},\partial_{x_{2}})$ is the horizontal gradient operator. Besides, we also consider the anisotropic regularity criterion for the weak solution of Navier-Stokes equations in $\mathbb{R}^3$. Finally, we also get a further regularity criterion, when give the sufficient condition on $\partial_3u_3$.
keywords: 3D Navier-Stokes equations Leray-Hopf weak solution Regularity criterion.
On the well-posedness of inhomogeneous hyperdissipative Navier-Stokes equations
Daoyuan Fang Ruizhao Zi
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3517-3541 doi: 10.3934/dcds.2013.33.3517
This paper is devoted to the study of the inhomogeneous hyperdissipative Navier-Stokes equations on the whole space $\mathbb{R}^N,N\geq3$. Compared with the classical inhomogeneous Navier-Stokes, the dissipative term $- Δ u$ here is replaced by $D^2u$, where $D$ is a Fourier multiplier whose symbol is $m(\xi)=|\xi|^{\frac{N+2}{4}}$. For arbitrary small positive constants $ε$ and $δ$, global well-posedness is showed for the data $(\rho_0, u_0)$ such that $(ρ_{0} - 1, u_0)∈ H^{\frac{N}{2}+ε} × H^{δ}$ with $\inf_{x\in \mathbb{R}^N}\rho_0>0$. To our best knowledge, this is the first result on the inhomogeneous hyperdissipative Navier-Stokes equations, and it can also be viewed as the high-dimensional generalization of the 2D result for classical inhomogeneous Navier-Stokes equations given by Danchin [Local and global well-posedness results for flows of inhomogeneous viscous fluids. Adv. Differential Equations 9 (2004), 353--386.]
keywords: inhomogeneous existence uniqueness. Navier-Stokes equations hyperdissipative
$L^2$-concentration phenomenon for Zakharov system below energy norm II
Sijia Zhong Daoyuan Fang
Communications on Pure & Applied Analysis 2009, 8(3): 1117-1132 doi: 10.3934/cpaa.2009.8.1117
In this paper, we will prove a $L^2$-concentration result of Zakharov system in space dimension two, with initial data $(u_0,n_0,n_1)\in H^s\times L^2\times H^{-1}$ ($\frac {1 2}{1 3} < s < 1$), when blow up of the solution happens, by resonant decomposition and I-method, which is an improvement of [13].
keywords: blow up $L^2$-concentration global existence. Zakharov system in space dimension two
Local and global existence results for the Navier-Stokes equations in the rotational framework
Daoyuan Fang Bin Han Matthias Hieber
Communications on Pure & Applied Analysis 2015, 14(2): 609-622 doi: 10.3934/cpaa.2015.14.609
Consider the equations of Navier-Stokes in $R^3$ in the rotational setting, i.e. with Coriolis force. It is shown that this set of equations admits a unique, global mild solution provided only the horizontal components of the initial data are small with respect to the norm the Fourier-Besov space $\dot{FB}_{p,r}^{2-3/p}(R^3)$, where $p \in [2,\infty]$ and $r \in [1,\infty)$.
keywords: global solution Rotational flows Fourier-Besov space Chemin-Lerner space. Littlewood-Paley decomposition
Regularity of 3D axisymmetric Navier-Stokes equations
Hui Chen Daoyuan Fang Ting Zhang
Discrete & Continuous Dynamical Systems - A 2017, 37(4): 1923-1939 doi: 10.3934/dcds.2017081

In this paper, we study the three-dimensional axisymmetric Navier-Stokes system with nonzero swirl. By establishing a new key inequality for the pair $(\frac{ω^{r}}{r},\frac{ω^{θ}}{r})$, we get several Prodi-Serrin type regularity criteria based on the angular velocity, $u^θ$. Moreover, we obtain the global well-posedness result if the initial angular velocity $u_{0}^{θ}$ is appropriate small in the critical space $L^{3}(\mathbb{R}^{3})$. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say $ω^3$ or $u^3$.

keywords: Navier-Stokes equations regularity criteria global well-posedness axisymmetric swirl
Free boundary problem for compressible flows with density--dependent viscosity coefficients
Ping Chen Daoyuan Fang Ting Zhang
Communications on Pure & Applied Analysis 2011, 10(2): 459-478 doi: 10.3934/cpaa.2011.10.459
In this paper, we consider the free boundary problem of the spherically symmetric compressible isentropic Navier--Stokes equations in $R^n (n \geq 1)$, with density--dependent viscosity coefficients. Precisely, the viscosity coefficients $\mu$ and $\lambda$ are assumed to be proportional to $\rho^\theta$, $0 < \theta < 1$, where $\rho$ is the density. We obtain the global existence, uniqueness and continuous dependence on initial data of a weak solution, with a Lebesgue initial velocity $u_0\in L^{4 m}$, $4m>n$ and $\theta<\frac{4m-2}{4m+n}$. We weaken the regularity requirement of the initial velocity, and improve some known results of the one-dimensional system.
keywords: density-dependent viscosity coefficients. Compressible Navier-Stokes equations
Dispersive effects of the incompressible viscoelastic fluids
Daoyuan Fang Ting Zhang Ruizhao Zi
Discrete & Continuous Dynamical Systems - A 2018, 38(10): 5261-5295 doi: 10.3934/dcds.2018233

We consider the Cauchy problem of the $N$-dimensional incompressible viscoelastic fluids with $N≥2$. It is shown that, in the low frequency part, this system possesses some dispersive properties derived from the one parameter group $e^{± it\Lambda}$. Based on this dispersive effect, we construct global solutions with large initial velocity concentrating on the low frequency part. Such kind of solution has never been seen before in the literature even for the classical incompressible Navier-Stokes equations.

keywords: Viscoelastic fluids global well-posedness large data

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