Searcharxiv⌕ Search

arXiv subjects

Daoyuan Fang

Publications and source records attributed to Daoyuan Fang.

At least 19 recordsLinked to original sources
arXiv2021

Long time existence for semilinear wave equations with the inverse-square potential

In this paper, we study the semilinear wave equations with the inverse-square potential. By transferring the original equation to a "fractional dimensional" wave equation and analyzing the properties of its fundamental solution, we establish a long-time existence result, for sufficiently small, spherically symmetric initial data. Together with the previously known blow-up result, we determine the critical exponent which divides the global existence and finite time blow-up. Moreover, the sharp lower bounds of the lifespan are obtained, except for certain borderline case. In addition, our technology allows us to handle an extreme case for the potential, which has hardly been discussed in literature.

math.AP↗
arXiv2020

Lifespan of solutions to the Strauss type wave system on asymptotically flat space-times

By assuming certain local energy estimates on $(1+3)$-dimensional asymptotically flat space-time, we study the existence portion of the \emph{Strauss} type wave system. Firstly we give a kind of space-time estimates which are related to the local energy norm that appeared in \cite{MR2944027}. These estimates can be used to prove a series of weighted \emph{Strichartz} and \emph{KSS} type estimates, for wave equations on asymptotically flat space-time. Then we apply the space-time estimates to obtain the lower bound of the lifespan when the nonlinear exponents $p$ and $q\ge 2$. In particular, our bound for the subcritical case is sharp in general and we extend the known region of $(p,q)$ to admit global solutions. In addition, the initial data are not required to be compactly supported, when $p, q>2$.

math.AP↗
arXiv2020

Critical regularity criteria for Navier-Stokes equations in terms of one directional derivative of the velocity

In this paper, we consider the 3D Navier-Stokes equations in the whole space. We investigate some new inequalities and \textit{a priori} estimates to provide the critical regularity criteria in terms of one directional derivative of the velocity field, namely $\partial_3 \mathbf{u} \in L^p((0,T); L^q(\mathbb{R}^3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}<q\leq 6$. Moreover, we extend the range of $q$ while the solution is axisymmetric, i.e. the axisymmetric solution $\mathbf{m}{u}$ is regular in $(0,T]$, if $ \partial_3 u^3 \in L^p((0,T); L^q(\mathbb{R}^3)), ~\frac{2}{p} + \frac{3}{q} = 2, ~\frac{3}{2}<q< \infty$.

math.AP↗
arXiv2019

Global existence and lifespan for semilinear wave equations with mixed nonlinear terms

Firstly, we study the equation $\square u = |u|^{q_c}+ |\partial u|^p$ with small data, where $q_c$ is the critical power of Strauss conjecture and $p\geq q_c.$ We obtain the optimal lifespan $\ln({T_\varepsilon})\approx\varepsilon^{-q_c(q_c-1)}$ in $n=3$, and improve the lower-bound of $T_\varepsilon$ from $\exp({c\varepsilon^{-(q_c-1)}})$ to $\exp({c\varepsilon^{-(q_c-1)^2/2}})$ in $n=2$. Then, we study the Cauchy problem with small initial data for a system of semilinear wave equations $\square u = |v|^q,$ $ \square v = |\partial_t u|^p$ in 3-dimensional space with $q<2$. We obtain that this system admits a global solution above a $p-q$ curve for spherically symmetric data. On the contrary, we get a new region where the solution will blow up.

math.AP↗
arXiv2018

Muliti-scale regularity of axisymmetric Navier-Stokes equations

By applying the delicate \textit{a priori} estimates for the equations of $(Φ,Γ)$, which is introduced in the previous work, we obtain some multi-scale regularity criteria of the swirl component $u^θ$ for the 3D axisymmetric Navier-Stokes equations. In particularly, the solution $\mathbf{u}$ can be continued beyond the time $T$, provided that $u^θ$ satiesfies $$ u^θ \in L^{p}_{T}L^{q_{v}}_{v}L^{q_{h},w}_{h},~~\frac{2}{p}+\frac{1}{q_{v}}+\frac{2}{q_{h}}\leq 1, ~2<q_{h}\leq\infty,~\frac{1}{q_{v}}+\frac{2}{q_{h}}<1. $$

math.AP↗
arXiv2016

Global solutions to the isentropic compressible Navier-Stokes equations with a class of large initial data

In this paper, we consider the global well-posedness problem of the isentropic compressible Navier-Stokes equations in the whole space $\R^N$ with $N\ge2$. In order to better reflect the characteristics of the dispersion equation, we make full use of the role of the frequency on the integrability and regularity of the solution, and prove that the isentropic compressible Navier-Stokes equations admit global solutions when the initial data are close to a stable equilibrium in the sense of suitable hybrid Besov norm. As a consequence, the initial velocity with arbitrary $\dot{B}^{\fr{N}{2}-1}_{2,1}$ norm of potential part $\Pe^\bot u_0$ and large highly oscillating are allowed in our results. The proof relies heavily on the dispersive estimates for the system of acoustics, and a careful study of the nonlinear terms.

math.AP↗
arXiv2015

Global axisymmetric solutions of 3D inhomogeneous incompressible Navier-Stokes Systems with nonzero swirl

In this paper, we investigate the global well-posedness for the 3-D inhomogeneous incompressible Navier-Stokes system with the axisymmetric initial data. We prove the global well-posedness provided that $$\|\frac{a_{0}}{r}\|_{\infty} \textrm{ and } \|u_{0}^θ\|_{3} \textrm{ are sufficiently small}. $$ Furthermore, if $\mathbf{u}_0\in L^1$ and $ru^θ_0\in L^1\cap L^2$, we have \begin{equation*} \|u^θ(t)\|_{2}^{2}+\langle t\rangle \|\nabla (u^θ\mathbf{e}_θ)(t)\|_{2}^{2}+t\langle t\rangle(\|u_{t}^θ(t)\|_{2}^{2}+\|Δ(u^θ\mathbf{e}_θ)(t)\|_{2}^{2}) \leq C \langle t\rangle^{-\frac{5}{2}},\ \forall\ t>0. \end{equation*}

math.AP↗
arXiv2015

Global solutions to the Oldroyd-B model with a class of large initial data

Consider a global wellposed problem for the incompressible Oldroyd-B model. It is shown that this set of equations admits a unique global solution provided the initial horizontal velocity $u^h_0$, the product $\om u^d_0$ of the coupling parameter $\om$ and initial the vertical velocity $u^d_0$, and initial symmetric tensor of constrains $τ_0$ are sufficient small in the scaling invariant Besov space $\dot{B}^{\fr{d}{2}-1}_{2,1}\times\dot{B}^{\fr{d}{2}}_{2,1}, d\ge2$. In particular, the result implies the global well-posedness of Oldroyd-B model with large initial vertical velocity $u_0^d$.

math.AP↗
arXiv2015

Regularity of 3D axisymmetric Navier-Stokes equations

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 $L^{3}(\R^{3})$. Furthermore, we also get several Prodi-Serrin type regularity criteria based on one component of the solutions, say $ω^3$ or $u^3$.

math.AP↗
arXiv2013

Some new regularity criteria for the 3D Navier-Stokes Equations

Several types of new regularity criteria for Leray-Hopf weak solutions $u$ to the 3D Navier-Stokes equations are obtained. Some of them are based on the third component $u_3$ of velocity under Prodi-Serrin index condition, another type is in terms of $ω_3$ and $\partial_3u_3$ with Prodi-Serrin index condition. And a very recent work of the authors, based on only one of the nine entries of the gradient tensor, is renovated.

math.AP↗
arXiv2012

Global existence in critical spaces for density-dependent incompressible viscoelastic fluids

In this paper we consider the local and global well-posedness to the density-dependent incompressible viscoelastic fluids. We first study some linear models associated to the incompressible viscoelastic system. Then we approximate the system by a sequence of ordinary differential equations, by means of the Friedrichs method. Some uniform estimates for those solutions will be obtained. Using compactness arguments, we will get the local existence up to extracting a subsequence by means of Ascoli's lemma. With the help of small data conditions and hybird Besov spaces, we finally derive the global existence.

math.AP↗
arXiv2012

Regularity criterion for 3D Navier-Stokes Equations in Besov spaces

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 $\partial_3u_3$.

math.AP↗
arXiv2012

The regularity criterion for 3D Navier-Stokes Equations

In this article, we establish sufficient conditions for the regularity of solutions of Navier-Stokes equations based on one of the nine entries of the gradient tensor. We improve the recently results of C.S. Cao, E.S. Titi (Arch. Rational Mech.Anal. 202 (2011) 919-932) and Y. Zhou, M. Pokorn$\acute{y}$ (Nonlinearity 23, 1097-1107 (2010)).

math.AP↗
arXiv2012

On the unconditional uniqueness for NLS in $H^s$

In this article, we study the unconditional uniqueness of $\dot H^s$, $0<s< 1$, solutions for the nonlinear Schrödinger equation $i\partial_t u +Δu+ c |u|^αu=0$ in ${\mathbb R}^n$. We give a unified proof of the previously known results in the subcritical cases and critical cases, and we also extend these results to some previously unsettled cases. Our proof uses in particular negative order Sobolev spaces (or Besov spaces), general Strichartz estimates, and the improved regularity property for the difference of two solutions.

math.AP↗
arXiv2011

Decay Estimates for Isentropic Compressible Navier-Stokes Equations in Bounded Domain

In this paper, under the hypothesis that $ρ$ is upper bounded, we construct a Lyapunov functional for the multidimensional isentropic compressible Navier-Stokes equations and show that the weak solutions decay exponentially to the equilibrium state in $L^2$ norm. This can be regarded as a generalization of Matsumura and Nishida's results in 1982, since our analysis is done in the framework of Lions 1998 and Feireisl et al. 2001, the higher regularity of $(ρ, u)$ and the uniformly positive lower bound of $ρ$ are not necessary in our analysis and vacuum may be admitted. Indeed, the upper bound of the density $ρ$ plays the essential role in our proof.

math.AP↗