Yi Zhou
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Yi Zhou has a total of 17 co-authors. Here are the top co-authors by number of shared publications:
- Zhen Lei
School of Mathematical Sciences School of Mathematical Sciences; LMNS and Shanghai Key Laboratory for Contemporary Applied Mathematics Fudan Unviersity; F.-H. Lin, Courant Institute of Mathematics New York University
9 shared publications - Wei Han
Department of Mathematics
4 shared publications - Chun Liu
School of Earth Sciences and Engineering, Nanjing University, Nanjing, China
3 shared publications - Jianli Liu
Fudan University 1 School of Mathematical Sciences Shanghai 200433, People's Republic of China
2 shared publications - Ning-An Lai
Shaoxing University
2 shared publications - Wei Xu
Fudan University
2 shared publications
Latest Publications
- Open Access
- 1 January 2016
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 36 (7) , 3845-3856
Global well-posedness of axisymmetric Navier-Stokes equations with one slow variable
This paper deals with the global well-posedness of axisymmetric Navier-Stokes equations with swirl. We prove that there exists a global solution of Navier-Stokes equations under some weighted energy for a class of large anisotropic initial data slowly varying in the vertical variable.
- 1 January 2016
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 36 (7) , 3845-3856
Open AccessGlobal well-posedness of axisymmetric Navier-Stokes equations with one slow variable
This paper deals with the global well-posedness of axisymmetric Navier-Stokes equations with swirl. We prove that there exists a global solution of Navier-Stokes equations under some weighted energy for a class of large anisotropic initial data slowly varying in the vertical variable.
- Open Access
Almost Global Existence for 2-D Incompressible Isotropic Elastodynamics
We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval $[0,{T}/{\epsilon}]$, for initial data of the form $\epsilon U_0$, where $T$ depends only on some Sobolev norm of $U_0$. We show that for such data there exists a unique solution on a time interval $[0, \exp{T}/{\epsilon}]$, provided that $\epsilon$ is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted $L^2$ estimates and the ghost weight introduced by Alinhac.Open AccessAlmost Global Existence for 2-D Incompressible Isotropic Elastodynamics
We consider the Cauchy problem for 2-D incompressible isotropic elastodynamics. Standard energy methods yield local solutions on a time interval $[0,{T}/{\epsilon}]$, for initial data of the form $\epsilon U_0$, where $T$ depends only on some Sobolev norm of $U_0$. We show that for such data there exists a unique solution on a time interval $[0, \exp{T}/{\epsilon}]$, provided that $\epsilon$ is sufficiently small. This is achieved by careful consideration of the structure of the nonlinearity. The incompressible elasticity equation is inherently linearly degenerate in the isotropic case; in other words, the equation satisfies a null condition. This is essential for time decay estimates. The pressure, which arises as a Lagrange multiplier to enforce the incompressibility constraint, is estimated in a novel way as a nonlocal nonlinear term with null structure. The proof employs the generalized energy method of Klainerman, enhanced by weighted $L^2$ estimates and the ghost weight introduced by Alinhac. - Open Access
Formulation of finite-time singularity for free-surface Euler equations
We give an extremely short proof that the free-surface incompressible, irrotational Euler equations with regular initial condition can form a finite time singularity in 2D or 3D. Thus, we provide a simple view of the problem studied by Castro, Cordoba, Fefferman, Gancedo, Lopez-Fernadez, Gomez-Serrano and Coutand, Shkoller.
Open AccessFormulation of finite-time singularity for free-surface Euler equations
We give an extremely short proof that the free-surface incompressible, irrotational Euler equations with regular initial condition can form a finite time singularity in 2D or 3D. Thus, we provide a simple view of the problem studied by Castro, Cordoba, Fefferman, Gancedo, Lopez-Fernadez, Gomez-Serrano and Coutand, Shkoller.
- Open Access
Blow up for some semilinear wave equations in multi-space dimensions
In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) = |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} and {equation} \label{0.2} \Box u(t,x) = |u_{t}(t,x)|^{p}, \ \ x\in R^{n} {equation} both have global existence for small initial data, while for the combined nonlinearity, the solutions to the Cauchy problem for the nonlinear wave equation {equation} \label{0.3} \Box u(t,x) = | u_{t}(t,x)|^{p} + |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} with small initial data will blow up in finite time. In the two dimensional case, we also find that if $ q=4$, the Cauchy problem for the equation \eqref{0.1} has global existence, and the Cauchy problem for the equation {equation} \label{0.4} \Box u(t,x) = u (t,x)u_{t}(t,x)^{2}, \ \ x\in R^{2} {equation} has almost global existence, that is, the life span is at least $ \exp (c\varepsilon^{-2}) $ for initial data of size $ \varepsilon$. However, in the combined nonlinearity case, the Cauchy problem for the equation {equation} \label{0.5} \Box u(t,x) = u(t,x) u_{t}(t,x)^{2} + u(t,x)^{4}, \ \ x\in R^{2} {equation} has a life span which is of the order of $ \varepsilon^{-18} $ for the initial data of size $ \varepsilon$, this is considerably shorter in magnitude than that of the first two equations. This solves an open optimality problem for general theory of fully nonlinear wave equations (see \cite{Katayama}).Open AccessBlow up for some semilinear wave equations in multi-space dimensions
In this paper, we discuss a new nonlinear phenomenon. We find that in $n\geq 2$ space dimensions, there exists two indexes $p$ and $q$ such that the cauchy problems for the nonlinear wave equations {equation} \label{0.1} \Box u(t,x) = |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} and {equation} \label{0.2} \Box u(t,x) = |u_{t}(t,x)|^{p}, \ \ x\in R^{n} {equation} both have global existence for small initial data, while for the combined nonlinearity, the solutions to the Cauchy problem for the nonlinear wave equation {equation} \label{0.3} \Box u(t,x) = | u_{t}(t,x)|^{p} + |u(t,x)|^{q}, \ \ x\in R^{n}, {equation} with small initial data will blow up in finite time. In the two dimensional case, we also find that if $ q=4$, the Cauchy problem for the equation \eqref{0.1} has global existence, and the Cauchy problem for the equation {equation} \label{0.4} \Box u(t,x) = u (t,x)u_{t}(t,x)^{2}, \ \ x\in R^{2} {equation} has almost global existence, that is, the life span is at least $ \exp (c\varepsilon^{-2}) $ for initial data of size $ \varepsilon$. However, in the combined nonlinearity case, the Cauchy problem for the equation {equation} \label{0.5} \Box u(t,x) = u(t,x) u_{t}(t,x)^{2} + u(t,x)^{4}, \ \ x\in R^{2} {equation} has a life span which is of the order of $ \varepsilon^{-18} $ for the initial data of size $ \varepsilon$, this is considerably shorter in magnitude than that of the first two equations. This solves an open optimality problem for general theory of fully nonlinear wave equations (see \cite{Katayama}). - 1 May 2012
- Published by Springer Nature in Chinese Annals of Mathematics, Series B
- Vol. 33 (3) , 437-452
Nonresonance and global existence of nonlinear elastic wave equations with external forces
- 1 May 2012
- Published by Springer Nature in Chinese Annals of Mathematics, Series B
- Vol. 33 (3) , 437-452
Nonresonance and global existence of nonlinear elastic wave equations with external forces
- Open Access
Global Solutions of Evolutionary Faddeev Model With Small Initial Data
We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space $\mathbb{R}^{1 + n}$ to the unit sphere $\mathbb{S}^2$, which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.Open AccessGlobal Solutions of Evolutionary Faddeev Model With Small Initial Data
We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space $\mathbb{R}^{1 + n}$ to the unit sphere $\mathbb{S}^2$, which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space. - 26 August 2011
- Published by Springer Nature in Chinese Annals of Mathematics, Series B
- Vol. 32 (5) , 771-780
Global classical solutions to partially dissipative quasilinear hyperbolic systems
- 26 August 2011
- Published by Springer Nature in Chinese Annals of Mathematics, Series B
- Vol. 32 (5) , 771-780
Global classical solutions to partially dissipative quasilinear hyperbolic systems
- 8 July 2011
- Published by Springer Nature in Chinese Annals of Mathematics, Series B
- Vol. 32 (4) , 521-526
Sharpness on the lower bound of the lifespan of solutions to nonlinear wave equations
- 8 July 2011
- Published by Springer Nature in Chinese Annals of Mathematics, Series B
- Vol. 32 (4) , 521-526
Sharpness on the lower bound of the lifespan of solutions to nonlinear wave equations
- Open Access
Life-Span of Solutions to Critical Semilinear Wave Equations
The final open part of the famous Strauss conjecture on semilinear wave equations of the form \Box u=|u|^{p}, i.e., blow-up theorem for the critical case in high dimensions was solved by Yordanov and Zhang, or Zhou independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. Recently, Takamura and Wakasa have obtained the sharp upper bound of the lifespan of the solution to the critical semilinear wave equations, and their method is based on the method in Yordanov and Zhang. In this paper, we give a much simple proof of the result of Takamura and Wakasa by using the method in Y. Zhou for space dimensions n\geq 2. Simultaneously, this estimate of the life span also proves the last open optimality problem of the general theory for fully nonlinear wave equations with small initial data in the case n=4 and quadratic nonlinearity(One can see Li and Chen for references on the whole history).
Open AccessLife-Span of Solutions to Critical Semilinear Wave Equations
The final open part of the famous Strauss conjecture on semilinear wave equations of the form \Box u=|u|^{p}, i.e., blow-up theorem for the critical case in high dimensions was solved by Yordanov and Zhang, or Zhou independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. Recently, Takamura and Wakasa have obtained the sharp upper bound of the lifespan of the solution to the critical semilinear wave equations, and their method is based on the method in Yordanov and Zhang. In this paper, we give a much simple proof of the result of Takamura and Wakasa by using the method in Y. Zhou for space dimensions n\geq 2. Simultaneously, this estimate of the life span also proves the last open optimality problem of the general theory for fully nonlinear wave equations with small initial data in the case n=4 and quadratic nonlinearity(One can see Li and Chen for references on the whole history).
- 11 February 2011
- Published by Springer Nature in Science China Mathematics
- Vol. 54 (2) , 205-220
Global existence of solutions of the critical semilinear wave equations with variable coefficients outside obstacles
- 11 February 2011
- Published by Springer Nature in Science China Mathematics
- Vol. 54 (2) , 205-220
Global existence of solutions of the critical semilinear wave equations with variable coefficients outside obstacles