Publications: (Short descriptions for highlighted papers with *)

1*. Zhiwu Lin, Instability of periodic BGK waves, Mathematical Research Letters, 8, 521-534 (2001).

(In this paper, I proved linear instability of any periodic BGK wave of Vlasov-Poisson to multiple periodic perturbations. For the proof, I introduced a family of dispersion operators and a continuation argument. This approach was later used in many other problems including [4] [8] [10] [13] [14] [15].)

2. Zhiwu Lin, Stability and instability of traveling solitonic bubbles, Advances in Differential Equations, 7, no. 8, 897-918 (2002).

3. Zhiwu Lin, Instability of some ideal plane flows, SIAM J. Math. Anal., 35, no. 2, 318--356 (2003).

4. Zhiwu Lin, Some stability and instability criteria for ideal plane flows, Comm. Math. Phys., 246, no.1, 87-112 (2004).

5*. Zhiwu Lin, Nonlinear instability of ideal plane flows, Inter. Math. Res. Not., 41, 2147-2178 (2004).

(In this paper, I proved nonlinear instability of any linearly unstable steady flow of 2D Euler equation. The main point is to remove the restriction on the linear growth rate as imposed in the previous works. For the proof, I introduced an averaged Liapunov exponent and used a two-layer approach to overcome the loss of derivative. This approach was also used in [6] [9].)

6. Zhiwu Lin, Nonlinear instability of periodic waves for Vlasov-Poisson system, Comm. Pure Appl. Math., 58, no. 4, 505-528 (2005).

7. Zhiwu Lin, Some recent results on instability of ideal plane flows, Contemp. Math. 371, 217-229 (2005).

8. Zhiwu Lin and Walter Strauss, Linear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure Appl. Math., 60, no. 5, 724-787 (2007).

9*. Zhiwu Lin and Walter Strauss, Nonlinear stability and instability of relativistic Vlasov-Maxwell systems, Comm. Pure Appl. Math., 60, no. 6, 789-837 (2007).

(In this paper, we proved a sharp nonlinear stability criterion for a class of electromagnetic equilibria of 1.5 D Vlasov-Maxwell systems. In particular, the first example of stable periodic equilibrium to general perturbations was given.)

10. Yan Guo and Zhiwu Lin, Unstable and Stable Galaxy Models, Comm. Math. Phys., 279, 789-813 (2008).

11. Vera Mikyoung Hur and Zhiwu Lin, Unstable surface waves in running water, Comm. Math. Phys., 282, no. 3, 733-796 (2008). (Erratum)

12. Zhiwu Lin and Yue Liu, Stability of peakons for the Degasperis-Procesi equation, Comm. Pure Appl. Math., 62, 125-146 (2009).

13*. Zhiwu Lin and Walter Strauss, A sharp stability criterion for the Vlasov-Maxwell system, Invent. Math., 173, no. 3, 497-546 (2008).

(In this paper, we proved a sharp linear stability criterion for 3D Vlasov-Maxwell systems with symmetry.)

14. Zhiwu Lin, Instability of nonlinear dispersive solitary waves, J. Funct. Anal., 255, 1191-1124 (2008).

15*. Zhiwu Lin, On Linear Instability of large solitary water waves, Inter. Math. Res. Not., 1247-1303 (2009).

(In this paper, I proved linear instability of large solitary waves close to the highest wave with 120- degree angle at the crest. More precisely, solitary waves higher than the one with the maximal energy (not the highest one) are proved to be unstable.)

16. Charles Y. Li and Zhiwu Lin, A Resolution of the Sommerfeld Paradox, SIAM J. Math. Anal., 43, No.4, 1923-1954 (2011).

17*. Zhiwu Lin and Chongchun Zeng, Small BGK waves and nonlinear Landau damping, Comm. Math. Phys., 306, 291-331 (2011).

(In this paper, we proved that nonlinear Landau damping fails for perturbations in the Sobolev space H^{s} (s<3/2), by constructing BGK waves arbitrarily close to the homogeneous states. The exponent 3/2 was shown to be critical in the sense that in the Sobolev space H^{s} (s>3/2), there exist no nontrivial invariant structures in the H^{s} (s>3/2) neighborhood of homogeneous states. Similar results were obtained near Couette flow of 2D Euler equation in [18].)

18. Zhiwu Lin and Chongchun Zeng, Inviscid dynamical structures near Couette flow, Arch. Rational Mech. Anal., 200, 1075-1097 (2011).

19.  Zhiwu Lin and Chongchun Zeng, Small BGK waves and nonlinear Landau damping (higher dimensions), Indiana Univ. Math. J., 61, 1711-1735 (2012).

20*. Zhiwu Lin and Chongchun Zeng, Unstable manifolds of Euler equations, Comm. Pure Appl. Math., 66, 1803-1936 (2013).

(In this paper, we constructed stable and unstable manifolds near unstable steady flows of Euler equations in any dimension, under a spectral gap condition which is satisfied for all unstable shear flows. To our knowledge, this work is the first construction of invariant manifolds for continuum models without dissipation. To overcome the loss of derivative, we used a combination of Lagrangian and Eulerian formulations of Euler equations.)

21. Zhiyu Wang, Yan Guo, Zhiwu Lin and Pingwen Zhang, Unstable galaxy models. Kinet. Relat. Models 6, 701-714 (2013).

22. Zhiwu Lin, Zhengping Wang and Chongchun Zeng, Stability of traveling waves of nonlinear Schrodinger equation with nonzero condition at infinity, Arch. Ration. Mech. Anal., 222, 143-212 (2016).

23*. Yan Guo and Zhiwu Lin, The existence of stable BGK waves, Commun. Math. Phys., 352, 1121-1152 (2017).

(In this work, we constructed the first class of stable BGK waves to perturbations of minimal period. Such stable BGK waves are believed to play an important role in the long time dynamics of Vlasov-Poisson system. But the proof of their existence has been open since 1958.)

24. Jincheng Yang and Zhiwu Lin, Linear Inviscid Damping for Couette Flow in Stratified Fluid, J. Math. Fluid Mech., 20, 445--472 (2018).

25*. Zhiwu Lin and Chongchun Zeng, Instability, index theorem, and exponential trichotomy for Linear Hamiltonian PDEs, to appear in Memoir of AMS.

(In this long paper, we studied the detailed properties of general linear Hamiltonian PDEs. The energy functional is assumed to have only finitely many negative directions. There is no restriction on the symplectic operator, which can be an arbitrary skew-symmetric operator. First, we gave a generalization of the index theorems obtained for various special cases in the literature. Moreover, we gave more precise information about these indexes. Second, we proved the exponential trichotomy of solutions of the linearized equation, which is an important step for constructing invariant manifolds of nonlinear Hamiltonian PDEs. Third, we gave sharp conditions the structural stability/instability of linear Hamiltonian system. This theory has been applied to examples including long wave models (KDV, BBM, Bousinesq etc.), Gross-Pitaevskii equation for superfluids, 2D Euler equation and 3D Vlasov-Maxwell systems.)

26. Jiayin Jin, Shasha Liao, and Zhiwu Lin, Nonlinear Modulational Instability of Dispersive PDE Models, Arch. Ration. Mech. Anal. 231, 1487--1530 (2019).

27. Zhiwu Lin and Ming Xu, Metastability of Kolmogorov flows and inviscid damping of shear flows, Arch. Ration. Mech. Anal. 231, 1811--1852 (2019).

28*. Jiayin Jin, Zhiwu Lin and Chongchun Zeng, Invariant Manifolds of traveling waves of the 3D Gross-Pitaevskii equation in the energy space, Comm. Math. Phys. 364, 981-1039 (2018).

(In this work, we study the local dynamics near general unstable traveling waves of the 3D Gross-Pitaevskii equation in the energy space by constructing smooth local invariant center-stable, center-unstable and center manifolds. We prove that (i) the center-unstable manifold attracts nearby orbits exponentially before they get away from the traveling waves along the center directions and (ii) if an initial data is not on the center-stable manifolds, then the forward flow will be ejected away from traveling waves exponentially fast. Furthermore, under a non-degenerate assumption, we show the orbital stability of the traveling waves on the center manifolds, which also implies the local uniqueness of the local invariant manifolds. Our approach based on a geometric bundle coordinates can be applied to construct invariant manifolds for a general class of Hamiltonian PDEs.)

29. Zhiwu Lin, Jincheng Yang and Hao Zhu, Barotropic instability of shear flows, accepted by Studies in Applied Mathematics.

30. Jiayin Jin, Zhiwu Lin and Chongchun Zeng, Dynamics near the solitary waves of the supercritical gKDV Equations, J. Differential Equations 267, 7213--7262 (2019).

31*. Mahir Hadzic, Zhiwu Lin, Gerhard Rein, Stability and instability of self-gravitating relativistic matter distributions, arXiv:1810.00809.

(We consider steady state solutions of the massive, asymptotically flat, spherically symmetric Einstein-Vlasov system, i.e., relativistic models of galaxies or globular clusters, and steady state solutions of the Einstein-Euler system, i.e., relativistic models of stars. Such steady states are embedded into one-parameter families parametrized by their central redshift. We prove their linear instability when the central redshift is sufficiently large, i.e., when they are strongly relativistic. This confirms the scenario of dynamic instability proposed in the 1960s by Zel'dovich and Podurets (for the Einstein-Vlasov system) and by Harrison, Thorne, Wakano, and Wheeler (for the Einstein-Euler system). Our results are in sharp contrast to the corresponding non-relativistic, Newtonian setting. Finally, in the case of the Einstein-Euler system we prove a rigorous version of the turning point principle which relates the stability of steady states along the one-parameter family to the winding points of the so-called mass-radius curve. In the proof, we used the Hamiltonian structures of the linearized equations and the theory developed in paper 25 for general linear Hamiltonian PDEs).

32*. Zhiwu Lin and Chongchun Zeng, Separable Hamiltonian PDEs and Turning point principle for stability of gaseous stars, arXiv: 2005.00973.

(We consider stability of non-rotating gaseous stars modeled by the Euler-Poisson system. Under general assumptions on the equation of states, we proved a turning point principle (TPP) that the stability of the stars is entirely determined by the mass-radius curve parameterized by the center density. In particular, the stability can only change at extrema (i.e. local maximum or minimum points) of the total mass. For very general equation of states, TPP implies that for increasing center density the stars are stable up to the first mass maximum and unstable beyond this point until next mass extremum (a minimum). Moreover, we get a precise counting of unstable modes and exponential trichotomy estimates for the linearized Euler-Poisson system. To prove these results, we develop a general framework of separable Hamiltonian PDEs. The general approach is flexible and can be used for many other problems including stability of rotating and magnetic stars, relativistic stars and galaxies.)