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Weekly seminar links: PDE

##### PDE

The PDE group in SMS consists of three professors (Baoxiang Wang, Zhifei Zhang and Shulin Zhou) and one assistant professor (Chao Wang). The group is mainly devoted to the mathematical theory of fluid mechanics equations including the Navier-Stokes equations, Euler equations and complex fluids, the well-posedness theory of dispersive equations and regularity theory of the elliptic and parabolic equations. Recent main achievements are as follows.

**1.
Harmonic Analysis Method Solving the Navier-Stokes Equations ( ZHANG Zhifei, WANG Chao, WANG Baoxiang)**

Global regularity or finite time blow-up
for the Navier-Stokes equations is still a challenging problem in the
mathematical fluid mechanics. It is also one of the Millennium Prize Problems.
Harmonic analysis has been an important tool solving the Navier-Stokes
equations. In this direction, an important theorem for incompressible
Navier-Stokes proved by Cannone, Meyer and Planchon states: if the initial
velocity is highly oscillating, then smooth solution of the incompressible
Navier-Stokes equations with this type of data is global in time. Along this
line, an important goal is to find more types of data generating global
solution of the Navier-Stokes equations by using more refined analysis
(Littlewood-Paley theory and profile decomposition, etc.) and the structure of
the equations. The proof of the above-mentioned results used the semigroup
method. It is difficult to generalize similar results to the compressible
Navier-Stokes equations, because semigroup method will lead to the loss of one
derivative for a quasilinear hyperbolic-parabolic system. Z. Zhang et al. [Chen-Miao-
Zhang, *CPAM* (2010)] developed a
semigroup method based on the paralinearized technique to prove a
Cannone-Meyer-Planchon theorem for compressible Navier-Stokes equations. C.
Wang, Z. Zhang et al. [Sun-Wang-Zhang, *JMPA* (2011)] proved a very surprising result on the blow-up mechanism for the
compressible Navier-Stokes equations conjectured by Nash in [Nash, *AJM* (1958)], which states that smooth
solution does not develop singularity if the density and the temperature do not
concentrate. Based on these results and harmonic analysis method, C. Wang, Z.
Zhang et al. [Wang-Wang-Zhang, *ARMA* (2014)]
proved the global well-posedness of compressible Navier-Stokes equations for a
class of initial data, which may have large oscillation for the density and
large energy for the velocity.

For the incompressible Navier-Stokes
equations (NS), Koch and Tataru in 2001 proved that it has a unique global
solution when the initial data is small enough in*BMO**-1**.* However, the well/ill posedness for NS in
critical Besov spaces*B*∞,*q*-1 remains open for many years. Bourgain and
Pavlovic in 2008, Yoneda in 2010 proved its ill-posedness in critical Besov
spaces*B*∞,q-1*, 2<q≤∞.* Baoxiang Wang in [Wang, *Adv. Math.* (2015)] finally showed that NS is ill-posed in all critical Besov spaces*B*∞,*q*-1 with1≤q<∞. The result in
the case1≤q≤2 seems surprising. Since*B*∞,*q*-1*⊂**BMO**-1**if 1≤q≤2, the small
initial data in**B*∞,*q*-1*with 1≤q≤2 can guarantee that NS has a unique solution in**BMO**-1*, however, the
solution can have an inflation in*B*∞,*q*-1*,* 1≤q≤2.

**2.
Free Boundary Problems in Fluid Mechanics (ZHANG Zhifei, WANG Chao)**

The hydrodynamic equations with free boundary have a long history and can be used to describe many physical phenomena, such as sea water flow, astronomical evolution and so on. This direction is also one of the hot directions of PDEs in recent years.

One of the important free problems in
fluid mechanics is the water-wave problem. The well-posedness of the water-wave
equations without vorticity was solved by S. Wu. P. Zhang and Z. Zhang [Zhang-Zhang, *CPAM* (2008)] used the Clifford
analysis to prove the well-posedness of water-wave equations with vorticity.
Most of well-posedness results required the intial data to be very smooth.
However, there are indeed some physical solutions such as stokes waves with
greatest height, which has low regularity at crest. Therefore, it is
interesting to study the well-posedness of the water-wave equation for low
regularity data. Recently, C. Wang and Z. Zhang et al. [Wang-Zhang-Zhao-Zheng, *arXiv* (2015)] proved the well-posedness
in the low regularity Sobolev space, in which the velocity is a Lipschtiz
function and the free surface is Holder continuous with exponent*3/2*+. By a recent
ill-posedness result by Bourgain and Li, the regularity threshold for the
velocity should be sharp. The question of whether smooth solution of the
water-wave equations develops singularity in finite time is a challenging
problem, although there are some important progress on the spash singularity in
recent years. C. Wang and Z. Zhang et al. [Wang-Zhang-Zhao- Zheng, *arXiv* (2015)] also present a break-down
criterion of smooth solution in terms of the gradient of the velocity, Taylor
sign condition and the mean curvature of the free surface.

Another important free problem in fluid mechanics is the vortex sheet, which may lead to Kelvin-Helmholtz instability. In 1953, Syrovatskij found that the magnetic field has the stabilizing effect on Kelvin-Helmholtz instability. He studied the linear stability of the current-vortex sheet and introduced the Syrovatskij linear stability condition. However, nonlinear stability has been a long-standing open problem. Z. Zhang et al. [CPAM (online)] proved nonlinear stability of current-vortex sheet in the ideal incompressible Magneto-Hydrodynamics under the Syrovatskij stability condition. This result gives a first rigorous confirmation of the stabilizing effect of the magnetic field on Kelvin-Helmholtz instability.

**3.
Linear Inviscid Damping (ZHANG Zhifei)**

Hydrodynamic stability has been one of the
hot issues in PDEs, whose history can be recounted to Rayleigh, Orr, Landau
etc. Euler equations are always an important model in the mathematics and
physics. To study its stability is always hot issues in the PDEs. Recently,
Mouhot and Vallani gave a groundbreaking proof of nonlinear Landau damping.
Later, Bedrossian and Masmoudi proved the inviscid damping for the
two-dimensional Euler equation when perturbation is close to the Couette flow
in the Gevrey class. In 1960, Case predicted the linear damping for monotone
shear flows. Z. Zhang et al. [Sun-Wang-Zhang, *CPAM* (2017)] proved the optimal decay estimates of the velocity for
the linearized Euler equations around a class of monotone shear flow, which
confirms the prediction of Case. Whether the linear damping holds for the flows
without monotonicity due to possible instability is a controversial problem. D.
Wei, Z. Zhang and W. Zhao [Wei-Zhang-Zhao, *arXiv* (2017)] also extended the linear inviscid damping to more general shear flows
including the well-know Poiseuille flow and Kolmogorov flow. Moreover, they
rigorously proved the vorticity depletion phenomena found by Bouchet and Morita
based on numerical analysis. The vorticity depletion instead of vorticity
mixing is an important mechanism leading to the linear damping for the
non-monotone flows.

**4.
Well-posedness of Dispersive Equations (WANG Baoxiang)**

Nonlinear Schrödinger equations (NLS), together with some other nonlinear dispersive equations are one of the important equations in nonlinear PDEs which has a strong physical background and valuable applications.

Baoxiang Wang and his collaborators have
developed a new method, the frequency uniform decomposition techniques to study
the Cauchy problem of the nonlinear dispersive equations with data in
modulation spaces, which were regarded as “a series of deep and pioneer works”
by some peers. B. Wang joint with Hudzik obtained the global well posedness for
the energy supercritical NLS with small data in modulation space*M*2,10, which contains a
class of initial data with infinite energy (the initial data can also be large
in*M*2,10 by scaling invariance of NLS). B. Wang joint
with Han and Huang [Wang-Han-Huang, *AIHP* (2009); Wang, *JFA* (2013)] obtained
the global well-posedness and scattering results for the non-elliptical
derivative nonlinear Schrödinger equation (DNLS) with small data in*M*2,1*s*, which seems the
first global well-posedness and scattering result for the non-elliptical DNLS.

**5.
Regularity of Quasilinear Elliptic Equation (ZHOU Shulin)**

Elliptic equation has a very long history and is also a fundamental direction in PDEs. The study of elliptic equation not only develops many tools and techniques, but also promotes the development of other disciplines such as hydrodynamic equations, geometric analysis, etc. Shulin Zhou has some profound results on the elliptic equations.

S. Zhou et al. [Yao-Zhou, *JFA* (2017)] proved Calderón-Zygmund
estimate for a class of quasilinear elliptic equations. In addition, for the
p-Laplacian equation with small coefficients in the BMO space, S. Zhou and his
collaborators used the improved Vitali covering lemma, the maximal function and
the appropriate localization technique introduced in their work [Zhang-Zhou, *JFA* (2014)] to obtain a new weighted
global gradient estimate in Lorentz spaces. Finally, under some assumptions on
the continuous function p(x), S. Zhou et al. [Zhang-Zhou, *JMAA* (2012)] obtained the local*C*1,*α*regularity of
solutions of the strong p(x)-Laplacian by using the integral estimates in
Campanato spaces.