Research works on compressible Euler equations
As one of the most important partial differential equations,
Compressible Euler equations are used to describe
gas dynamics and have wide applications such as in the aircraft manufacturing. The
major difficulty in analyzing the solution of this system is the formation of discontinu
ities, known as shock waves, even when initial data are smooth. Currently, solutions for
compressible Euler equations and hyperbolic conservation laws with small total varia
tion in one space dimension (1D) are fairly well understood. However, wellposedness
and behaviors for 1D solutions with large amplitude (large data) and multiD solutions,
on which my research focuses, are still widely open.
A picture for shock formation of 1D solution u(t,x).
I systematically studied
shock formation and structure of large shockfree solutions for
compressible Euler equations in 1D and in a varying duct. Especially,
in my very recent papers, by providing a sharp time decay of density
lower bound in the order of O(1/t) for classical solutions with large
initial data away from vacuum, we generalized Lax's theory (1964) on
the shock formation for isentropic Euler equations to include all
physical cases with arbitrarily large initial data, and then to full
1D Euler equations.
 A complete resolution on Shock formation for 1D large solution:

Singularity formation for compressible Euler
equations, submitted, (with Ronghua Pan and Shengguo Zhu).
pdf
 Shock formation in the compressible Euler equations and related
systems, J. Hyperbolic Differential Equations, 10 (2013), no. 1,
149172 (with Robin Young and Qingtian Zhang). pdf

Smooth solutions and singularity formation for the inhomogeneous
nonlinear wave equation, J. Differential Equations, 252 (2012), no. 3,
25802595, (with Robin Young).
 Formation of singularity and smooth wave propagation for the
nonisentropic compressible Euler equations, J. Hyperbolic Differential
Equations, 8 (2011), no. 4, 671690.
 Sharp lower bound on density in the order of O(1/t):
 Optimal timedependent lower bound on density for classical solutions of 1D
compressible Euler equations,
to appear on Indiana University Mathematics Journal. pdf
 Structure and long time behavior of shock free solutions:

Shock formation and exact solutions
for the compressible Euler equations, Arch. Ration. Mech. Anal., 217:3 (2015), 12651293. (with Robin Young). pdf
The existence of large BV (bounded total variation) solutions for the
initial value problems to 1D isentropic (psystem) and full Euler
equations is a notoriously difficult and important open problem in the
field of fluid dynamics and hyperbolic conservation laws. We obtained
several progresses in this field.
 By providing a front tracking approximate solution away from vacuum
including finite time BV norm blowup, Bressan, Zhang and I showed that
currently available approximate solutions will not provide good total
variation estimate for even finite time. This work is inspired by an
earlier paper with Jenssen providing partial answer to this question.
The proof of these results relies on the careful study of wave
interactions of psystem.

No BV bounds for approximate solutions to psystem with general pressure law,
to appear on J. Hyperbolic Differential Equations, (with Alberto Bressan, Qingtian Zhang and Shengguo Zhu). pdf

Lack of BV Bounds for Approximate Solutions to
the psystem with Large Data, J. Differential Equations, 256 (2014), 30673085 (with Alberto Bressan and Qingtian Zhang). pdf

No TVD fields for 1d isentropic gas flow, Comm. Partial Differential
Equations, 38 (2013), no. 4, 629657 (with Helge Kristian Jenssen). pdf
 For 1D full Euler system, we completely resolved pairwise wave interactions.

Pairwise wave interactions in ideal polytropic gases, Arch. Ration.
Mech. Anal., 204 (2012), no. 3, 787836, (with Erik Endres and Helge
Kristian Jenssen).
Research works on compressible NavierStokes equations
A
new field I recently entered is the compressible NavierStokes
equations. For multiD solutions on some nonisentropic compressible
NavierStokes equations, in collaboration with Zhang and Zhu, we proved
the existence of unique local strong solutions for initial boundary
value problem possibly including vacuum. Moreover, we proved a minimum
principle on the temperature, The paper is available at pdf.