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 (1-D) are fairly well understood. However, well-posedness
and behaviors for 1-D solutions with large amplitude (large data) and multi-D solutions,
on which my research focuses, are still widely open.
A picture for shock formation of 1-D solution u(t,x).
I systematically studied
shock formation and structure of large shock-free solutions for
compressible Euler equations in 1-D 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
1-D Euler equations.
- A complete resolution on Shock formation for 1-D large solution:
Sharp lower bound on density in the order of O(1/t):
Singularity formation for compressible Euler
equations, submitted, (with Ronghua Pan and Shengguo Zhu).
- Shock formation in the compressible Euler equations and related
systems, J. Hyperbolic Differential Equations, 10 (2013), no. 1,
149-172 (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,
2580-2595, (with Robin Young).
- Formation of singularity and smooth wave propagation for the
non-isentropic compressible Euler equations, J. Hyperbolic Differential
Equations, 8 (2011), no. 4, 671-690.
Structure and long time behavior of shock free solutions:
- Optimal time-dependent lower bound on density for classical solutions of 1-D
compressible Euler equations,
to appear on Indiana University Mathematics Journal. pdf
The existence of large BV (bounded total variation) solutions for the
initial value problems to 1-D isentropic (p-system) 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.
Shock formation and exact solutions
for the compressible Euler equations, Arch. Ration. Mech. Anal., 217:3 (2015), 1265-1293. (with Robin Young). pdf
- 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 p-system.
For 1-D full Euler system, we completely resolved pairwise wave interactions.
No BV bounds for approximate solutions to p-system 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 p-system with Large Data, J. Differential Equations, 256 (2014), 3067-3085 (with Alberto Bressan and Qingtian Zhang). pdf
No TVD fields for 1-d isentropic gas flow, Comm. Partial Differential
Equations, 38 (2013), no. 4, 629-657 (with Helge Kristian Jenssen). pdf
Pairwise wave interactions in ideal polytropic gases, Arch. Ration.
Mech. Anal., 204 (2012), no. 3, 787-836, (with Erik Endres and Helge
Research works on compressible Navier-Stokes equationsA
new field I recently entered is the compressible Navier-Stokes
equations. For multi-D solutions on some nonisentropic compressible
Navier-Stokes 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.