FOCUSED RESEARCH GROUP (FRG) PROJECT, 20032007:

MultiDimensional Problems for the Euler Equations of
Compressible Fluid Flow and Related Problems in Hyperbolic
Conservation Laws
Supported by NSF: $1million.
PROJECT SUMMARY
The project is devoted to a mathematical study of the Euler equations
governing the motion of an inviscid compressible fluid and
related problems. Compressible fluids occur all around us in nature,
e.g. gases and plasmas, whose study is crucial to understanding
aerodyanmics, atmospheric sciences, thermodynamics, etc.
While the onedimensional fluid flows are rather well understood,
the general theory for multidimensional flows is comparatively mathematically
underdeveloped. The goal of this collaborative research
is to advance the mathematical understanding of the multidimensional
equations of inviscid compressible fluid dynamics
and introduce a new generation of researchers to the outstanding problems in the field.
Multidimensional Conservation Laws,
SIAM News: Volume 39, Number 5, June 2006.
PRINCIPAL INVESTIGATORS
Some pictures.
SEMESTERS OF EMPHASIS, WORKSHOPS, AND SUMMER SCHOOL
 Fall 2003: Semester of emphasis and workshop, University of Pittsburgh.
 Fall 2004: Semester of emphasis and workshop, Stanford University.
 Summer 2005: Summer school, University of Wisconsin  Madison.
 Spring 2006: Semester of emphasis and workshop, University of Houston.
 Summer 2007: Workshop, Stanford University.
RELATED PUBLICATIONS AND PREPRINTS
 G.Q. Chen, M. Feldman,
Multidimensional transonic shocks and free boundary problems for
nonlinear equations of mixed type.
J. Amer. Math. Soc. 16 (2003), no. 3, 461494
 S. Canic, E. H. Kim,
A class of quasilinear degenerate elliptic problems.
J. Differential Equations 189 (2003), no. 1, 7198.
 C. M. Dafermos,
Entropy for hyperbolic conservation laws. Entropy, 107120,
Princeton Ser. Appl. Math., Princeton Univ. Press, Princeton, NJ, 2003.
 L. Hong, J. Hunter,
Singularity formation and instability in the unsteady inviscid and
viscous Prandtl equations. Commun. Math. Sci. 1 (2003), no. 2, 293316.
 T.P. Liu, S.H. Yu,
Energy method for equations in gas dynamics.
Hyperbolic problems: theory, numerics, applications, 5360,
Springer, Berlin, 2003.
 S.Y. Ha and M. Slemrod,
Global existence of plasma ionsheaths and their dynamics,
Comm. Math. Phys. 238 (2003), no. 12, 149186.
 T. Sideris, B. Thomases, D. Wang,
Long time behavior of solutions to the 3D compressible Euler equations
with damping,
Comm. Partial Differential Equations 28 (2003), no. 34, 795816.
 Y. Zheng, A global solution to a twodimensional Riemann problem
involving shocks as free boundaries.
Acta Math. Appl. Sin. Engl. Ser. 19 (2003), no. 4, 559572.
 J. Ryan, C.W. Shu,
On a onesided postprocessing technique for the discontinuous
Galerkin methods. Methods Appl. Anal. 10 (2003), no. 2, 295307.
 G.Q. Chen, M. Feldman,
Steady transonic shocks and free boundary problems for the Euler equations
in infinite cylinders. Comm. Pure Appl. Math. 57 (2004), no. 3, 310356.
 T.P. Liu, S.H. Yu,
Boltzmann equation: micromacro decompositions and positivity of
shock profiles. Comm. Math. Phys. 246 (2004), no. 1, 133179.
 J. Hunter, A. Tesdall, Weak shock reflection. A celebration of mathematical modeling, 93112, Kluwer Acad. Publ., Dordrecht, 2004.
 G.Q. Chen, M. Feldman,
Free boundary problems and transonic shocks for the Euler equations
in unbounded domains.
Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 3 (2004), no. 4, 827869.
 S. Canic, D. Lamponi, A. Mikelic, J. Tambaca,
Selfconsistent effective equations modeling blood flow in
mediumtolarge compliant arteries.
Multiscale Model. Simul. 3 (2005), no. 3, 559596.
 M. Feldman, S.Y. Ha, and M. Slemrod,
Exact selfsimilar solutions for the twodimensional plasmaion sheath system. J. Phys. A 38 (2005), no. 32, 71977204.
 G.Q. Chen, M. Torres,
Divergencemeasure fields, sets of finite perimeter, and conservation laws.
Arch. Ration. Mech. Anal. 175 (2005), no. 2, 245267.
 M. Feldman, S.Y. Ha, and M. Slemrod,
A geometric levelset formulation of a plasmasheath interface. Arch. Ration. Mech. Anal. 178 (2005), no. 1, 81123.
 G.Q. Chen, S.X. Chen, D. Wang, Z. Wang,
A multidimensional piston problem for the Euler equations
for compressible flow.
Discrete Contin. Dyn. Syst. 13 (2005), no. 2, 361383.
 V. Elling, T.P. Liu, The ellipticity principle for selfsimilar potential flows. J. Hyperbolic Differ. Equ. 2 (2005), no. 4, 909917.
 G.Q. Chen, M. Feldman,
Potential theory for shock reflection by a largeangle wedge.
Proc. Natl. Acad. Sci. USA 102 (2005), no. 43, 1536815372.
 S. Canic, B. Keyfitz, E. H. Kim,
Free boundary problems for nonlinear wave systems: Mach stems for interacting shocks. SIAM J. Math. Anal. 37 (2006), no. 6, 19471977.
 G.Q. Chen, Y. Zhang, D. Zhu,
Existence and stability of supersonic Euler flows past Lipschitz wedges. Arch. Ration. Mech. Anal. 181 (2006), no. 2, 261310.
 J. Hunter, Shorttime existence for scaleinvariant Hamiltonian waves. J. Hyperbolic Differ. Equ. 3 (2006), no. 2, 247267.
 C.S. Chou, C.W. Shu, High order residual distribution conservative finite difference WENO schemes for steady state problems on nonsmooth meshes. J. Comput. Phys. 214 (2006), no. 2, 698724.
 M. Feldman, S.Y. Ha, and M. Slemrod,
Selfsimilar isothermal irrotational motion for the Euler, EulerPoisson systems and the formation of the plasma sheath. J. Hyperbolic Differ. Equ. 3 (2006), no. 2, 233246.
 T. Li, D. Wang,
Blowup phenomena of solutions to the Euler equations for compressible fluid flow. J. Differential Equations 221 (2006), no. 1, 91101.
 Y. Zheng, Twodimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sin. Engl. Ser. 22 (2006), no. 2, 177210.
 G.Q. Chen, C. Dafermos, M. Slemrod, and D. Wang,
On twodimensional sonicsubsonic flow,
Commun. Math. Phys. 271 (2007), 635647.
 G.Q. Chen, M. Feldman,
Global solutions of shock reflection by largeangle wedges for potential flow,
to appear in Annals of Mathematics.
 G.Q. Chen, M. Slemrod, and D. Wang,
Vanishing viscosity method for transonic flow,
to appear in Archive for Rational Mechanics and Analysis.