NSF FOCUSED RESEARCH GROUP (FRG) PROJECT, 2003-2007:

Multi-Dimensional 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 one-dimensional fluid flows are rather well understood, the general theory for multi-dimensional flows is comparatively mathematically underdeveloped. The goal of this collaborative research is to advance the mathematical understanding of the multi-dimensional 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

Suncica Canic, University of Houston
Gui-Qiang Chen, Northwestern University
Constantine Dafermos, Brown University
John Hunter, University of California - Davis
Tai-Ping Liu, Stanford University
Chi-Wang Shu, Brown University
Marshall Slemrod, University of Wisconsin - Madison
Dehua Wang, University of Pittsburgh
Yuxi Zheng, Pennsylvania State University

Some pictures.

SEMESTERS OF EMPHASIS, WORKSHOPS, AND SUMMER SCHOOL

Fall 2003: Semester of emphasis and workshop, University of Pittsburgh.
- Topics Course: Topics in Applied Mathematics
- Workshop on Multi-Dimensional Euler Equations and Conservation Laws: November 6-9, 2003.
Fall 2004: Semester of emphasis and workshop, Stanford University.
- FRG meeting and Summer Workshop on Kinetic Theory and Conservation Law, July 17-19 and July 20-31, 2004
Summer 2005: Summer school, University of Wisconsin - Madison.
- Madison Workshop on Multi-dimensional Hyperbolic Conservation Laws, June 8-12, 2005.
Spring 2006: Semester of emphasis and workshop, University of Houston.
- The Houston FRG Workshop on Multi-dimensional Hyperbolic Conservation Laws, March 1-5, 2006.
Summer 2007: Workshop, Stanford University.
- Workshop on Kinetic Theory and Conservation Laws, June 26-July 5, 2007.

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, 461-494
S. Canic, E. H. Kim, A class of quasilinear degenerate elliptic problems. J. Differential Equations 189 (2003), no. 1, 71-98.
C. M. Dafermos, Entropy for hyperbolic conservation laws. Entropy, 107-120, 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, 293-316.
T.-P. Liu, S.-H. Yu, Energy method for equations in gas dynamics. Hyperbolic problems: theory, numerics, applications, 53-60, Springer, Berlin, 2003.
S.-Y. Ha and M. Slemrod, Global existence of plasma ion-sheaths and their dynamics, Comm. Math. Phys. 238 (2003), no. 1-2, 149-186.
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. 3-4, 795-816.
Y. Zheng, A global solution to a two-dimensional Riemann problem involving shocks as free boundaries. Acta Math. Appl. Sin. Engl. Ser. 19 (2003), no. 4, 559-572.
J. Ryan, C.-W. Shu, On a one-sided post-processing technique for the discontinuous Galerkin methods. Methods Appl. Anal. 10 (2003), no. 2, 295-307.
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, 310-356.
T.-P. Liu, S.-H. Yu, Boltzmann equation: micro-macro decompositions and positivity of shock profiles. Comm. Math. Phys. 246 (2004), no. 1, 133-179.
J. Hunter, A. Tesdall, Weak shock reflection. A celebration of mathematical modeling, 93--112, 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, 827--869.
S. Canic, D. Lamponi, A. Mikelic, J. Tambaca, Self-consistent effective equations modeling blood flow in medium-to-large compliant arteries. Multiscale Model. Simul. 3 (2005), no. 3, 559--596.
M. Feldman, S.-Y. Ha, and M. Slemrod, Exact self-similar solutions for the two-dimensional plasma-ion sheath system. J. Phys. A 38 (2005), no. 32, 7197--7204.
G.-Q. Chen, M. Torres, Divergence-measure fields, sets of finite perimeter, and conservation laws. Arch. Ration. Mech. Anal. 175 (2005), no. 2, 245--267.
M. Feldman, S.-Y. Ha, and M. Slemrod, A geometric level-set formulation of a plasma-sheath interface. Arch. Ration. Mech. Anal. 178 (2005), no. 1, 81--123.
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, 361--383.
V. Elling, T.-P. Liu, The ellipticity principle for self-similar potential flows. J. Hyperbolic Differ. Equ. 2 (2005), no. 4, 909--917.
G.-Q. Chen, M. Feldman, Potential theory for shock reflection by a large-angle wedge. Proc. Natl. Acad. Sci. USA 102 (2005), no. 43, 15368--15372.
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, 1947--1977.
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, 261--310.
J. Hunter, Short-time existence for scale-invariant Hamiltonian waves. J. Hyperbolic Differ. Equ. 3 (2006), no. 2, 247--267.
C.-S. Chou, C.-W. Shu, High order residual distribution conservative finite difference WENO schemes for steady state problems on non-smooth meshes. J. Comput. Phys. 214 (2006), no. 2, 698--724.
M. Feldman, S.-Y. Ha, and M. Slemrod, Self-similar isothermal irrotational motion for the Euler, Euler-Poisson systems and the formation of the plasma sheath. J. Hyperbolic Differ. Equ. 3 (2006), no. 2, 233--246.
T. Li, D. Wang, Blowup phenomena of solutions to the Euler equations for compressible fluid flow. J. Differential Equations 221 (2006), no. 1, 91--101.
Y. Zheng, Two-dimensional regular shock reflection for the pressure gradient system of conservation laws. Acta Math. Appl. Sin. Engl. Ser. 22 (2006), no. 2, 177--210.
G.-Q. Chen, C. Dafermos, M. Slemrod, and D. Wang, On two-dimensional sonic-subsonic flow, Commun. Math. Phys. 271 (2007), 635-647.
G.-Q. Chen, M. Feldman, Global solutions of shock reflection by large-angle 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.