The scalar Riemann problem in two spatial dimensions: Piecewise smoothness of solutions and its breakdown
Consider the scalar quasilinear equation par. delta..mu..(t,x)/par. deltat+,,t,,/sup n//sub i=1/ par. deltaf/sub i/(..mu..(t,x))/par. deltax/sub i/=0, for n=1,2,f/sub i/element ofC/sup 2/: R..-->..R. For n=2, we define the two-dimensional Riemann problem and show the unique (in the sense of Kruzkov) solutions are piecewise smooth for f/sub 1/ identical to f/sub 2/ identical to f, f purely convex or having a single inflection point. A mechanism leading to a presumed loss of piecewise smoothness is presented for f having three or more inflection points. The analysis is based on a study of the generalization of the one-dimensional Riemann problem to allow for initial data having a finite number or jump discontinuities with constant data or rarefaction waves between jumps.
- Research Organization:
- Courant Institute of Mathematical Sciences, New York Univ., New York, NY 10012
- OSTI ID:
- 5635372
- Journal Information:
- SIAM J. Math. Anal.; (United States), Journal Name: SIAM J. Math. Anal.; (United States) Vol. 17:5; ISSN SJMAA
- Country of Publication:
- United States
- Language:
- English
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