TY - JOUR
T1 - Selection problems of Z2 -periodic entropy solutions and viscosity solutions
AU - Soga, Kohei
PY - 2017/8/1
Y1 - 2017/8/1
N2 - Z2-periodic entropy solutions of hyperbolic scalar conservation laws and Z2-periodic viscosity solutions of Hamilton–Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton–Jacobi equations. Bessi (Commun Math Phys 235:495–511, 2003) investigated the convergence of approximate Z2-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural Z2-periodic solution with the aid of Aubry–Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which may depend on the approximation parameter.
AB - Z2-periodic entropy solutions of hyperbolic scalar conservation laws and Z2-periodic viscosity solutions of Hamilton–Jacobi equations are not unique in general. However, uniqueness holds for viscous scalar conservation laws and viscous Hamilton–Jacobi equations. Bessi (Commun Math Phys 235:495–511, 2003) investigated the convergence of approximate Z2-periodic solutions to an exact one in the process of the vanishing viscosity method, and characterized this physically natural Z2-periodic solution with the aid of Aubry–Mather theory. In this paper, a similar problem is considered in the process of the finite difference approximation under hyperbolic scaling. We present a selection criterion different from the one in the vanishing viscosity method, which may depend on the approximation parameter.
KW - 35L65
KW - 37J50
KW - 49L25
KW - 60G50
KW - 65M06
UR - http://www.scopus.com/inward/record.url?scp=85024367785&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85024367785&partnerID=8YFLogxK
U2 - 10.1007/s00526-017-1208-7
DO - 10.1007/s00526-017-1208-7
M3 - Article
AN - SCOPUS:85024367785
VL - 56
JO - Calculus of Variations and Partial Differential Equations
JF - Calculus of Variations and Partial Differential Equations
SN - 0944-2669
IS - 4
M1 - 119
ER -