• Bulletin of the Korean Mathematical Society
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• v.34 no.1
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• pp.43-49
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• 1997

In this paper, we study an error of vanishing viscosity method for the first-order Hamilton-Jacobi equations.

• Bulletin of the Korean Mathematical Society
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• v.61 no.3
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• pp.699-715
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• 2024

We study the Riemann problem for the pressureless Euler system with the source term depending on the time. By means of the variable substitution, two kinds of Riemann solutions including deltashock and vacuum are constructed. The generalized Rankine-Hugoniot relation and entropy condition of the delta-shock are clarified. Because of the source term, the Riemann solutions are non-self-similar. Moreover, we propose a time-dependent viscous system to show all of the existence, uniqueness and stability of solutions involving the delta-shock by the vanishing viscosity method.

• Bulletin of the Korean Mathematical Society
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• v.52 no.6
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• pp.1945-1962
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• 2015

This paper is mainly concerned with the formation of delta standing wave for the scalar conservation law with a linear flux function involving discontinuous coefficients by using the self-similar viscosity vanishing method. More precisely, we use the self-similar viscosity to smooth out the discontinuous coefficient such that the existence of approximate viscous solutions to the delta standing wave for the Riemann problem is established and then the convergence to the delta standing wave solution is also obtained when the viscosity parameter tends to zero. In addition, the Riemann problem is also solved with the standard method and the instability of Riemann solutions with respect to the specific small perturbation of initial data is pointed out in some particular situations.

• Journal of the Korean Mathematical Society
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• v.35 no.1
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• pp.85-114
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• 1998

We prove the existence of solutions of the Reimann problem for a system of conservation laws of mixed type using the method of vanishing viscosity term.