The Riemann problem for the transportation equations in gas dynamics

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“…with the initial data (1.4), the δ-shock wave type solution is defined as a measure-valued solution (see also [25]). …”

“…4, to solve the Cauchy problem for system (1.9), in the initial data (1.13) we introduce the initial velocity of singularity, and instead of the initial data (1.13), we use the initial data Formulas for the trajectory of a singularity φ(t) and for the coefficient e(t) of the δ-function in Theorem 4.3 and Corollary 4.4 coincide with the analogous formulas from [1], [16], [25] if we identify the velocity on the discontinuity line x = φ(t) in formula (1.12) with the phase velocity of nonlinear wave:…”

Vanishing viscosity limit for Riemann solutions to zero-pressure gas dynamics with flux perturbation

“…Weinan et al [7] discussed the existence of global weak solution and the behavior of such global solution with random initial data. The 1-D and 2-D Riemann problems were constructively solved by Sheng and Zhang [8], and a new kind of discontinuity, called delta shock wave, was found in the Riemann solutions. A delta shock wave is a generalization of an ordinary shock wave, on which at least one of the state variables may develop an extreme concentration in the form of a weighted Dirac delta function with the discontinuity as its support.…”

“…In addition, the shocks constructed by this method are physical ones, since they satisfy the entropy inequalities. Specially, in order to obtain the stability of the delta shock wave of (1.1), by using the vanishing viscosity method, Sheng and Zhang [8] considered the following regularized system: ⎧ ⎨ ⎩ ρ t + (ρu) x = 0, (ρu) t + (ρu 2 ) x = εtu xx , (1.5) where ε > 0 is a small parameter. All of the existence, uniqueness, and stability of solutions were investigated to viscous perturbations.…”

“…We propose to include this parameter in hope of investigating the effect of a flux approximation to the stability of the delta-shock and vacuum state solutions to the zero-pressure gas dynamics under viscosity approach. Obviously, in contrast to the previous works in [8,12,18], we here develop a viscosity approach which contains flux approximation in the considered systems. So this paper to some extent extends the results and proofs in [8].…”