Multidimensional delta-shock waves and the transportation and concentration processes
| dc.creator | Shelkovich, V. M. | |
| dc.date | 2008-03-25 | |
| dc.date.accessioned | 2026-07-07T09:28:18Z | |
| dc.date.available | 2026-07-07T09:28:18Z | |
| dc.description | {\it $δ$-Shock wave type solutions} in the multidimensional system of conservation laws $$ ρ_t + \nabla\cdot(ρF(U))=0, \qquad (ρU)_t + \nabla\cdot(ρN(U))=0, \quad x\in \bR^n, $$ are studied, where $F=(F_j)$ is a given vector field, $N=(N_{jk})$ is a given tensor field, $F_j, N_{kj}:\bR^n \to \bR$, $j,k=1,...,n$; $ρ(x,t)\in \bR$, $U(x,t)\in \bR^n$. The well-known particular cases of this system are zero-pressure gas dynamics in a standard form $$ ρ_t + \nabla\cdot(ρU)=0, \quad (ρU)_t + \nabla\cdot(ρU\otimes U)=0, $$ and in the relativistic form $$ ρ_t + \nabla\cdot(ρC(U))=0, \quad (ρU)_t + \nabla\cdot(ρU\otimes C(U))=0, $$ where $C(U)=\frac{c_0U}{\sqrt{c_0^2+|U|^2}}$, $c_0$ is the speed of light. We introduce the integral identities which constitute definition of $δ$-shocks for the above systems and using this definition derive the Rankine--Hugoniot conditions for curvilinear $δ$-shocks. We show that $δ$-shocks are connected with {\em transportation processes and concentration processes} and derive the $δ$-shock balance laws describing mass and momentum transportation between the volume outside the wave front and the wave front. In the case of zero-pressure gas dynamics the transportation process is the concentration process. We also prove that energy of the volume outside the wave front and total energy are {\em nonincreasing quantities}. The possibility of the {\em effect of kinematic self-gravitation} and the {\em effect of dimensional bifurcations of $δ$-shock} in zero-pressure gas dynamics are discussed. | |
| dc.identifier | https://arxiv.org/abs/0803.3549 | |
| dc.identifier | http://arxiv.org/abs/0803.3549 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/157403 | |
| dc.subject | Analysis of PDEs | |
| dc.subject | Mathematical Physics | |
| dc.subject | 35L65 (Primary); 35L67, 76L05 (Secondary) | |
| dc.title | Multidimensional delta-shock waves and the transportation and concentration processes | |
| dc.type | text |