Structures of non-classical discontinuities in solutions of hyperbolic systems of equations. (English. Russian original) Zbl 1490.74064
Russ. Math. Surv. 77, No. 1, 47-79 (2022); translation from Usp. Mat. Nauk 77, No. 1, 55-90 (2022).
Summary: Discontinuity structures in solutions of a hyperbolic system of equations are considered. The system of equations has a rather general form and, in particular, can describe the longitudinal and torsional non-linear waves in elastic rods in the simplest setting and also one-dimensional waves in unbounded elastic media. The properties of discontinuities in solutions of these equations have been investigated earlier under the assumption that only the relations following from the conservation laws for the longitudinal momentum and angular momentum about the axis of the rod and the displacement continuity condition hold on the discontinuities. The shock adiabat has been studied. This paper deals with stationary discontinuity structures under the assumption that viscosity is the main governing mechanism inside the structure. Some segments of the shock adiabat are shown to correspond to evolutionary discontinuities without structure. It is also shown that there are special discontinuities on which an additional relation must hold, which arises from the condition that a discontinuity structure exists. The additional relation depends on the processes in the structure. Special discontinuities satisfy evolutionary conditions that differ from the well-known Lax conditions. Conclusions are discussed, which can also be of interest in the case of other systems of hyperbolic equations.
MSC:
| 74J40 | Shocks and related discontinuities in solid mechanics |
| 74J30 | Nonlinear waves in solid mechanics |
| 74K10 | Rods (beams, columns, shafts, arches, rings, etc.) |
| 35Q74 | PDEs in connection with mechanics of deformable solids |
| 35L67 | Shocks and singularities for hyperbolic equations |
Keywords:
shock adiabat; nonlinear wave; elastic rod; non-classical discontinuity; vanishing viscosityReferences:
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