The invariant solution with blow-up for the gas dynamics equations from one-parameter three-dimensional subalgebra consisting of space, Galilean and pressure translations

Dilara Siraeva; Dilara Siraeva

doi:10.3934/math.2024006

AIMS Mathematics

2024, Volume 9, Issue 1: 89-98. doi: 10.3934/math.2024006

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The invariant solution with blow-up for the gas dynamics equations from one-parameter three-dimensional subalgebra consisting of space, Galilean and pressure translations

Dilara Siraeva ^,

Mavlyutov Institute of Mechanics–Subdivision of the Ufa Federal Research Centre of the Russian Academy of Sciences, 71 Prospekt Oktyabrya St, Ufa, Republic of Bashkortostan, Russia

Received: 14 July 2023 Revised: 09 November 2023 Accepted: 17 November 2023 Published: 27 November 2023
MSC : 76M60, 35B06

In this paper, new results were presented on the symmetry reduction of gas dynamics system of partial differential equations following the general framework of Lev Ovsyannikov's article "The "podmodeli" program, Gas dynamics." We considered the gas dynamics equations with an equation of state prescribing the pressure as the sum of density function and entropy function. This system has a 12-dimensional Lie algebra and we considered its certain three-dimensional subalgebra generated by space translations, Galilean translations and pressure translation. For this subalgebra, the symmetry reduction of the original system leads to a system of ordinary differential equations. We obtained a family of exact solutions for this system, which describes the motion of particles with a linear velocity field and non-homogeneous deformation in the 3D-space. For these solutions, the trajectories of all points are either parabolas or rays. At $ t = 0 $ an instantaneous collapse occurs when all of the particles accumulate in a plane with infinitely many particles at every point of the plane. For a fixed period of time, the particles were emitted from the same point on a plane and ended up on the same line. The gas motion was vortex. A one-dimensional subalgebra embedded into three-dimensional subalgebra was considered. The invariants were written in a consistent form. It was shown that the submodel of rank one was embedded in the submodel of rank three.
- gas dynamics equations,
- Lie algebra,
- invariant,
- submodel,
- exact solution
Citation: Dilara Siraeva. The invariant solution with blow-up for the gas dynamics equations from one-parameter three-dimensional subalgebra consisting of space, Galilean and pressure translations[J]. AIMS Mathematics, 2024, 9(1): 89-98. doi: 10.3934/math.2024006

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Abstract

In this paper, new results were presented on the symmetry reduction of gas dynamics system of partial differential equations following the general framework of Lev Ovsyannikov's article "The "podmodeli" program, Gas dynamics." We considered the gas dynamics equations with an equation of state prescribing the pressure as the sum of density function and entropy function. This system has a 12-dimensional Lie algebra and we considered its certain three-dimensional subalgebra generated by space translations, Galilean translations and pressure translation. For this subalgebra, the symmetry reduction of the original system leads to a system of ordinary differential equations. We obtained a family of exact solutions for this system, which describes the motion of particles with a linear velocity field and non-homogeneous deformation in the 3D-space. For these solutions, the trajectories of all points are either parabolas or rays. At $ t = 0 $ an instantaneous collapse occurs when all of the particles accumulate in a plane with infinitely many particles at every point of the plane. For a fixed period of time, the particles were emitted from the same point on a plane and ended up on the same line. The gas motion was vortex. A one-dimensional subalgebra embedded into three-dimensional subalgebra was considered. The invariants were written in a consistent form. It was shown that the submodel of rank one was embedded in the submodel of rank three.

References

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