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Non-Euclidean geometry and differential equations

A. Popov (1996)

Banach Center Publications

In this paper a geometrical link between partial differential equations (PDE) and special coordinate nets on two-dimensional smooth manifolds with the a priori given curvature is suggested. The notion of G-class (the Gauss class) of differential equations admitting such an interpretation is introduced. The perspective of this approach is the possibility of applying the instruments and methods of non-Euclidean geometry to the investigation of differential equations. The equations generated by the...

Non-existence of global classical solutions to 1D compressible heat-conducting micropolar fluid

Jianwei Dong, Junhui Zhu, Litao Zhang (2024)

Czechoslovak Mathematical Journal

We study the non-existence of global classical solutions to 1D compressible heat-conducting micropolar fluid without viscosity. We first show that the life span of the classical solutions with decay at far fields must be finite for the 1D Cauchy problem if the initial momentum weight is positive. Then, we present several sufficient conditions for the non-existence of global classical solutions to the 1D initial-boundary value problem on [ 0 , 1 ] . To prove these results, some new average quantities are...

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