Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings
- 29 December 2021
- journal article
- research article
- Published by IOP Publishing in Journal of Physics A: Mathematical and Theoretical
- Vol. 55 (3), 035203
- https://doi.org/10.1088/1751-8121/ac42aa
Abstract
The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes (GCs) and dynamics of mappings of R-n -> R-n associated with the n-dimensional homogeneous Euler equation. Several characteristic features of the multi-dimensional case (n > 1) are described. Existence or nonexistence of blow-ups in different dimensions, boundness of certain linear combinations of blow-up derivatives and the first occurrence of the GC are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimension n. Several concrete examples in two- and three-dimensional cases are analysed. Properties of R-(u) under bar(n)-> R-(x) under bar(n) mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.Keywords
Funding Information
- European Union Marie Sklodowska-Curie Grant IPaDEGAN (778010)
This publication has 17 references indexed in Scilit:
- On the dispersionless Kadomtsev–Petviashvili equation inn+1 dimensions: exact solutions, the Cauchy problem for small initial data and wave breakingJournal of Physics A: Mathematical and Theoretical, 2011
- On the solutions of the dKP equation: the nonlinear Riemann Hilbert problem, longtime behaviour, implicit solutions and wave breakingJournal of Physics A: Mathematical and Theoretical, 2008
- Burgers turbulencePhysics Reports, 2007
- Singular sector of the Burgers Hopf hierarchy and deformations of hyperelliptic curvesJournal of Physics A: General Physics, 2002
- Two-Dimensional Riemann Problem for Scalar Conservation LawsJournal of Differential Equations, 2002
- General solutions of the Monge-Ampère equation in n-dimensional spaceJournal of Geometry and Physics, 1995
- Formation of singularities on the free surface of an ideal fluidPhysical Review E, 1994
- The large-scale structure of the universe: Turbulence, intermittency, structures in a self-gravitating mediumReviews of Modern Physics, 1989
- ON MATRICES DEPENDING ON PARAMETERSRussian Mathematical Surveys, 1971
- On Singularities of Mappings of Euclidean Spaces. I. Mappings of the Plane into the PlaneAnnals of Mathematics, 1955