scholarly journals Homogeneous Euler equation: blow-ups, gradient catastrophes and singularity of mappings

2021 ◽  
Author(s):  
Boris G. Konopelchenko ◽  
Giovanni Ortenzi
Keyword(s):  
Euler Equation ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Dimensional Case ◽  
Linear Combinations ◽  
First Occurrence ◽  
Different Dimensions ◽  
Characteristic Features

Abstract The paper is devoted to the analysis of the blow-ups of derivatives, gradient catastrophes and dynamics of mappings of ℝn → ℝn 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 gradient catastrophe are among of them. It is shown that the potential solutions of the Euler equations exhibit blow-up derivatives in any dimenson n. Several concrete examples in two- and three-dimensional cases are analysed. Properties of ℝnu → ℝ nx mappings defined by the hodograph equations are studied, including appearance and disappearance of their singularities.

scholarly journals Formation and construction of a shock wave for 3-D compressible Euler equations with the spherical initial data

2004 ◽  
Vol 175 ◽  
pp. 125-164 ◽  
Author(s):  
Huicheng Yin
Keyword(s):  
Shock Wave ◽  
Initial Data ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Lipschitz Continuous ◽  
Compressible Euler ◽  
Nondegeneracy Conditions ◽  
Weak Entropy Solution

AbstractIn this paper, the problem on formation and construction of a shock wave for three dimensional compressible Euler equations with the small perturbed spherical initial data is studied. If the given smooth initial data satisfy certain nondegeneracy conditions, then from the results in [22], we know that there exists a unique blowup point at the blowup time such that the first order derivatives of a smooth solution blow up, while the solution itself is still continuous at the blowup point. From the blowup point, we construct a weak entropy solution which is not uniformly Lipschitz continuous on two sides of a shock curve. Moreover the strength of the constructed shock is zero at the blowup point and then gradually increases. Additionally, some detailed and precise estimates on the solution are obtained in a neighbourhood of the blowup point.

Exact, infinite energy, blow-up solutions of the three-dimensional Euler equations

Nonlinearity ◽  
2003 ◽  
Vol 16 (5) ◽  
pp. 1823-1831 ◽  
Author(s):  
J D Gibbon ◽  
D R Moore ◽  
J T Stuart
Keyword(s):  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Infinite Energy

Effects of signal-dependent motilities in a Keller–Segel-type reaction–diffusion system

2017 ◽  
Vol 27 (09) ◽  
pp. 1645-1683 ◽  
Author(s):  
Youshan Tao ◽  
Michael Winkler
Keyword(s):  
Initial Data ◽  
Weak Solutions ◽  
Blow Up ◽  
Three Dimensional ◽  
Reaction Diffusion ◽  
Large Data ◽  
Dimensional Case ◽  
Stripe Pattern ◽  
Bounded Convex Domain ◽  
Flux Boundary Conditions

This work considers the Keller–Segel-type parabolic system [Formula: see text] in a smoothly bounded convex domain [Formula: see text], [Formula: see text], under no-flux boundary conditions, which has recently been proposed as a model for processes of stripe pattern formation via so-called “self-trapping” mechanisms. In the two-dimensional case, in stark contrast to the classical Keller–Segel model in which large-data solutions may blow up in finite time, for all suitably regular initial data the associated initial value problem is seen to possess a globally-defined bounded classical solution, provided that the motility function [Formula: see text] is uniformly positive. In the corresponding higher-dimensional setting, it is shown that certain weak solutions exist globally, where in the particular three-dimensional case this solution actually is bounded and classical if the initial data are suitably small in the norm of [Formula: see text]. Finally, if still [Formula: see text] but merely the physically interpretable quantity [Formula: see text] is appropriately small, then the above-weak solutions are proved to become eventually smooth and bounded.

scholarly journals The blowup of solutions for 3-D axisymmetric compressible Euler equations

1999 ◽  
Vol 154 ◽  
pp. 157-169 ◽  
Author(s):  
Huicheng Yin ◽  
Qingjiu Qiu
Keyword(s):  
Asymptotic Expansion ◽  
Initial Data ◽  
Euler Equations ◽  
Blow Up ◽  
Three Dimensional ◽  
Compressible Euler Equations ◽  
Classical Solutions ◽  
Complete Asymptotic Expansion ◽  
Blowup Of Solutions ◽  
Compressible Euler

AbstractIn this paper, for three dimensional compressible Euler equations with small perturbed initial data which are axisymmetric, we prove that the classical solutions have to blow up in finite time and give a complete asymptotic expansion of lifespan.

Analysis of three-dimensional aerospace configurations using the Euler equations

1989 ◽  
Author(s):  
N. KROLL ◽  
C. ROSSOW ◽  
S. SCHERR ◽  
J. SCHOENE ◽  
G. WICHMANN
Keyword(s):  
Euler Equations ◽  
Three Dimensional
Sign in / Sign up

Export Citation Format

Share Document