Perturbational Blowup Solutions to the 1-dimensional Compressible Euler Equations

Abstract

We study the construction of analytical non-radially solutions for the 1-dimensional compressible adiabatic Euler equations in this article. We could design the perturbational method to construct a new class of analytical solutions. In details, we perturb the linear velocity:% \begin{equation} u=c(t)x+b(t) \end{equation} and substitute it into the compressible Euler equations. By comparing the coefficients of the polynomial, we could deduce the corresponding functional differential system of $(c(t),b(t),{\rho }^{\gamma -1}(0,t)).$ Then by skillfully applying the Hubble's transformation: \begin{equation} c(t)=\frac{\dot{a}(t)}{a(t)}, \end{equation} the functional differential equations can be simplified to be the system of $(a(t),b(t),{\rho }^{\gamma -1}(0,t))$. After proving the existence of the corresponding ordinary differential equations, a new class of blowup or global solutions can be shown. Here, our results fully cover the previous known ones by choosing $b(t)=0$.

11 pages, Key Words: Euler Equations, Navier-Stokes Equations, Perturbation, Linear Velocity, Perturbational Method, Non-Radial Symmetry, Construction of Solutions, Global Solutions, Blowup Solutions, Free Boundary

References in corpus (3)

• Analytical Blowup Solutions to the Pressureless Navier-Stokes-Poisson Equations with Density-dependent Viscosity in R^N
• Self-Similar Blowup Solutions to the 2-Component Camassa-Holm Equations
• Perturbational Blowup Solutions to the 2-Component Camassa-Holm Equations

Cited by in corpus (5)

• Self-Similar Solutions with Elliptic Symmetry for the Compressible Euler and Navier-Stokes Equations in R^{N}
• Vortical and Self-similar Flows of 2D Compressible Euler Equations
• Exact, Rotational, Infinite Energy, Blowup Solutions to the 3-Dimensional Euler Equations
• Perturbational Blowup Solutions to the 2-Component Camassa-Holm Equations
• Drifting Solutions with Elliptic Symmetry for the Compressible Navier-Stokes Equations with Density-dependent Viscosity