# Backward self-similar solutions for compressible Navier–Stokes equations

@article{Germain2021BackwardSS, title={Backward self-similar solutions for compressible Navier–Stokes equations}, author={Pierre Germain and Tsukasa Iwabuchi and Tristan L{\'e}ger}, journal={Nonlinearity}, year={2021}, volume={34}, pages={868 - 893} }

This article is devoted to backward self-similar blow up solutions of the compressible Navier–Stokes equations with radial symmetry. We show that such solutions cannot exist if they either have finite energy, or satisfy appropriate smallness conditions. Furthermore, numerical simulations lead us to the conjecture that such solutions do not exist.

## One Citation

On Self-similar Solutions to Degenerate Compressible Navier–Stokes Equations

- Physics, Mathematics
- 2019

We study cavitating self-similar solutions to compressible Navier-Stokes equations with degenerate density-dependent viscosity. We prove both existence of expanders and non-existence of small…

## References

SHOWING 1-10 OF 39 REFERENCES

Self-similar solutions of the compressible Navier–Stokes equations

- Mathematics, Physics
- 2019

We construct forward self-similar solutions (expanders) for the compressible Navier-Stokes equations. Some of these self-similar solutions are smooth, while others exhibit a singularity do to…

Self-similar solutions for navier-stokes equations in

- Mathematics
- 1996

We construct self-similar solutions for three-dimensional incompressible Navier-Stokes equations, providing some examples of functional spaces where this can be done. We apply our results to a…

Forward Discretely Self-Similar Solutions of the Navier–Stokes Equations

- Mathematics
- 2014

Extending the work of Jia and Šverák on self-similar solutions of the Navier–Stokes equations, we show the existence of large, forward, discretely self-similar solutions.

Self-similar solutions to the isothermal compressible Navier–Stokes equations

- Mathematics
- 2006

We investigate the self-similar solutions to the isothermal compressible Navier-Stokes equations. The aim of this paper is to show that there exist neither forward nor backward self-similar solutions…

Forward Self-Similar Solutions of the Navier-Stokes Equations in the Half Space

- Mathematics
- 2014

For the incompressible Navier-Stokes equations in the 3D half space, we show the existence of forward self-similar solutions for arbitrarily large self-similar initial data.

Remarks on Self-Similar Solutions to the Compressible Navier-Stokes Equations of a 1D Viscous Polytropic Ideal Gas

- Physics
- 2008

This paper is concerned with the self-similar solutions to the compressible Navier-Stokes equations of a 1D viscous polytropic ideal gas. Our results show that there exist neither forward nor…

Self-Similar Solutions of the Compressible Flow in One-Space Dimension

- Mathematics, Computer ScienceJ. Appl. Math.
- 2013

It is proved that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy for the isentropic compressible fluids in one-space dimension.

On the Existence of Globally Defined Weak Solutions to the Navier—Stokes Equations

- Mathematics
- 2001

Abstract. We prove the existence of globally defined weak solutions to the Navier—Stokes equations of compressible isentropic flows in three space dimensions on condition that the adiabatic constant…

Local-in-space estimates near initial time for weak solutions of the Navier-Stokes equations and forward self-similar solutions

- Mathematics
- 2012

We show that the classical Cauchy problem for the incompressible 3d Navier-Stokes equations with (−1)-homogeneous initial data has a global scale-invariant solution which is smooth for positive…

On Leray's Self‐Similar Solutions of the Navier‐Stokes Equations Satisfying Local Energy Estimates

- Mathematics
- 1998

Abstract.This paper proves that Leray's self‐similar solutions of the three‐dimensional Navier‐Stokes equations must be trivial under very general assumptions, for example, if they satisfy local…