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For the isentropic compressible fluids in one-space dimension, we prove that the Navier-Stokes equations with density-dependent viscosity have neither forward nor backward self-similar strong solutions with finite kinetic energy. Moreover, we obtain the same result for the nonisentropic compressible gas flow, that is, for the fluid dynamics of the Navier-Stokes equations coupled with a transport equation of entropy. These results generalize those in Guo and Jiang's work (2006) where the one-dimensional compressible fluids with constant viscosity are considered.

Self-similar solutions have attracted much attention in mathematical physics because understanding them is fundamental and important for investigating the well-posedness, regularity, and asymptotic behavior of differential equations in physics. Since the pioneering work of Leray [

In one-space dimension, the isentropic compressible fluid flow is governed by the Navier-Stokes equations:

When considering an ideal compressible gas flow, particularly in the thermodynamic analysis with exergy loss and entropy generation, both the viscosity and pressure rely on the entropy, so it is necessary to extend the nonisentropic fluid dynamics to include the transport of entropy (see [

Navier-Stokes equations enjoy a scaling property: if

Mellet and Vasseur [

The main result for the self-similar solutions of the isentropic compressible Navier-Stokes equations is as follows.

Assume that

There is no self-similar strong solution satisfying the global-energy estimate (

If there is a forward (backward) self-similar strong solution satisfying the local-energy estimate (

For the self-similar solutions of the coupled system of the nonisentropic compressible Navier-Stokes equations with an entropy transport equation, the main result is as follows.

Assume that

There is no self-similar strong solution satisfying the global-energy estimate (

If there is a forward (backward) self-similar strong solution satisfying the local-energy estimate (

Theorem

Any self-similar solution of (

If

From (

If

Similar to the proof of Lemma

If

Fix

If

Recalling the proofs of Lemmas

Now, Theorem

If

Lions [

Suppose that

The case of backward self-similar solutions can be proved similarly, so Theorem

The authors declare that there is no conflict of interests regarding the publication of this paper.

This work is partially supported by Zhejiang Provincial Natural Science Foundation of China (no. LQ13G030018) and National Natural Science Foundation of China (nos. 11001049 and 11226184).