Blow-up criteria for the full compressible Navier-Stokes equations involving temperature in Vishik Spaces

: In this paper, we consider the conditional regularity for the 3D incompressible Navier-Stokes equations in Vishik spaces. These results will be regarded an improvement of the results given by Huang-Li-Xin, (SIAM J. Math. Anal., 2011) and Jiu-Wang-Ye,(J. Evol. Equ., 2021).

We note that the condition (1.5) is in scaling invariant norm in the sense of (1.4) for the temperature.
Remark 1.2. Without the restriction λ < 3µ, in the case away from vacuum, through the argument in [9] and our proof, we obtain the similar results [9,Theorem 1.3] of what the authors in [9] says in Vishik space.
Next, we consider the full compressible Navier-Stokes equations without temperature.
where ρ, u, and P are the density, velocity and pressure respectively. The equation of state is given by The constants µ and λ are the shear viscosity and the bulk viscosity coefficients respectively. They satisfy the following physical restrictions: µ > 0, 3λ + 2µ ≥ 0. Through a similar scheme in Theorem 1.1, we also obtain the following result for the equations (1.6).
for some r ∈ (3, ∞) and the compatibility condition: If T * < ∞ is the maximal time of existence, then both

Notations and some auxiliary lemmas
We follow the notation of [6] and [9]. For 1 ≤ p ≤ ∞, L p (R 3 ) represents the usual Lebesgue .
A function f belongs to the homogeneous Sobolev spaces D k,l if u ∈ L 1 loc (R 3 ) : ∇ k u L l < ∞. C > 0 is an absolute constant which may be different from line to line unless otherwise stated in this paper. We also now introduce a Banach spaceV s p,σ,θ (R 3 ) which is larger than the homogeneous Besov space; see [10,17].
As mentioned in [20], we remind that the following continuous embeddings hold: In what follows, for simplicity, we write 3. Proof of Theorem 1.1 We will prove Theorem 1.1 by a contradiction argument. Therefore, we assume that Proof. From Lemma 2.3 and Lemma 3.1 in [9], we know that d dt and Multiplying the inequality (3.3) by (C + 1) and adding the result to the inequality (3.2), we have d dt For the second term in the right hand side of (3.4), we note that Now, let's control each term sequentially by Hölder's inequality, interpolation inequality(for the term II below), Sobolev embedding theorem, Berstein's inequality and Young's inequalities: (The term (I)): (3.6) (The term (III)): Summing up the estimates above, we have By similar above arguments, we get (3.8) Substituting (3.7) and (3.8) into (3.4), we obtain d dt where we used the fact that for a sufficiently large constant C > 0. Now, choosing N > 0 sufficiently large such that C2 −N 2 θ 2 L 2 ≤ 1 128 , the estimate (3.9) becomes d dt Then, Grönwall's inequality and (3.10) enables us to obtain that the desired results.
Proof of Theorem 1.1. In the proof in Theorem 1.1 in [9], as long as Lemma 3.2 in [9] is only replaced by Lemma 3.1 in present paper, the proof is completed.

Proof of Theorem 1.2
Let (ρ, u) be a strong solution to the problem (1.6)-(1.7) as described in Theorem 1.2. Then the standard energy estimate yields We first prove Theorem 1.2 by a contradiction argument. Otherwise, there exists some constant The first key estimate on ∇u will be given in the following lemma.
Proof. It follows from the momentum equations in (1.6) that wherev := v t + u · ∇v, G := (2µ + λ)divu − P(ρ), ω := ∇ × u are the material derivative of f, the effective viscous flux G and the vorticity ω, respectively. In particular, for the effective viscous flux, it is well-known that ∇G L p ≤ ρu L p , ∀p ∈ (1, +∞), and ∇G L 2 + ∇ω L 2 ≤ C( ρu t L 2 + ρu · ∇u L 2 ). (4.4) Multiplying the momentum equation (1.6) 2 by u t and integrating the resulting equation over R 3 gives From (1.6) 1 , we note that For the first term in the right hand side of (4.5), one has Substituting (4.6) into (4.5), we have d dt For the second term in the right hand side of (4.5), we have In a similar way to (3.5), we let control each term sequentially.
Proof of Theorem 1.2. In the proof in Theorem 1.1 in [6], as long as Lemma 3.1 in [6] is only replaced by Lemma 4.1 in our paper, the proof is completed.

Appendix
For the convenience of the reader, we give the proof for (4.6), given in [6].

Discussion
Our result is focused on the full compressible Navier-Stokes equationss. However, it is believed that our results can be expanded in various ways for the coupled equations or system. In this regard, we think of it as a future study and intend to produce more meaningful results.

Conclusions
The current paper results are Blow-up criteria for solutions in Vishik Space which is a weaker space to Besov space and Lebesgue space. It seems to be a meaningful result in this regard, and it is new.