On formation of singularity for non-isentropic Navier-Stokes equations without heat-conductivity

It is known that smooth solutions to the non-isentropic Navier-Stokes equations without heat-conductivity may lose their regularities in finite time in the presence of vacuum. However, in spite of the recent progress on such blowup phenomenon, it remain to give a possible blowup mechanism. In this paper, we present a simple continuation principle for such system, which asserts that the concentration of the density or the temperature occurs in finite time for a large class of smooth initial data, which is responsible for the breakdown of classical solutions. It also give an affirmative answer to a strong version of conjecture proposed by J.Nash in 1950s

The constant viscosity coefficients µ and λ satisfy the physical restrictions Positive constants c v , κ, and ν are respectively the heat capacity, the ratio of the heat conductivity coefficient over the heat capacity. The compressible Navier-Stokes system (1.1) consists of a set of equations describing compressible viscous heat-conducting flows. Indeed, the equations (1.1) 1 , (1.1) 2 , and (1.1) 3 respectively describe the conservation of mass, momentum, and energy.
There is a considerable body of literature on the multi-dimensional compressible Navier-Stokes system (1.1) by physicists and mathematicians because of its physical importance, complexity, rich phenomena, and mathematical challenges; see [2, 4, 5, 9, 13, 15-17, 20, 21, 25] and the references cited therein. However, many physically important and mathematically fundamental problems are still open due to the lack of smoothing mechanism and the strong nonlinearity. For example, although the local strong solutions to the compressible Navier-Stokes system (1.1) for general initial data with nonnegative density were respectively obtained by [2], whether the unique local strong solution can exist globally in time is an outstanding challenging open problem in contrast to the isentropic case [13].
In the presence of vacuum, as pointed out by Xin [25], non-isentropic Navier-Stokes equations without heat-conductivity will develop finite time singularity, see also reference [1]. Indeed, very recently, Xin-Yan [26] further proved that any classical solutions of viscous compressible fluids with or without heat conduction will blow up in finite time, as long as the initial data has an isolated mass group. Their results hold for the whole space and bounded domains, yet the blowup mechanism is not clarified. It is the main purpose if this paper to resolve this key issue. Theorem 1.1 reveals that the concentration of the density or the temperature must be responsible for the loss of regularity in finite time.
Although vacuum will lead to breakdown of smooth solutions in finite time , it is also important to study the mechanism of blowup and structure of possible singularities of general strong (or smooth) solutions to the compressible Navier-Stokes system.
The pioneering work can be traced to Serrin's criterion [22] on the Leray-Hopf weak solutions to the three-dimensional incompressible Navier-Stokes equations, which can be stated that if a weak solution u satisfies then it is regular.
Recently, Huang-Li-Xin [11] extended the Serrin's criterion (1.3) to the barotropic compressible Navier-Stokes equations and showed that if T * < ∞ is the maximal time of existence of a strong (or classical) solution (ρ, u), then with r and s as in (1.3). For more information on the blowup criteria of barotropic compressible flow, we refer to [7,11,12,14,23] and the references therein.
When it comes to the full compressible Navier-Stokes system (1.1), the problem is more complicated. In [18], Nash proposed a conjecture on the possible blowup of compressible heat-conductive flows. He wrote "This should give existence, smoothness, and continuation (in time) of flows, conditional on the non-appearance of certain gross type of singularity, such as infinities of temperature or density." Under the condition that λ < 7µ, (1.5) Fan-Jiang-Ou [3] obtained the following blowup criterion Recently, under just the physical restrictions (1.2), Huang-Li [8] and Huang-Li-Xin [12] established the following blowup criterion: where D(u) is the deformation tensor. Later, in the absence of vacuum, Sun-Wang-Zhang [24] established the following blowup criterion for bounded domains with positive heat-conductivity κ > 0 that provided that (1.2) and (1.5) hold true. As a consequence, Nash's conjecture is partially verified as [24] can't rule out the possibility of appearance of vacuum.
Recently, for κ > 0, we [10] establish a blowup criterion allowing initial vacuum, which is independent of temperature, as follows where r, s satisfy (1.3). As a matter of fact, the blowup criterion (1.7) further implies as long as (1.5) holds true. This makes Nash's conjecture as an immediately corollary for positive heat-conductivity flows. Our main goal in this paper is to give an affirmative answer to a strong version of Nash's conjecture without heat-conduction. We will assume that κ = 0, and without loss of generality, take c v = R = 1. The system (1.1) is reduced to (1.9) The system (1.9) is supplemented with the following initial conditions: (1.10) (ρ, u, P ) satisfies the far field condition: (ρ, u, P )(x, t) → (0, 0, 0) as |x| → ∞; (1.11) To state the main result, we will use the following notations and conventions.
Notations. For 1 ≤ p ≤ ∞ and integer k ≥ 0, the standard homogeneous and inhomogeneous Sobolev spaces in R 3 are denoted by: Denote byḟ The strong solutions to the Cauchy problem (1.9)-(1.11) are defined as follows.
Then the main result in this paper can be stated as follows: 12) and the compatibility conditions: . Let (ρ, u, P ) be the strong solution to the compressible Navier-Stokes system (1.9) in R 3 . If T * < ∞ is the maximal time of existence, then A few remarks are in order: Under the conditions of Theorem 1.1, the local existence of the strong solutions was guaranteed in [2]. Thus, the assumption T * makes sense.
Remark 1.2. The main contribution of Theorem 1.1 asserts that Nash's conjecture even holds for zero heat-conductivity flows. In the case κ = 0, the formation of singularity is only due to the concentration of either the density or temperature. In this sense, we give an affirmative answer to a strong version of Nash's conjecture. Remark 1.4. It's easy to prove a same continuation principle for two-dimensional problem without any restrictions on µ, λ. Since the proof is analogous and simpler, we omit it for simplicity.
We may also investigate the following different boundary conditions.
(1) Ω = R 3 and constantsρ,P ≥ 0, (ρ, u, P ) satisfies the far field condition either vacuum or non-vacuum: (1. 16) and initial condition Our next theorem asserts that Nash's conjecture also holds for different boundary conditions. Theorem 1.2. Let (ρ, u, P ) be the strong solution to the full compressible Navier-Stokes system (1.9) together with We now make some comments on the analysis of this paper.
Let (ρ, u, θ) be a strong solution described in Theorem 1.1. Suppose that (1.23) were false, that is, (1.25) One needs to show that Higher order derivatives estimates for above quantities then follow easily from above regularities.
Let's say a few words on the regularity criterion (1.23). In the absence of heatconductivity, the equation for the temperature changes its form from parabolic to hyperbolic type, thus resulting in loss of regularity benefiting from the smooth effect of heat dissipation. But fortunately, it enjoys a same nonlinear structure as the density equation. Since the methods in all previous works [? , 24] depend crucially on Hoff's a priori estimates. The main point is how to avoid terms involving density gradient in calculations. It turns out to be possible to treat the terms arising from pressure gradient. Some new ideas are needed to recover all the a priori estimates, that is, instead of the temperature θ and pressure P , we treat the total energy E = 1 2 |u| 2 + θ. Finally, the a priori estimates on both the L ∞ t L p x -norm of the density gradient and pressure gradient along with the L 1 t L ∞ x -norm of the velocity gradient can be obtained simultaneously by solving a logarithm Gronwall inequality based on a logarithm estimate (see Lemma 2.4) and the a priori estimates we have just derived.
The rest of the paper is organized as follows: In the next section, we collect some elementary facts and inequalities that will be needed later. The main result, Theorem 1.1, is proved in Section 3.

Preliminaries
In this section, we recall some known facts and elementary inequalities that will be used later.
We start with the standard energy estimate Lemma 3.1.
Next, a high energy estimate holds under the condition (1.25). Proof. It follows from (1.25) that Multiplying (1.1) 2 by q|u| q−2 u, and integrating over Ø, one obtains by using lemma 2.1 that
Also, θ is always non-negative before blowup time T * . Lemma 3.3. As long as θ 0 ≥ 0, it holds that Proof. The equation for P can be rewritten as Since the solution is smooth , we can always define particle path before blowup time. (3.11) Consequently, along particle path, one has d dt P (X(x, t), t) = −2P divu + F Hence, θ ≥ 0 follows immediately from (3.13).
Before proving Theorem 1.1, we state some a priori estimates under the condition (1.25).
Let E be the specific energy defined by (3.14) Let G, ω be the effective viscous flux, vorticity respectively given by Then, the momentum equations can be rewritten as Then, we derive the following crucial estimate on the L ∞ (0, T ; L 2 )-norm of ∇u.
Then, we will estimate the last term on the righthand side of (3.18). First, it follows from (1.1) that E satisfies It follows from (3.14) and (3.19) that (3.21) Cauchy's and Sobolev's inequalities yield that Integration by parts and recall G = (2µ + λ)divu − P also gives (3.23) In view of (3.16), which implies where Consequently, Gronwall's inequality together with Lemma 3.2 implies (3.17). This finishes the proof of Lemma 3.4.
Next lemma deals with ∇u. (3.28) Proof. Make use the factḟ to obtain that Multiplying the above equation byu and integrating over R 3 show that First, recalling that One can get after integration by parts and using the above equation that Integration by parts leads to (3.34) Similarly,  Note that ∇u 4 4 ≤ ∇u 2 ∇u 3 Substituting this estimate into (3.36) and once again recalling that we conclude by Gronwall's inequality that Finally, the following Lemma 3.6 will deal with the higher order estimates of the solutions which are needed to guarantee the extension of local strong solution to be a global one. Proof. In view of (3.38) and (3.28), one has Applying the standard L p -estimate to (3.16) gives which shows for some β ∈ (0, 1).

Outline of Theorem 1.2
The main idea is quite analogous to section 3.
We need only to redefine the effective flux G as G = (2µ + λ)divu − P +P , For any q ∈ [1, ∞) and under the condition (1.25), G satisfies G L q ≤ C ∇u L q + P L q + C (4.6) and ∇G L q + ω L q ≤ C ρu L q . (4.7) Then the higher regularity of the density, velocities and temperature can be obtained without difficulty.
Case III. Ω is a bounded domain.
Since there is no boundary condition for effective viscous flux. We will outline the proof of Lemma 3.4.
We then finish the proof of Theorem 1.2 for bounded domain by adapting a same procedure as Theorem 1.1 with the help of Lemma 2.4.