Open Access

Blow-up criteria for a two-fluid model of the truncated Euler equations

Journal of Inequalities and Applications20132013:461

https://doi.org/10.1186/1029-242X-2013-461

Received: 4 March 2013

Accepted: 18 September 2013

Published: 7 November 2013

Abstract

In this paper, we study a two-fluid model of the truncated Euler equations with partial viscosity. We obtain new blow-up criteria for a solution of the system in terms of vorticity in the homogeneous Besov space B ˙ , 1 .

Keywords

blow-up criteriontruncated Euler equationsBesov space

1 Introduction and main results

In this paper, we are concerned with the regularity of the following two-fluid model of the truncated Euler equations with partial viscosity:
{ u t Δ u + u u + p = θ e 3 , ( x ; t ) R 3 × ( 0 ; 1 ) , θ t + u θ = | u | 2 , u = 0 , u ( x , 0 ) = u 0 , θ ( x , 0 ) = θ 0 ,
(1.1)

where u = ( u 1 ( x , t ) , u 2 ( x , t ) , u 3 ( x , t ) ) denotes the fluid velocity vector field, p = p ( x , t ) is the scalar pressure, e 3 = ( 0 , 0 , 1 ) T , while u 0 and θ 0 are given initial velocity and initial temperature, respectively, with u 0 = 0 . For more detailed background of the system, we refer the readers to [1] and the references therein. When θ = 0 , the system reduces to the well-known Navier-Stokes equations.

It is well known that the question of global existence or finite time blow-up of smooth solutions for the 3D incompressible Navier-Stokes equations is still unsolved. This challenging problem has attracted significant attention. Therefore, it is interesting to study the blow-up criterion of the solutions for system (1.1). Recently, Sun and Fan [2] first proved the following blow-up criterion for system (1.1):
curl u L 1 ( 0 , T ; B ˙ , 0 ( R 3 ) ) , u L 2 ( 0 , T ; B ˙ , 0 ( R 3 ) ) .

Our purpose in this paper is to establish blow-up criteria of a strong solution for the two-fluid model of the truncated Euler equations in terms of vorticity in the homogeneous Besov space B ˙ , 1 . The main difficulty is without viscosity in the second equation for system (1.1).

Now we state our main results as follows.

Theorem 1.1 Let ( u 0 , θ 0 ) H 2 ( R 3 ) , ( u ( , t ) , θ ( , t ) ) be a strong solution to system (1.1) with the initial data ( u 0 , θ 0 ) for 0 t < T . If the solution u satisfies the following condition:
curl u L 2 ( 0 , T ; B ˙ , 1 ) ,
(1.2)

then the solution ( u , θ ) can be extended smoothly beyond t = T .

2 Preliminaries and lemmas

Throughout this paper, we introduce some function spaces, notations and important inequalities.

Let e t denote the heat semigroup defined by
e t f = K t f , K t = ( 4 π t ) 3 2 exp ( | x | 2 4 t )

for t > 0 and x R 3 , where denotes the convolution of functions defined on R 3 .

We now recall the definition of the homogeneous Besov space with negative indices B ˙ , α on R 3 and the homogeneous Sobolev space H ˙ q α of exponent α > 0 . It is known (p.192 of Ref. [3]) that f S ( R 3 ) belongs to B ˙ , α if and only if e t Δ f L for all t > 0 and t α 2 e t Δ f L ( 0 , ; L ) . The norm of B ˙ , α is defined, up to equivalence, by
f B ˙ , α = sup t > 0 ( t α 2 e t Δ f ) .
We introduce now the homogeneous Sobolev space H ˙ q α ( R 3 ) , which is defined as the set of functions f L r ( R 3 ) , 1 r = 1 q α 3 such that ( Δ ) α 2 f L q ( R 3 ) . This space is endowed with the norm
f H ˙ q α = ( Δ ) α 2 f L q ,

and when q = 2 , we just let H ˙ 2 α ( R 3 ) = H ˙ α ( R 3 ) .

Lemma 2.1 [4]

Let 1 < p < q < and s = α ( q p 1 ) > 0 . Then there exists a constant depending only on α, p and q such that the estimate
f L q C ( Δ ) s 2 f L p p q f B ˙ , α 1 p q
(2.1)

holds for all f H ˙ p α ( R 3 ) B ˙ , α ( R 3 ) .

In particular, for s = 1 , p = 2 and q = 4 , we get α = 1 and
f L 4 C f H ˙ 1 1 2 f B ˙ , 1 1 2 .
(2.2)

Lemma 2.2 [5]

For any function f W 1 , s ( R 3 ) ( s 1 ), and r 1 , we have
f L γ C f L 2 1 α f L 2 α ,
(2.3)
where
α = 1 r 1 γ 1 3 1 s 1 r

and C is a constant independent of f.

3 Proof of the main results

In this section, we prove Theorem 1.1.

Proof of Theorem 1.1 First, we multiply the both sides of the first equation of (1.1) by u and the second equation of (1.1) by θ, respectively; after integration by parts over R 3 , we get
1 2 d d t u 2 d x + | u | 2 d x | u θ | d x ,
(3.1)
1 2 d d t θ 2 d x = θ | u | 2 d x .
(3.2)
Next, we multiply the both sides of the first equation of (1.1) by ( Δ u ) ; after integration by parts over R 3 , we get
1 2 d d t | u | 2 d x + | Δ u | 2 d x u u Δ u d x + | Δ u | θ d x .
(3.3)
Using the Young inequality and adding (3.2) to (3.3), we obtain
d d t ( | u | 2 d x + θ 2 d x ) + | Δ u | 2 d x u u Δ u d x + θ | u | 2 d x + θ 2 d x .
(3.4)
It is easy to see that
u u Δ u d x = i , k k u i u k u d x u L 2 u u L 2 u L 2 u L 4 2 u L 2 u B ˙ , 1 Δ u L 2 C u L 2 2 × u B ˙ , 1 2 + 1 2 Δ u L 2 2 ,
(3.5)

where we use the Hölder inequality, the Young inequality and (2.2).

Similarly,
θ | u | 2 d x θ L 2 u u L 2 θ L 2 u L 4 2 θ L 2 u B ˙ , 1 Δ u L 2 C θ L 2 2 × u B ˙ , 1 2 + 1 2 Δ u L 2 2 .
(3.6)
Putting (3.5) and (3.6) into (3.4) and applying the Gronwall inequality yields
u L ( 0 , T ; H 1 ) L 2 ( 0 , T ; H 2 ) , θ L ( 0 , T ; L 2 ) .
Multiplying the both sides of the second equation of (1.1) by | θ | p 2 θ ( 2 < p < ), after integration by parts over R 3 , we deduce that
1 p d d t θ p d x | θ | p 1 | u | 2 d x θ L p p 1 u L 2 p 2 θ L p p 1 u H 2 2 ,
which implies that
θ L ( 0 , T ; L p ) ( 2 < p < ) .
(3.7)
Taking curl on the both sides of the first equation of (1.1), we obtain the following vorticity formulation for the vorticity field ω = curl u :
t ω + u ω Δ ω = curl ( θ e 3 ) + ω u ,
(3.8)
and then multiplying the both sides of the vorticity equation by | ω | p 2 ω ( p > 2 ), after integration by parts over R 3 , we infer that
d d t v 2 d x + | v | 2 d x C | θ | | | ω | p 2 ω | d x + C u | ω | p d x C | θ | | v | p 2 p | v | d x + C u | ω | p 2 | ω | p 2 d x C θ L 4 p p + 2 v L 4 1 2 p v L 2 + C u L 2 v L 4 2 C v L 4 1 2 p v L 2 + C v L 2 1 2 v L 2 3 2 C v L 2 1 2 1 p v L 2 3 2 1 p + C v L 2 1 2 v L 2 3 2 C v L 2 2 + 1 2 v L 2 2 ,
(3.9)
where v : = | ω | p 2 . Using the Gronwall inequality, we obtain
ω L ( 0 , T ; L p ) ( p > 2 ) ,
which implies that
u L ( 0 , T ; L p ) ( p > 2 ) .
(3.10)
On the other hand, since u is a solution of the Stokes system
u t Δ u + p = u u + θ e 3 L ( 0 , T ; L p ) ,
it follows from the W 2 , p -theory of the Stokes system that
u L p ( 0 , T ; W 2 , p ) ( p > 2 ) .
(3.11)
So,
u L p ( 0 , T ; L ) ( p > 2 ) .
(3.12)
Differentiating the second equation of (1.1) with respect to x i ( 1 i 3 ) and multiplying the resulting equation by i θ , and then integrating by parts over R 3 , we get
1 2 d d t | θ | 2 d x C | u | | θ | 2 d x + C | u | | Δ u | | θ | d x C u L θ L 2 2 + C u L Δ u L 2 θ L 2 .
By (3.12) and the Gronwall inequality, we have
θ L ( 0 , T ; H 1 ) .
(3.13)
We multiply the both sides of the equation of (3.8) by ( Δ ω ) ; after integration by parts over R 3 , we get
1 2 d d t | ω | 2 d x + | Δ ω | 2 d x | u | | ω | | Δ ω | d x + | Δ ω | | θ | d x u L 4 Δ ω L 2 ω L 4 + θ L 2 Δ ω L 2 C Δ ω L 2 1 2 ω L 2 3 2 + C Δ ω L 2 C ω L 2 2 + 1 2 Δ ω L 2 2 + C ,
the Gronwall inequality yields
u L ( 0 , T ; H 2 ) L 2 ( 0 , T ; H 3 ) .
(3.14)
In the following calculations, we use the following commutator estimate and bilinear estimate [6]:
Λ s ( f g ) f Λ s g L p ( f L p 1 Λ s 1 g L q 1 + Λ s f L p 2 g L q 2 ) ,
(3.15)
Λ s ( f g ) L p ( f L p 1 Λ s g L q 1 + Λ s f L p 2 g L q 2 ) ,
(3.16)
with s > 0 , Λ s = ( Δ ) s 2 and 1 p = 1 p 1 + 1 q 1 = 1 p 2 + 1 q 2 . Taking the operation Λ 2 on both sides of the second equation of (1.1), then multiplying them by Λ 2 θ , and integrating by parts over R 3 , we have
1 2 d d t | Λ 2 θ | 2 d x | ( Λ 2 ( u θ ) u Λ 2 θ ) d x | + Λ 2 ( | u | 2 ) Λ 2 θ d x C ( u L Λ 2 θ L 2 + u L Λ 2 θ L 2 u L 3 θ L 6 ) Λ 2 θ L 2 + C u L Λ 3 u L 2 Λ 2 θ L 2 .
Combining (3.12) with (3.14) and using the Gronwall inequality, we obtain
θ L ( 0 , T ; H 2 ) .

This completes the proof. □

Declarations

Acknowledgements

Research supported by the National Natural Science Foundation of China (11161041), the Fundamental Research Funds for the Central Universities (No. 31920130006) and Middle-Younger Scientific Research Fund (No. 12XB39).

Authors’ Affiliations

(1)
Mathematics and Computer College, North-western Minorities University
(2)
Foundation Department, Zhejiang Industry and Trade Vocational College
(3)
Foundation Department, Shandong Transport Vocational College

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Copyright

© Xu et al.; licensee Springer. 2013

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