arxiv:1805.07v1 [math.ap] 6 May 018 Sufficient conditions on Liouville type theorems for the D steady Navier-Stokes euations G. Seregin, W. Wang May 8, 018 Abstract Our aim is to prove Liouville type theorems for the three dimensional steadystate Navier-Stokes euations provided the velocity field belongs to some Lorentz spaces. The corresponding statement contains several known results as a particular case. Keywords: Liouville theorem, Navier-Stokes euations, Lorentz spaces 1 Introduction The classical Liouville type problem is to describe all bounded solutions to the three dimensional steady-state Navier-Stokes euations u+u u = p, divu = 0 (1.1) in the entire space R. This is still an open problem. Another Liouville type problem is to show that all solutions to system (1.1) belonging to the space J 1, which is the closure of the set of all smooth divergence free compactly supported functions, denoted by C0,0 (R ), with respect to the semi-norm ( u L (R ) = R u dx, are identically eual to zero. This problem is related to the name of J. Leray and, to the best of authors s knowledge, has not been solved yet. However, there are several sufficient conditions providing that all solutions u to (1.1) are eual zero. Let us list the most interesting ones. Oxford University, UK PDMI, RAS, Russia Dalian University of Technology, China 1
We start with Galdi s result. Galdi proved the above Liouville type theorem under the assumption that u L (R ) in [5]. Another interesting result belongs to Chae. In [1], he showed the condition u L 6 5 (R ) is sufficient for u 0 in R. Also, Chae-Wolf gave a logarithmic improvement of Galdi s result in [], assuming that N(u) := u {ln(+1/ u )} 1 dx <. R Letusnoticetwoothersufficientconditions. Ithasbeenshownin[8]thatthecondition u BMO 1 (R ) implies u 0 as well. Moreover, Kozono, et al., proved in [7] that u 0 if the vorticity w = o( x 5 ) for sufficiently large x or u L, (R ) δd(u/ for a small constant δ. More references, we refer to [4, 6, ] and the references therein. One of our aims is to relax the restriction imposed on the norm u L in [7]. Let, (R ) us remind the definition of the Lorentz spaces. Suppose that Ω R n and 1 p <, 1 l. It is said that a measurable function f belongs to the Lorentz space L p,l (Ω) if f L p,l (Ω) < +, where f L p,l (Ω) := ( 0 sup σ>0 Given u, define the following uantity σ l 1 {x Ω : f(x) > σ} l p dσ l σ {x Ω : f(x) > σ} 1 p if l = +. if l < +, where B(R) = B(0,R). Our result is as follows: M γ,,l (R) := R γ u L,l (B R \B R/ ) Theorem 1.1. Let u and p be a smooth solution to (1.1). (i) For >, l (or = l = ), γ =, assume that lim inf R M,,l(R) < (1.)
then Moreover, if D(u) := R M R u dx c(,l)liminf lim inf R M for some 0 < δ < 1/c(,l), then u 0. (ii) For 1/5 < <, 1 l, γ > 1 + 1, suppose that holds then u 0 as well.,,l(r). (1.),,l(R) δd(u) (1.4) lim inf R M γ,,l(r) < (1.5) Remarks 1.. (i) Letting = l = and assuming that u (R ), we observe that L M (R) 0 as R. So, Galdi s result follows from Theorem 1.1.,, (ii) For = and l =, condition (1.4) can be regarded as a generalisation a result proved by Kozono-Terasawa-Wakasugi in [7]. (iii) If we let N(u) =, then N(v) = for the function v = x [lnln( x +e)] ν with 0 < ν. However, if we assume that u C x [lnln( x +e)] ν for the same ν, we can easily check the following fact u L, (B(R)\B(R/)) 0 as R. The latter shows that statement (i) of Theorem 1.1 does not follow from the result of Chae-Wolf []. (iii) The second statement of the theorem is an improvement of one of the results in [10], see Theorem 1.8, where it is assume that γ > 4 6. Caccioppoli Type Ineualities We start with an auxiliary lemma about Caccioppoli type ineuality for the system (1.1). Proposition.1. Let u and p be the smooth solution of (1.1). Then the following Caccioppoli type ineualities hold: if > and 1 l, then u dx CR u dx+cr u L,l (B(R)\B(R/)) ; (.1) B(R/) B(R)\B(R/)
if 0 < δ 1, > > 6( δ), then 6 δ u dx C R u dx+ B(R/) B(R)\B(R/) ( ) +C(δ) u δ R δ δ δ L, (B(R)\B(R/)). (.) Proof. Given R > 0, fix numbers and r so that R/4 < r R. Now, let us pick up a cut-off function φ(x) C0 (B(R)) satisfying the following conditions: 0 φ 1, φ(x) = 1 if x B( ), φ(x) = 0 if x c, and φ(x) c/(r ). We also may assume that function φ(x) = η( x ), i.e., it depends on the distance to the origin only. In this case, it is easy to check that φ udx = 0. \B(r/) Then, by Theorem III.4 in [5] and by scaling, for any 1 < s <, there exist a constant c 0 (s) and a function w Ws 1 () such that div w = φ u in, w = 0 on B(r/), and w s dx C 0 (s) φ u s dx. \B(r/) \B(r/) The function w is extended by zero outside the set \B(r/). Moreover, it is actually smooth as u is smooth. According to the general Marcinkiewicz interpolation theorem, we find w L,l (\B(r/)) C 0 () φ u L,l (\B(r/)). for any 1 < < and any 1 l. Multiplication of both sides of the euation (1.1) by (φu w) and integration by parts give: φ u dx = = u : ( φ u)dx+ w : udx u : (φu u)dx+ + u : (u w)dx = I 1 + +I 4. 4
and Obviously, since R r > R/4 > R/, I 1 C 1 ( ( u dx r ρ C 1 ( r ρ ( I C ( u dx \B( ) B(R)\B(R/) u dx u dx u dx w L (\B(r/)) C 1 r ρ u L () u L (B(R)\B(R/)). Now, our aim is to prove ineuality (.1). To this end, assuming that > and l, let us estimate I, using integration by parts and Hölder ineuality in Lorentz spaces. Indeed, I = 1 u φ u dx \B( ) C u φ L, l l (\B( )) u L, l (\B( )) C 1 r u L,l (B(R)\B(R/)) 1 L, l l (B(R)\B(R/)) C 1 r ρ u L,l (B(R)\B(R/)) R The uantity I 4 is evaluated similarly, if we use the estimate for the gradient of w with suitable exponents: I 4 = w : u udx \B(r/) C w L, l l (\B(r/)) u L, l (\B(r/) C u φ L, l l (\B(r/)) u L,l (\B(r/)) C τ ρ u L,l (B(R)\B(R/)) R. Hence, we get u dx 1 u C ( dx+ (r ρ) B(ρ). B(R)\B(R/) + C r ρ u L,l (B(R)\B(R/)) R 5, ) u dx +
which yields the ineuality (.1) by the standard iteration. Now, let us prove the second ineuality of the proposition. To this end, we introduce ū = u [u] \B(r/), where [u] Ω is the mean value of u over a domain Ω. Applying integration by parts, we find I = 1 φu ( u [u] \B(r/) )dx = = 1 φu ( u )dx = 1 (u φ)( u [u] \B(r/) )dx \B( ) and, since r/ < R/4, I C r \B(r/) u ū u+[u] \B(r/) dx. Under our assumptions on numbers and δ, the following is true 0 < β = 1 δ δ 6 < 1. So, applying the Hölder ineuality for Lorentz spaces, we show I C u ū 1 δ ū δ u+[u] \B(r/) dx r \B(r/) C r u L, (\B(r/)) ū (1 δ) L 1 δ, (\B(r/)) ū δ L6δ (\B(r/)) 1 L 1 β, 6 6 δ (\B(r/)) u+[u] \B(r/) L, (\B(r/)) C r u L, (\B(R/)) ū 1 δ L, (\B(r/)) ū δ L 6 (\B(r/)) R β u+[u] \B(r/) L, (\B(r/)). By Gagliardo-Nireberg-Sobolev ineuality and by the ineuality [u] \B(r/) L, (\B(r/)) c u L, (\B(r/)), we can transform the estimate of I to the following final form I C r Rβ u δ L, (\B(R/)) u δ L (\B(r/)) C r Rβ u δ L, (\B(R/)) u δ L (\B(R/)) 6
1 \B(R/) ( R u β ) dx+c(δ) r u δ δ L, (\B(R/)). Now, our aim is to evaluate I 4. Using similar arguments, we have I 4 = (u u) wdx = (u w) ūdx \B(r/) \B(r/) w L, (\B(r/)) ū 1 δ L, (\B(r/)) ū δ L 6 (\B(r/)) R β u L, (\B(r/)). Taking into account the bound for the gradient of w, we arrive at the same type estimate as in the case of I Conseuently, combining bounds of I 1,,I 4 we get B( ) u dx 1 u dx+ C ( (r ρ) B(R)\B(R/) ( R β ) +C(δ) (r ) u δ δ L, (B(R)\B(R/)) for any R < τ R. Hence, the ineuality (.1) follows. 4 Proof of Theorem 1.1 ) u dx We start with a proof of the statement (i). It is easy to check that, for < < 6, the following estimate is valid: ( ) R u dx C()R 1 6 u L, (B(R)\B(R/)) B(R)\B(R/) C()R 1 M,,l(R). Taking into account condition (1.), we find (1.) and then (1.4). Now, our goal is to prove the statement (ii). Applying the Hölder ineuality to the first term on the right hand side in (.1), we find the following: ( ) u dx CR 1 M 1,, (R)+C(δ) u δ +1 L, (B(R)\B(R/)) Rβ 1 δ B(R/) ( CR 1 M 1,, (R)+C M δ )( δ)) δ +1 γ,, (R)Rβ 1 (γ ( ) = CR 1 M 1,, (R)+C M δ +1 γ,, (R)R δ γ( δ) δ. 7 = (.1)
Now, fix ]1/5,[. Then we can find 1 having the following properties: > 1 > 1 5, γ > 1 + 1 1 > 1 + 1. Given 1, there exists a number δ ]0,1[ such that 1 = 6( δ) 6 δ <. It remains to notice that But γ > 1 + 1 1 and thus a := δ γ( δ) = ( 1) 6 1 = + 1 1 γ 6 1. ( γ 6( ) 1) = 6 1 a = 1 1 + 1 1 γ 6 1 < 0 Passing to the limit as R, we complete the proof of the theorem. Acknowledgments. The first author is supported by the grant RFBR 17-01- 000-a. W. Wang was supported by NSFC under grant 11671067, the Fundamental Research Funds for the Central Universities and China Scholarship Council. References [1] D. Chae, Liouville-type theorem for the forced Euler euations and the Navier- Stokes euations. Commun. Math. Phys.6: 7-48 (014). [] D. Chae, T. Yoneda, On the Liouville theorem for the stationary Navier-Stokes euations in a critical space, J. Math. Anal. Appl. 405 (01), no., 706-710. [] D. Chae, J. Wolf, On Liouville type theorems for the steady Navier- Stokes euations in R, arxiv:1604.0764. [4] Chae, G., Weng, S., Liouville type theorems for the steady axially symmetric Navier-Stokes and magnetohydrodynamic euations, Discrete And Continuous Dynamical Systems, Volume 6, Number 10, 016 Pp. 567-585. [5] G. P. Galdi, An introduction to the mathematical theory of the Navier- Stokes euations. Steady-state problems. Second edition. Springer Monographs in Mathematics. Springer, New York, 011. xiv+1018 pp. [6] Koch, G., Nadirashvili, N., Seregin, G., Sverak, V., Liouville theorems for the Navier-Stokes euations and applications, Acta Mathematica, 0 (00), 8-105. 8
[7] H. Kozono, Y. Terasawab, Y. Wakasugib, A remark on Liouville-type theorems for the stationary NavierCStokes euations in three space dimensions, Journal of Functional Analysis, 7(017), 804-818. [8] G. Seregin, Liouville type theorem for stationary Navier-Stokes euations, Nonlinearity, (016), 11-15. [] G. Seregin, A liouville type theorem for steady-state Navier-Stokes euations, arxiv:1611.0156 and J. E. D. P. (016), Expos no IX, 5 p. [10] G. Seregin, Remarks on Liouville type theorems for steady-state Navier-Stokes euations, arxiv:170.108v1 and Algebra i Analiz, 018, Vol. 0, no.., 8-48.