On a Type I singularity condition in terms of the pressure for the Euler equations in \mathbb R^3
aa r X i v : . [ m a t h . A P ] D ec On a Type I singularity condition in terms ofthe pressure for the Euler equations in R ∗ and Peter Constantin † ∗ Department of MathematicsChung-Ang UniversitySeoul 06974, Republic of Koreae-mail: [email protected] † Department of MathematicsPrinceton UniversityPrinceton, NJ 08544, USAe-mail: [email protected]
Abstract
We prove a blow up criterion in terms of the Hessian of the pressure of smoothsolutions u ∈ C ([0 , T ); W ,q ( R )), q > t = T happens only if Z T Z t (cid:26)Z s k D p ( τ ) k L ∞ dτ exp (cid:18)Z ts Z σ k D p ( τ ) k L ∞ dτ dσ (cid:19)(cid:27) dsdt = + ∞ . As consequences of this criterion we show that there is no blow up at t = T if k D p ( t ) k L ∞ ≤ c ( T − t ) with c < t ր T . Under the additional assumption of R T k u ( t ) k L ∞ ( B ( x ,ρ )) dt < + ∞ , we obtain localized versions of these results. AMS Subject Classification Number: keywords: type I singularity, blow up criterion, Hessian of the pressure
We are concerned with the homogeneous incompressible Euler equation on R × [0 , T ).( E ) ( u t + u · ∇ u = −∇ p, ∇ · u = 0 , u ( x,
0) = u ( x )where u ( x, t ) = ( u ( x, t ) , u ( x, t ) , u ( x, t )) is the fluid velocity and p = p ( x, t ) is thescalar pressure. The local in time well-posedness of the Cauchy problem of (E) for1mooth initial data u is well established in various function spaces (see e.g. [13, 7]and the reference therein). In this paper we are interested in the problem of finitetime blow up of local smooth solutions u ∈ C ([0 , T ); W ,q ( R )), for q >
3, in whichcases the local well-posedness is established in [10]. There are very many studies ofthis problem, in particular establishing blow up criteria (see e.g. [1, 11, 8, 7]). Wemention that there are also important results of showing apparition of singularity atthe boundary points in the domains having boundaries [12, 9]. Our main concern is onthe possibility of spontaneous appearance of interior singularity starting from a smoothinitial data. Below we first establish a global in space blow up criterion in terms ofthe Hessian of the pressure, which is sharper than any previously known ones in theliterature. As an immediate consequence of this criterion we are able to obtain a sharp‘small type I condition’ in terms of the Hessian of the pressure, which is consistentwith hyperbolic scaling. The second result is a localization of the first result, showingthat certain conditions in terms of the Hessian of the pressure in a ball imply no blowup in the ball.
Theorem 1.1
Let ( u, p ) ∈ C ( R × (0 , T )) be a solution of the Euler equation (E) with u ∈ C ([0 , T ); W ,q ( R )) , for some q > .(i) If Z T Z t (cid:26)Z s k D p ( τ ) k L ∞ dτ exp (cid:18)Z ts Z σ k D p ( τ ) k L ∞ dτ dσ (cid:19)(cid:27) dsdt < + ∞ , (1.1) then lim sup t → T k u ( t ) k W ,q < + ∞ . (ii) If lim sup t → T ( T − t ) k D p ( t ) k L ∞ < , (1.2) then lim sup t → T k u ( t ) k W ,q < + ∞ . Remark 1.1
A blow up criterion of the Euler equations in terms of the Hessian ofpressure was obtained in [3] in a different form. Let S = ( S ij ) with S ij = ( ∂ i u j + ∂ j u i )the symmetric part of the velocity gradient matrix, and we set the unit vectors ξ = ω/ | ω | , ζ = Sξ/ | Sξ | . Define ζ · P ξ = µ, where P = ( ∂ i ∂ j p ) is the Hessian of the pressure.Then, it is shown [3, Theorem 5.1] that there is no blow up at t = T if Z T exp (cid:18)Z t k µ ( s ) k L ∞ ds (cid:19) dt < + ∞ . (1.3)We note that there exists no mutual implication relation between the conditions (1.1)and (1.3), although a stronger condition than (1.3) (hence, the result of the criterionitself is weaker) Z T exp (cid:18)Z t k D p ( s ) k L ∞ ds (cid:19) dt < + ∞ . (1.4)2mplies (1.1). A related result is found in [6]. Remark 1.2
Comparing (1.2) with the ‘small type I condition’ in terms of the velocity,lim sup t → T ( T − t ) k Du ( t ) k L ∞ < ∂ i ∂ j p = X k,m =1 R i R j ( ∂ k u m ∂ m u k ) , we see that (1.2) is an optimal ‘small type I condition’ in terms of the pressure consistentwith hyperbolic scaling. Here, R j , j = 1 , ,
3, are the Riesz tranforms in R (see e.g.[14])We also note that from the condition (1.4) it is impossible to deduce the correct smalltype I condition (1.2) guaranteeing absence of blow up. Theorem 1.2
Let ( u, p ) ∈ C ( B ( x , ρ ) × ( T − ρ, T )) be a solution to (E) with u ∈ C ([ T − ρ, T ); W ,q ( B ( x , ρ ))) ∩ L ∞ ( T − ρ, T ; L ( B ( x , ρ ))) for some q ∈ (3 , ∞ ) .(i) If Z TT − ρ k u ( t ) k L ∞ ( B ( x ,ρ )) dt < + ∞ (1.6) and Z TT − ρ (cid:26)Z tT − ρ Z sT − ρ k D p ( τ ) k L ∞ ( B ( x ,ρ )) dτ × exp (cid:18)Z ts Z σT − ρ k D p ( τ ) k L ∞ ( B ( x ,ρ )) dτ dσ (cid:19) ds (cid:27) dt < + ∞ , (1.7) then for all r ∈ (0 , ρ ) lim sup t ր T k u ( t ) k W ,q ( B ( x ,r )) < + ∞ . (1.8) (ii) If (1.6) holds, and lim sup t → T ( T − t ) k D p ( t ) k L ∞ ( B ( x ,ρ )) < , (1.9) then for all r ∈ (0 , ρ ) we have lim sup t ր T k u ( t ) k W ,q ( B ( x ,r )) < + ∞ . (1.10) Remark 1.3
A remark similar to Remark 1.2 above holds, comparing (1.9) with thelocal version of ‘small Type I condition’lim sup t → T ( T − t ) k∇ u ( t ) k L ∞ ( B ( x ,ρ )) < , (1.11)which was obtained in [5]. 3 The Proof of the Main Theorems
Proof of Theorem 1.1:
Proof of (i)
We use the particle trajectory mapping α X ( α, t ) from R into R generated by u = u ( x, t ), which means the solution of theordinary differential equation, ∂X ( α, t ) ∂t = u ( X ( α, t ) , t ) on (0 , T ) ,X ( α,
0) = α ∈ R . (2.1)Taking curl of (E), we obtain the vorticity equations ω t + u · ∇ ω = ω · ∇ u, ω = ∇ × u. (2.2)The equation (2.2) can be rewritten in terms of the particle trajectory as ∂∂t ω ( X ( α, t ) , t ) = ω ( X ( α, t ) , t ) · ∇ u ( X ( α, t ) , t ) . (2.3)Therefore, ∂ ∂t ω ( X ( α, t ) , t ) = ∂∂t ω ( X ( α, t ) , t ) · ∇ u ( X ( α, t ) , t )+ ω ( X ( α, t ) , t ) · ∂∂t ∇ u ( X ( α, t ) , t ) . (2.4)From (E) we also compute ∂∂t ∂ j u k ( x, t ) + u · ∇ ∂ j u k ( x, t ) = − X m =1 ∂ j u m ∂ m u k − ∂ j ∂ k p, which can be written as ∂∂t ∂ j u k ( X ( α, t ) , t ) = − X m =1 ∂ j u m ( X ( α, t ) , t ) ∂ m u k ( X ( α, t ) , t ) − ∂ j ∂ k p ( X ( α, t ) , t ) . (2.5)Substituting (2.3) and(2.5) into (2.4), one has ∂ ∂t ω k ( X ( α, t ) , t ) = X j =1 ∂∂t ω j ( X ( α, t ) , t ) ∂ j u k ( X ( α, t ) , t )+ X j =1 ω j ( X ( α, t ) , t ) ∂∂t ∂ j u k ( X ( α, t ) , t )4 X j,m =1 ω m ( X ( α, t ) , t ) ∂ m u j ( X ( α, t ) , t ) ∂ j u k ( X ( α, t ) , t ) − X j,m =1 ω j ( X ( α, t ) , t ) ∂ j u m ( X ( α, t ) , t ) ∂ m u k ( X ( α, t ) , t ) − X j =1 ω j ( X ( α, t ) , t ) ∂ j ∂ k p ( X ( α, t ) , t )= − X j =1 ω j ( X ( α, t ) , t ) ∂ j ∂ k p ( X ( α, t ) , t ) , (2.6)from which, after a double integral in time, we obtain ω k ( X ( α, t ) , t ) = ω ,k ( α ) + ∂ω k ( X ( α, t ) , t ) ∂t (cid:12)(cid:12)(cid:12) t =0 + t − X j =1 Z t Z s ω j ( X ( α, σ ) , σ ) ∂ j ∂ k p ( X ( α, σ ) , σ ) dσds = ω ,k ( α ) + X j =1 ω ,j ( α ) ∂ j u ,k ( α ) t − X j =1 Z t Z s ω j ( X ( α, σ ) , σ ) ∂ j ∂ k p ( X ( α, σ ) , σ ) dσds, (2.7)where ω = ∇ × u , and we used (2.3) to compute ∂ω k ( X ( α, t ) , t ) ∂t (cid:12)(cid:12)(cid:12) t =0 + = X j =1 ω ,j ( α ) ∂ j u ,k ( α ) . This leads us into | ω ( X ( α, t ) , t ) | ≤ | ω ( α ) | + | ω ( α ) · ∇ u ( α ) | t + Z t Z s | D p ( X ( α, σ ) , σ ) || ω ( X ( α, σ ) , σ ) | dσds. (2.8)Since the right hand side of (2.8) is monotone non-decreasing with respect to t > , t ), we havesup <τ The hypothesis (2.14) implies there exists t ∈ (0 , T ) and η ∈ (0 , 1) suchthat sup t <τ The starting point of the argument, equation (2.6), can also be derivedfrom the Lagrangian form of the Euler equations, ∂ X ( α, t ) ∂t = −∇ p ( X ( α, t ) , t ) . (2.16)Indeed, taking the gradient the both sides of (2.16), and multiplying them by ω ( α ),and then using the Cauchy formula ω ( X ( α, t ) , t ) = ∇ X ( α, t ) ω ( α ), we have (2.6). Proof of Theorem 1.2 : We consider a sequence { t k } k ∈ N ⊂ ( T − ρ, T ) such that t k 3. Now, given η > 0, we choose δ > Z TT − δ k u ( t ) k L ∞ ( B ( x ,ρ )) dt < η , Then, for such δ > | α − β | small enough to have | α − β | (cid:18) Z T − δ k∇ u ( t ) k L ∞ ( B ( x ,r )) e R tT − ρ k∇ u ( s ) k L ∞ ( B ( x ,r )) ds dt (cid:19) < η . Then, (2.17) shows that | X ( α, T ) − X ( β, T ) | < η . The claim X ( · , T ) ∈ C ( B ( x , r )) isproved.By the continuity of the trajectory mapping X ( · , t ) for t ∈ [0 , T ] for each r ∈ (0 , ρ )there exists ε > B (cid:18) x , r + ρ − r (cid:19) ⊂ X (cid:18) B (cid:18) x , r + ρ − r (cid:19) , T − t (cid:19) ⊂ B ( x , ρ ) ∀ t ∈ [0 , ε ] . (2.18)where X ( α, t ) is the extension of the particle trajectory to t = T defined by thefollowing ordinary differential equations ∂X ( α, t ) ∂t = u ( X ( α, t ) , t ) on [ T − ε, T ) ,X ( α, T − ε ) = α ∈ B ( x , r ) . (2.19)Then, we have from (2.12) k ω ( t ) k L ∞ ( B ( x ,r + ρ − r ) ) ≤ sup α ∈ B ( x ,r + ρ − r ) {| ω ( α, T − ε ) | + | ω ( α, T − ε ) ||∇ u ( α, T − ε ) t } ×× (cid:20) Z tT − ε (cid:26)Z sT − ε | D p ( X ( α, τ ) , τ ) | dτ exp (cid:18)Z ts Z σT − ε | D p ( X ( α, τ ) , τ ) | dτ dσ (cid:19) ds (cid:27)(cid:21) ≤ (cid:8) k ω k L ∞ ( B ( x ,ρ )) + k ω ∇ u k L ∞ ( B ( x ,ρ )) T (cid:9) ×× (cid:20) Z tT − ε (cid:26)Z sT − ε k D p ( τ ) k L ∞ ( B ( x ,ρ )) dτ exp (cid:18)Z ts Z σT − ε k D p ( τ ) k L ∞ ( B ( x ,ρ )) dτ dσ (cid:19) ds (cid:27)(cid:21) . (2.20)8ntegrating this over [ T − ε, T ], we find Z TT − ε k ω ( t ) k L ∞ ( B ( x ,r + ρ − r ) ) dt ≤ (cid:8) k ω k L ∞ ( B ( x ,ρ )) + k ω ∇ u k L ∞ ( B ( x ,ρ )) T (cid:9) ×× (cid:20) T + Z TT − ε Z tT − ε (cid:26)Z sT − ε k D p ( τ ) k L ∞ ( B ( x ,ρ )) dτ ×× exp (cid:18)Z ts Z σT − ε k D p ( τ ) k L ∞ ( B ( x ,ρ )) dτ dσ (cid:19) ds (cid:27) dt (cid:21) , which is finite by the hypothesis (1.7). 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