Long time existence of smooth solutions to 2D compressible Euler equations of Chaplygin gases with non-zero vorticity
aa r X i v : . [ m a t h . A P ] F e b Long time existence of smooth solutions to 2D compressibleEuler equations of Chaplygin gases with non-zero vorticity
Fei Hou , ∗ Huicheng Yin , , *
1. Department of Mathematics, Nanjing University, Nanjing 210093, China2. School of Mathematical Sciences and Mathematical Institute,Nanjing Normal University, Nanjing 210023, China
Abstract
For the 2D compressible isentropic Euler equations of polytropic gases with an initial perturba-tion of size ε of a rest state, it has been known that if the initial data are rotationnally invariant orirrotational, then the lifespan T ε of the classical solutions is of order O ( ε ) ; if the initial vorticityis of size ε α ( ≤ α ≤ ), then T ε is of O ( ε α ) . In the present paper, for the 2D compressibleisentropic Euler equations of Chaplygin gases, if the initial data are a perturbation of size ε , and theinitial vorticity is of any size δ with < δ ≤ ε , we will establish the lifespan T δ = O ( δ ) . Forexamples, if δ = e − ε or δ = e − e ε are chosen, then T δ = O ( e ε ) or T δ = O ( e e ε ) although theperturbations of the initial density and the divergence of the initial velocity are only of order O ( ε ) .Our main ingredients are: finding the null condition structures in 2D compressible Euler equations ofChaplygin gases and looking for the good unknown; establishing a new class of weighted space-time L ∞ - L ∞ estimates for the solution itself and its gradients of 2D linear wave equations; introducingsome suitably weighted energies and taking the L p (1 < p < ∞ ) estimates on the vorticity. Keywords.
Compressible Euler equations, Chaplygin gases, vorticity, null condition, weighted L ∞ - L ∞ estimates, ghost weight, A p weight. Contents g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.2 Estimates of the auxiliary energy X m ( t ) . . . . . . . . . . . . . . . . . . . . . . . . . . 15 * Fei Hou ( fhou @ nju.edu.cn ) and Huicheng Yin ( huicheng @ nju.edu.cn , @ njnu.edu.cn ) are sup-ported by the NSFC (No. 11731007). ( σ, u ) . . . . . . . . . . . . 174.2 Weighted L ∞ - L ∞ estimates for the linear wave equation . . . . . . . . . . . . . . . . . 194.3 Improved pointwise estimates of u . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 φ The 2D compressible isentropic Euler equations are ( ∂ t ρ + div( ρu ) = 0 (Conservation of mass) ,∂ t ( ρu ) + div( ρu ⊗ u ) + ∇ P = 0 (Conservation of momentum) , (1.1)where ( t, x ) = ( t, x , x ) ∈ R = [0 , ∞ ) × R , ∇ = ( ∂ , ∂ ) = ( ∂ x , ∂ x ) , and u = ( u , u ) , ρ, P stand for the velocity, density, pressure respectively. In addition, the pressure P = P ( ρ ) is a smoothfunction of ρ when ρ > , moreover, P ′ ( ρ ) > for ρ > .For the polytropic gases (see [9]), P ( ρ ) = Aρ γ , (1.2)where A and γ ( < γ < ) are some positive constants.For the Chaplygin gases (see [9] or [12]), P ( ρ ) = P − Bρ , (1.3)where P > and B > are constants.If ( ρ, u ) ∈ C is a solution of (1.1) with ρ > , then (1.1) is equivalent to the following form ∂ t ρ + div( ρu ) = 0 ,∂ t u + u · ∇ u + c ( ρ ) ρ ∇ ρ = 0 , (1.4)where the sound speed c ( ρ ) := p P ′ ( ρ ) .Consider the initial data of (1.1) as follows ( ρ (0 , x ) , u (0 , x )) = (¯ ρ + ρ ( x ) , u ( x )) , (1.5)where ¯ ρ > is a constant, ¯ ρ + ρ ( x ) > , and ρ ( x ) , u ( x ) = ( u ( x ) , u ( x )) ∈ C ∞ . When curl u ( x ) := ∂ u − ∂ u ≡ , (1.6)as long as ( ρ, u ) ∈ C for ≤ t ≤ T , then curl u ≡ always holds for ≤ t ≤ T . In thiscase, one can introduce the potential function φ such that u = ∇ φ , then the Bernoulli’s law implies ∂ t φ + |∇ φ | + h ( ρ ) = 0 with h ′ ( ρ ) = c ( ρ ) ρ and h (¯ ρ ) = 0 . By the implicit function theorem due to h ′ ( ρ ) > for ρ > , then the density function ρ can be expressed as ρ = h − (cid:18) − ∂ t φ − |∇ φ | (cid:19) =: H ( ∂φ ) , (1.7)where ∂ = ( ∂ t , ∇ ) . Substituting (1.7) into the mass conservation equation in (1.1) yields that ∂ t ( H ( ∂φ )) + X i =1 ∂ i (cid:0) H ( ∂φ ) ∂ i φ (cid:1) = 0 . (1.8)For any C solution φ , (1.8) can be rewritten as the following second order quasilinear equation ∂ t φ + 2 X k =1 ∂ k φ∂ tk φ − c ( ρ )∆ φ + X i,j =1 ∂ i φ∂ j φ∂ ij φ = 0 , (1.9)where c ( ρ ) = c ( H ( ∂φ )) , and the Laplace operator ∆ := X i =1 ∂ i . Without loss of generality and forsimplicity, c (¯ ρ ) = 1 can be supposed, and then c ( ρ ) = 1 − ρc ′ (¯ ρ ) ∂ t φ + O ( | ∂φ | ) . Especially, in thecase of the Chaplygin gases, (1.9) is ∂ t φ − △ φ + 2 X k =1 ∂ k φ∂ t ∂ k φ − ∂ t φ △ φ + X i,j =1 ∂ i φ∂ j φ∂ ij φ − |∇ φ | △ φ = 0 . (1.10)When k ρ ( x ) k H + k u ( x ) k H ≤ ¯ ε and ¯ ε > is sufficiently small, if follows from Theorem 6.5.3of [15] and equation (1.9) that the lifespan T ¯ ε of smooth solution ( ρ, u ) to (1.4) fulfills T ¯ ε ≥ C ¯ ε , where C > is a constant depending only on the initial data. In addition, for the polytropic gases, since thefirst null condition does not hold for equation (1.9), then T ¯ ε ≤ ˜ C ¯ ε holds for suitably positive constant ˜ C (see [4] and [21]); for the Chaplygin gases, note that both the first null condition and the second nullcondition hold for equation (1.10), then T ¯ ε = + ∞ holds (see [5]).When curl u ( x ) , (1.11)if curl u ( x ) = O (¯ ε α ) , where α ≥ is a constant, then by Theorem 1 and Theorem 2 of [27] that the lifespan T ¯ ε of smoothsolution ( ρ, u ) to (1.4) satisfies T ¯ ε ≥ C ¯ ε min { α, } . (1.12)Note that T ¯ ε in (1.12) is also optimal for the polytropic gases, i.e., T ¯ ε ≤ ¯ C ¯ ε min { α, } holds for someconstant ¯ C > C , one can see [1]- [2], [21] and [27]. With respect to more results on the blowup or theblowup mechanism of (1.1) for polytropic gases, the papers [7]- [8], [11], [14], [23], [25], [26], [29] canbe referred. In the present paper, we intend to study the long time existence of smooth solutions to (1.4)for the Chaplygin gases with initial data (1.5) and curl u ( x ) , curl u ( x ) = O ( δ ) , (1.13)where δ > and δ = o ( ε ) . For this end, we introduce the following quantities on the initial data andvorticity ε = X k ≤ N k ( h| x |i∇ ) k ( u ( x ) , ρ ( x ) ρ ( x ) + ¯ ρ ) k L x + X k ≤ N − kh| x |i ( h| x |i∇ ) k ∆ − div u ( x ) k L x + X k ≤ N − kh| x |i ( h| x |i∇ ) k (∆ − curl( u ( x ) curl u ( x )) , u ( x ) , ρ ( x ) ρ ( x ) + ¯ ρ ) k L x , (1.14)and δ = X k ≤ N − kh| x |i ( h| x |i∇ ) k curl u ( x ) k L x + X k ≤ N X p = , , kh| x |i ( h| x |i∇ ) k curl u ( x ) k L px , (1.15)where h| x |i = p | x | , the integers N and N fulfill N ≥ and N + 2 ≤ N ≤ N − . Inaddition, the state equation in (1.3) is conveniently written as P ( ρ ) = P − ¯ ρ ρ . (1.16) Theorem 1.1.
There exists three constants ε , δ , κ > such that when the initial data ( ρ , u ) satisfies ε ≤ ε and δ ≤ δ , then (1.1) with (1.16) admits a solution ( ρ − ¯ ρ, u ) ∈ C ([0 , T δ ]; H N ( R )) with T δ = κδ . Remark 1.1.
In Theorem 1.1, when δ = ε α with ≤ α ≤ and < ε ≤ ε , then T δ = κδ has beenshown in Theorem 2 of [27]. Hence we can assume δ ≤ ε in Theorem 1.1 without loss of generality. Remark 1.2.
If we choose δ = ε ℓ with ℓ ≥ or e − εp with p > in Theorem 1.1, then the existencetime of smooth solution ( ρ, u ) to (1.1) for the Chaplygin gases is larger than κε ℓ or κe εp . Remark 1.3. If ( ρ ( x ) , u ( x )) ∈ C ∞ ( B (0 , R )) , where B (0 , R ) is a ball with the center at the ori-gin and the radius R > , then kh| x |i ( h| x |i∇ ) k (∆ − curl( u ( x ) curl u ( x )) , u ( x ) , ρ ( x ) ρ ( x )+¯ ρ ) k L x and kh| x |i ( h| x |i∇ ) k ∆ − div u ( x ) k L x in (1.14) can be replaced by k∇ k (∆ − curl( u ( x ) curl u ( x )) , u ( x ) , ρ ( x ) ρ ( x )+¯ ρ ) k L x ( B (0 ,R )) and k∇ k ∆ − div u ( x ) k L x ( B (0 ,R )) respec-tively. The reasons are: • from the Helmholtz decomposition of initial velocity u , one has u = P u + P u , where P u = −∇ ( − ∆) − div u , P u = −∇ ⊥ ( − ∆) − curl u and ∇ ⊥ = ( − ∂ x , ∂ x ) . When supp u ⊂ B (0 , R ) ,we choose a smooth cut-off function η ( x ) such that η | supp u = 1 and supp η ⊂ B (0 , R ) , and then u = ηP u + ηP u . In this case, the related term ∆ − div u ( x ) can be thought to be supported in B (0 , R ) . • from (A.2) in Appendix A, we have ∂ t φ + | u | + σ − ( σ ) = − ( − ∆) − curl( u curl u ) ,where P u = ∇ φ and σ = ρ ρ +¯ ρ . When supp u , supp ρ ⊂ B (0 , R ) , then supp φ , supp σ ⊂ B (0 , R ) . As in the above, ∆ − curl( u curl u ) can be thought to be supported in B (0 , R ) . Remark 1.4.
Consider the 2D full compressible Euler equations of Chaplygin gases ∂ t ρ + div( ρu ) = 0 (Conservation of mass) ,∂ t ( ρu ) + div( ρu ⊗ u ) + ∇ P = 0 (Conservation of momentum) ,∂ t ( ρe + 12 ρ | u | ) + div(( ρe + 12 ρ | u | + P ) u ) = 0 (Conservation of energy) ,ρ (0 , x ) = ¯ ρ + ερ ( x ) , u (0 , x ) = εu ( x ) , S (0 , x ) = ¯ S + εS ( x ) , (1.17)where P = P ( ρ, S ) , e = e ( ρ, S ) , S stand for the pressure, inner energy and entropy respectively. Inaddition, ε > is small, and ( ρ ( x ) , u ( x ) , S ( x )) ∈ C ∞ ( R ) . If ρ ( x ) = ρ ( r ) , S ( x ) = S ( r ) and u ( x ) = f ( r ) xr + g ( r ) x ⊥ r with x ⊥ = ( − x , x ) and r = | x | , then we have shown (1.17) has a globalsmooth solution ( ρ, u, S ) in [17] and [18] (when g ( r ) ≡ , the global existence of smooth symmetricsolutions ( ρ, u, S ) to 2D and 3D systems (1.17) has been established in [10] and [12] respectively). Bycombining the methods in the paper and [18], we can actually establish that for the perturbed problem of(1.17) ∂ t ρ + div( ρu ) = 0 ,∂ t ( ρu ) + div( ρu ⊗ u ) + ∇ p = 0 ,∂ t ( ρe + 12 ρ | u | ) + div(( ρe + 12 ρ | u | + P ) u ) = 0 ,ρ (0 , x ) = ¯ ρ + ερ ( x ) + δρ ( x ) , u (0 , x ) = εu ( x ) + δu ( x ) ,S (0 , x ) = ¯ S + εS ( x ) + δS ( x ) , (1.18)where curl u ( x ) ≡ (or ρ ( x ) = ρ ( r ) , S ( x ) = S ( r ) and u ( x ) = f ( r ) xr + g ( r ) x ⊥ r ), δ = o ( ε ) ,and ( ρ ( x ) , u ( x ) , S ( x )) ∈ C ∞ ( R ) , then (1.18) admits a smooth solution ( ρ, u, S ) for t ∈ [0 , T δ ] with T δ = κδ as in Theorem 1.1. Remark 1.5.
We mention the interesting works on the Euler-Maxwell system, which are related ourresult. The global smooth, small-amplitude, irrotational solution to the Euler-Maxwell two-fluid systemwas proved in [13]. Considering the influence of the vorticity, Ionescu and Lie in [20] have shown thatthe existence time is larger than Cδ , where δ is the size of initial vorticity and C is some positive constant. Remark 1.6.
It is well known that the compressible Euler system is symmetric hyperbolic with respectto the time t when the vacuum does not appear. A. Majda posed the following conjecture on Page 89of [24]: Conjecture.
If the multidimensional nonlinear symmetric system is totally linearly degenerate, thenit typically has smooth global solutions when the initial data are in H s ( R n ) with s > n + 1 unless thesolution itself blows up in finite time. Note that the compressible Euler system of Chaplygin gases is totally linearly degenerate, A. Majda’sconjecture together with the opinion in [3] can yield the following open question:
Open question.
For the n − dimensional ( n ≥ ) full compressible Euler equations of Chaplygingases ∂ t ρ + div( ρu ) = 0 ,∂ t ( ρu ) + div( ρu ⊗ u ) + ∇ P = 0 ,∂ t ( ρe + 12 ρ | u | ) + div(( ρe + 12 ρ | u | + P ) u ) = 0 ,ρ (0 , x ) = ¯ ρ + ερ ( x ) , u (0 , x ) = εu ( x ) , S (0 , x ) = ¯ S + εS ( x ) , where ε > is small, and ( ρ ( x ) , u ( x ) , S ( x )) ∈ C ∞ ( R ) with u ( x ) = ( u ( x ) , · · · , u n ( x )) , then T ε = + ∞ . Although this open question has not been solved so far, our result in Theorem 1.1 illustrates that theorder of lifespan T ε is only essentially influenced by the initial vorticity.We next give some comments on the proof of Theorem 1.1. From now on, ¯ ρ = 1 is always assumed.Introducing the perturbed sound speed σ ( t, x ) = 1 − ρ ( t,x ) as the new unknown, then (1.1) is reduced to ( ∂ t σ + div u = Q := σ div u − u · ∇ σ,∂ t u + ∇ σ = Q := σ ∇ σ − u · ∇ u, (1.19)where the i -th component of the vector Q is Q i = σ∂ i σ − u · ∇ u i ( i = 1 , ). In addition, we alsodefine the good unknown g in the region | x | > as follows g := ( g , g ) = u − ωσ with g i = u i − ω i σ , i = 1 , , (1.20)where ω = ( ω , ω ) := ( x | x | , x | x | ) ∈ S . We point out that the introduction of g is motivated by thesecond order wave equations although (1.1) admits the non-zero and small higher order vorticity: bythe potential equation (1.10), then u i = ∂ i φ and σ = − ∂ t φ + higher order error terms of ∂φ , whichderives g i = ( ∂ i + ω i ∂ t ) φ + higher order error terms of ∂φ . It is well known that ( ∂ i + ω i ∂ t ) φ is thegood derivative in the study of the nonlinear wave equation (1.10) (see [6]) since ( ∂ i + ω i ∂ t ) φ will admitmore rapid space-time decay rates. By some ideas and methods dealing with the null condition structuresin [17–19], we obtain the better L ∞ space-time decay rates of g . Based on this, the elementary energy E N ( t ) can be estimated (see Lemma 5.1 in Section 5.1). Nevertheless, we have to overcome otheressential difficulties which are arisen by the interaction between the irrotational part of the velocity andthe vorticity. To solve the resulting difficulties, our ingredients are: • By the transport equation ( ∂ t + u · ∇ ) (cid:0) curl uρ (cid:1) = 0 and the careful analysis, we can derive that thekey influence of the vorticity is concentrated in the interior of the outgoing light cone. • Near the outgoing light conic surface, our first observation is that the system (1.1) can be changedinto the second order potential flow equation. However, the optimal time-decay rate of solutions to the 2Dfree wave equation is merely (1 + t ) − , which is far to derive the existence time T δ = κδ in Theorem 1.1.The reason is due to: for example, when δ = e − e ε is chosen, then the integral Z T δ dt √ t = O ( e e ε ) is sufficiently large as ε → , which leads to that the related energy E ( t ) can not be controlled well bythe corresponding energy inequality E ( t ) ≤ E (0) + Cε √ t E ( t ) . To overcome this difficulty, our secondobservation is that the velocity is the gradient of the potential and the potential satisfies a second orderquasilinear wave equation with the first and the second null conditions. By establishing a new type ofweighted L ∞ - L ∞ estimate for the derivatives of the potential, the better space-time decay rate of u canbe obtained (see Corollary 4.9 in Section 4).Based on the key estimates in the above, we eventually get the L and other L p (for some suitablenumbers p with p = 2 ) energy estimates of the vorticity and further close the bootstrap assumptions inSection 2.This paper is organized as follows. In Section 2, we will introduce the basic bootstrap assumptions,Helmholtz decomposition and some elementary pointwise estimates. The estimates of the good unknown g and some auxiliary energies are derived in Section 3. In Section 4, by establishing a new type of theweighted L ∞ - L ∞ for the 2D wave equations, the required pointwise estimates of the solution ( σ, u ) with suitable space-time decay rates are derived. Based on the pointwise estimates in Section 4, theHardy inequality and the ghost weight method in [5], we get the related energy estimates in Section 5. InSection 6, with the previous energy inequalities and Gronwall’s inequalities, the proof of Theorem 1.1 isfinished. Define the spatial rotation vector field
Ω := x ∂ − x ∂ . For the vector-valued function U = ( U , U ) , denote ˜Ω U := Ω U − U ⊥ = (Ω U + U , Ω U − U ) . Let ˜Ω U k = ( ˜Ω U ) k be the k -component of ˜Ω U rather than the operator ˜Ω act on the component U k .According to the definitions of Ω and ˜Ω , it is easy to check that for the scalar function f and thevector-valued functions U, V , Ω div U = div ˜Ω U, Ω curl U = curl ˜Ω U, ˜Ω ∇ f = ∇ Ω f, ˜Ω( U · ∇ V ) = U · ∇ ( ˜Ω V ) + ( ˜Ω U ) · ∇ V, Ω( U · V ) = ( ˜Ω U ) · V + U · ˜Ω V. (2.1)The spatial derivatives can be decomposed into the radial and angular components for r = | x | 6 = 0 : ∇ = ω∂ r + ω ⊥ | x | Ω , where ω ⊥ := ( − ω , ω ) . For convenience and simplicity, we often denote this decomposition as ∂ i = ω i ∂ r + 1 | x | Ω . (2.2)For the multi-index a , set S := t∂ t + r∂ r , Γ a = S a s Z a z , Z ∈ { ∂, Ω } , ˜Γ a = S a s ˜ Z a z , ˜ Z ∈ { ∂, ˜Ω } . (2.3)By acting ( S + 1) a s Z a z and ( S + 1) a s ˜ Z a z on the equations in (1.19), respectively, we then have ∂ t Γ a σ + div ˜Γ a u = Q a := X b + c = a C abc Q bc ,∂ t ˜Γ a u + ∇ Γ a σ = Q a := X b + c = a C abc Q bc , (2.4)where C abc are constants ( C aa = C a a = 1 ) and Q bc :=Γ b σ div ˜Γ c u − ˜Γ b u · ∇ Γ c σ,Q bc :=Γ b σ ∇ Γ c σ − ˜Γ b u · ∇ ˜Γ c u. (2.5)It is convenient to introduce the specific vorticity w := curl uρ = (1 − σ ) curl u (2.6)since ( ∂ t + u · ∇ ) w = 0 holds. For integer m ∈ N , we define E m ( t ) := X | a |≤ m k (˜Γ a u, Γ a σ )( t, x ) k L x , X m ( t ) := X | a |≤ m − kh| x | − t i ( ∂ t ˜Γ a u, div ˜Γ a u, ∇ Γ a σ, ∂ t Γ a σ )( t, x ) k L x ,W m ( t ) := X | a |≤ m kh| x |i Γ a w ( t, x ) k L x , W m ( t ) := X | a |≤ m kh| x |i Γ a curl u ( t, x ) k L x , W m ( t ) := X | a |≤ m n kh| x |i Γ a curl u ( t, x ) k L x + X p = , kh| x |i Γ a curl u ( t, x ) k L px o , W m ( t ) := X | a |≤ m n kh| x |i Γ a w ( t, x ) k L x + X p = , kh| x |i Γ a w ( t, x ) k L px o (2.7)and X ( t ) = 0 .Throughout the whole paper, we will make the following bootstrap assumptions: for tδ ≤ κ, E N ( t ) + X N ( t ) ≤ M ε (1 + t ) M ′ ε ,E N − ( t ) + X N − ( t ) ≤ M ε,W N − ( t ) + W N − ( t ) + W N ( t ) + W N ( t ) ≤ M δ (1 + t ) M ′ ε ,W N − ( t ) + W N − ( t ) + W N − ( t ) + W N − ( t ) ≤ M δ,δ ≤ ε , M ( ε + κ ) ≤ , M ≥ , M ′ > , (2.8)where the constant M ≥ will be chosen, M ′ > is some fixed constant. In Section 6, we will provethat the constant M on the right hands of the first four lines in (2.8) can be improved to M . For the rapidly decaying vector U = ( U , U ) with respect to the space variable x , we divide it intothe curl-free part P U (irrotational) and the divergence-free part P U (solenoidal), which is called theHelmholtz decomposition P U = ∇ Φ , P U = ∇ ⊥ Ψ , ∆Φ = div U, ∆Ψ = curl U,U = P U + P U = −∇ ( − ∆) − div U − ∇ ⊥ ( − ∆) − curl U, (2.9)where ∇ ⊥ = ( − ∂ x , ∂ x ) and ∆ is the Laplacian operator. It is easy to know k U k L = k P U k L + k P U k L . (2.10)Next, we give some divergence-curl inequalities. Lemma 2.1.
For any vector function U , < p < ∞ and ≤ β < p − , it holds that kh| x |i β ∇ U k L p . kh| x |i β div U k L p + kh| x |i β curl U k L p , kh| x | − t i∇ U k L . k U k L + kh| x | − t i div U k L + kh| x | − t i curl U k L . (2.11) Proof.
From the second line of (2.9), we get ∇ U = −∇ ( − ∆) − div U − ∇∇ ⊥ ( − ∆) − curl U, where ∇ ( − ∆) − and ∇∇ ⊥ ( − ∆) − are the bounded operators from L p to L p ( < p < ∞ ). On theother hand, h| x |i β belongs to A p class with ≤ β < p − (see [28]). Therefore, the first inequalityin (2.11) is derived.Since the second inequality in (2.11) follows from the direct integration by parts, we omit the detailshere. Lemma 2.2 (Commutator) . For the vector fields ˜Γ defined in (2.3) , one has [˜Γ , P ] := ˜Γ P − P ˜Γ = 0 , [˜Γ , P ] = 0 .Proof. We only prove ˜Γ P U = P (˜Γ U ) , since it always holds that [˜Γ , P ] = [˜Γ , Id − P ] = 0 . Note that [˜Γ , P ] = 0 is obvious for ˜Γ ∈ { ∂ } , we now focus on the case of ˜Γ ∈ { ˜Ω , S} .According to (2.1) and the first line of (2.9), one has − ∆ P ( ˜Ω U ) = −∇ div ˜Ω U = − ˜Ω ∇ div U = ˜Ω( − ∆) P U = − ∆( ˜Ω P U ) . This derives P ( ˜Ω U ) = ˜Ω P U by the uniqueness of the solution to ∆ since U fulfills the rapid decayassumption and P ( ˜Ω U ) , ˜Ω P U decay for the space variable.Analogously, P ( S U ) = S P U follows from − ∆ P ( S U ) = −∇ div S U = − ( S + 2) ∇ div U = ( S + 2)( − ∆) P U = − ∆( S P U ) . Lemma 2.3.
Let f ( t, x ) be a scalar function, then it holds that h| x |i p | f ( t, x ) | . X j =0 2 − j X | a | =0 k∇ a Ω j f ( t, y ) k L py , < p < ∞ , (2.12) h| x |i h| x | − t i| f ( t, x ) | . X j =0 2 − j X | a | =0 kh| y | − t i∇ a Ω j f ( t, y ) k L y , (2.13) h| x |i h| x | − t i | f ( t, x ) | . X j =0 n k Ω j f ( t, y ) k L y + − j X | a | =1 kh| y | − t i∇ a Ω j f ( t, y ) k L y o , (2.14) k f ( t, x ) k L ∞ x . ln (2 + t ) k∇ f ( t, y ) k L y + h t i − ( k f ( t, y ) k L y + k∇ f ( t, y ) k L y ) , (2.15) h| x |i | f ( t, x ) | . kh| y |i ∇ Ω ≤ f ( t, y ) k L y + X p = , k∇ Ω ≤ f ( t, y ) k L py , (2.16) h| x |i| f ( t, x ) | . k f ( t, y ) k H y + k Ω ≤ f ( t, y ) k L y + kh y i∇ Ω ≤ f ( t, y ) k L y , (2.17) where Ω ≤ f stands for P j ≤ Ω j f . Proof.
The inequality (2.12) with p = 2 is just (3.1) of [27]. We now deal with the general case of p ∈ (1 , ∞ ) . Note that (2.12) in the region | x | ≤ follows from the Sobolev embedding W ,p ( R ) ֒ → L ∞ ( R ) .For | x | ≥ , by the Sobolev embedding on the unit circle W ,p ( S ) ֒ → L ∞ ( S ) and the Newton-Leibniz formula in the radial direction, we arrive at | x || f ( t, x ) | p = | x || f ( t, | x | ω ) | p . | x | Z S | Ω ≤ f ( t, | x | ω ) | p dω . | x | Z ∞| x | Z S | Ω ≤ f ( t, rω ) | p − | ∂ r Ω ≤ f ( t, rω ) | dωdr . k Ω ≤ f ( t, y ) k pL py + k∇ Ω ≤ f ( t, y ) k pL py , which yields (2.12).For the inequalities (2.13)–(2.15), see (3.2), (3.4) of [27] and (3.4) of [22], respectively.Next, we start to prove (2.16). By the Sobolev embedding W , ( R ) ֒ → L ∞ ( R ) and ˙ W , ( R ) ֒ → L ( R ) , we have | f ( t, x ) | . k f ( t, y ) k L y + k∇ f ( t, y ) k L y . k∇ f ( t, y ) k L y + k∇ f ( t, y ) k L y , this implies (2.16) when | x | ≤ .For | x | ≥ , it follows from the Sobolev embedding W , ( S ) ֒ → L ∞ ( S ) that | f ( t, x ) | = | f ( t, | x | ω ) | . Z S | Ω ≤ f ( t, | x | ω ) | dω . Z ∞| x | Z S | Ω ≤ f ( t, rω ) | | ∂ r Ω ≤ f ( t, rω ) | dωdr. Multiplying this inequality by | x | and then applying the H ¨older inequality infer | x | | f ( t, x ) | . kh| y |i ∇ Ω ≤ f ( t, y ) k L y (cid:13)(cid:13)(cid:13)(cid:12)(cid:12) Ω ≤ f ( t, y ) (cid:12)(cid:12) (cid:13)(cid:13)(cid:13) L y . kh| y |i ∇ Ω ≤ f ( t, y ) k L y k Ω ≤ f ( t, y ) k L y . kh| y |i ∇ Ω ≤ f ( t, y ) k L y k∇ Ω ≤ f ( t, y ) k L y , where we have used the Sobolev embedding ˙ W , ( R ) ֒ → L ( R ) . Therefore, we achieve (2.16).Finally, we turn to the proof of (2.17). For the case of | x | ≤ , (2.17) is a direct result of the Sobolevembedding H ( R ) ֒ → L ∞ ( R ) .For | x | ≥ , similarly to the proof of (2.16), applying W , ( S ) ֒ → L ∞ ( S ) instead leads to | x | | f ( t, x ) | = | x | | f ( t, | x | ω ) | . | x | Z S | Ω ≤ f ( t, | x | ω ) | dω . | x | Z ∞| x | Z S | Ω ≤ f ( t, rω ) || ∂ r Ω ≤ f ( t, rω ) | dωdr . Z | Ω ≤ f ( t, y ) ||h| y |i∇ Ω ≤ f ( t, y ) | dy. Thus, we derive the desired inequality (2.17).1
Lemma 2.4.
For any multi-indices a, b with | a | ≤ N − and | b | ≤ N − , it holds that h| x |i | Γ a w ( t, x ) | . W | a | +2 ( t ) , h| x |i | Γ a curl u ( t, x ) | . W | a | +2 ( t ) , (2.18) h| x |i h| x | − t i ( | ˜Γ b P u ( t, x ) | + | Γ b σ ( t, x ) | ) . E | b | +2 ( t ) + X | b | +2 ( t ) , h| x |i h| x | − t i ( |∇ ˜Γ a P u ( t, x ) | + |∇ Γ a σ ( t, x ) | ) . E | a | +3 ( t ) + X | a | +3 ( t ) . (2.19) Furthermore, for | x | ≤ h t i / , | ˜Γ b P u ( t, x ) | + | Γ b σ ( t, x ) | . h t i − ln (2 + t ) (cid:8) E | b | +2 ( t ) + X | b | +2 ( t ) (cid:9) , (2.20) Proof.
The proof of (2.18) follows from (2.12) with p = 2 directly. The inequalities in (2.19) can beconcluded from (2.10), (2.11), (2.13) and (2.14).To achieve (2.20), we introduce the cutoff function χ ( s ) ∈ C ∞ such that ≤ χ ( s ) ≤ , χ ( s ) = ( , s ≤ / , , s ≥ / . (2.21)Choosing f ( t, x ) = χ (cid:0) | x |h t i (cid:1) ˜Γ b P u ( t, x ) and χ (cid:0) | x |h t i (cid:1) Γ b σ ( t, x ) in (2.15), we then get (2.20) by (2.10) and(2.11). Lemma 2.5.
For any multi-indices a, b with | a | ≤ N − and | b | ≤ N − , it holds that h| x |i | P ˜Γ a u ( t, x ) | . W | a | +1 ( t ) , (2.22) h| x |i | Γ a curl u ( t, x ) | . W | a | +1 ( t ) , (2.23) h| x |i |∇ Γ a P u ( t, x ) | . W | a | +1 ( t ) , (2.24) h| x |i| P ˜Γ b u ( t, x ) | . E | b | +2 ( t ) + W | b | +1 ( t ) . (2.25) Proof.
Applying (2.16) to P U yields that h| x |i | P U ( t, x ) | . kh| y |i ∇ Ω ≤ P U ( t, y ) k L y + X p = , k∇ Ω ≤ P U ( t, y ) k L py . kh| y |i Ω ≤ curl U ( t, y ) k L y + X p = , k Ω ≤ curl U ( t, y ) k L py , where we have used (2.11). Subsequently, by choosing U = ˜Γ a u , (2.22) is then derived.The inequality (2.23) is a direct result of the Sobolev embedding W , ( R ) ֒ → L ∞ ( R ) .Analogously, we can conclude from (2.11) that h| x |i |∇ Γ a P u ( t, x ) | . kh| y |i ∇∇ ≤ Γ a P u ( t, y ) k L y . kh| y |i ∇ ≤ Γ a curl u ( t, y ) k L y . W | a | +1 ( t ) , which yields (2.24).At last, we turn to the proof of (2.25). Let f ( t, x ) = P ˜Γ b u ( t, x ) in (2.17) and then it concludes from(2.11) that h| x |i| P ˜Γ b u ( t, x ) | . E | b | +2 ( t ) + kh x i∇ P Ω ≤ ˜Γ b u ( t, x ) k L . E | b | +2 ( t ) + kh x i curl Ω ≤ ˜Γ b u ( t, x ) k L , which implies (2.25).2 Combining Lemma 2.4 and 2.5, we obtain the following pointwise estimates. Corollary 2.6.
For any multi-indices a, b with | a | ≤ N − and | b | ≤ N − , it holds that | Γ a σ ( t, x ) | + | ˜Γ a u ( t, x ) | . h| x |i − h| x | − t i − (cid:8) E | a | +2 ( t ) + X | a | +2 ( t ) (cid:9) + h| x |i − W | a | +1 ( t ) , |∇ Γ a σ ( t, x ) | + |∇ ˜Γ a u ( t, x ) | . h| x |i − h| x | − t i − (cid:8) E | a | +3 ( t ) + X | a | +3 ( t ) (cid:9) + h| x |i − W | a | +1 ( t ) , (2.26) and | Γ b σ ( t, x ) | + | ˜Γ b u ( t, x ) | . h| x |i − h| x | − t i − (cid:8) E | b | +2 ( t ) + X | b | +2 ( t ) (cid:9) + h| x |i − (cid:8) E | b | +2 ( t ) + W | b | +1 ( t ) (cid:9) . (2.27) Furthermore, for | x | ≤ h t i / , | Γ a σ ( t, x ) | + | ˜Γ a u ( t, x ) | . h| x |i − W | a | +1 ( t ) + h t i − ln (2 + t ) (cid:8) E | a | +2 ( t ) + X | a | +2 ( t ) (cid:9) . (2.28) g In this subsection, several estimates of the good unknown g will be established. Lemma 3.1.
For m ≤ N − , it holds that G m ( t ) . E m +1 ( t ) + X m +1 ( t ) + W m ( t ) + X | b | + | c |≤ m kh| x |i Q bc k L ( | x |≥h t i / , (3.1) where G m ( t ) := X | a |≤ m kh| x |i∇ ˜Γ a g ( t, x ) k L ( | x |≥h t i / . (3.2) Proof.
Note that r∂ r ˜Γ a g i = x j ∂ j ˜Γ a ( u − σω ) i , where the Einstein summation convention is used. Itfollows from direct computation that there exist the bounded smooth functions f a,bi ( x ) and f a,bij ( x ) in | x | ≥ / such that ˜Γ a ( σω ) i = ω i Γ a σ + 1 | x | X b + c ≤ a f a,bi ( x )Γ c σ∂ j ˜Γ a ( σω ) i = ω i ∂ j Γ a σ + 1 | x | X b + c ≤ a h f a,bi ( x ) ∂ j Γ c σ + f a,bij ( x )Γ c σ i . (3.3)Then we have r∂ r ˜Γ a g i + ω j X b + c ≤ a [ f a,bi ( x ) ∂ j Γ c σ + f a,bij ( x )Γ c σ ]= x j ∂ j ˜Γ a u i − ω i x j ∂ j Γ a σ = x j ( ∂ j ˜Γ a u i − ∂ i ˜Γ a u j ) + ( x j ∂ i − x i ∂ j )˜Γ a u j + x i div ˜Γ a u − ω i S Γ a σ + ω i t∂ t Γ a σ = x j ǫ ji Γ a curl u + ǫ ji Ω(˜Γ a u j ) + x i Q a + ω i ( t − | x | ) ∂ t Γ a σ − ω i S Γ a σ, (3.4)3where the volume form ǫ ji is the sign of the arrangement { ji } and we have used the facts of ∂ j U i − ∂ i U j = ǫ ji curl U and x j ∂ i − x i ∂ j = ǫ ji Ω . Taking the L ( | x | ≥ h t i / norm on the both sides of (3.4)yields X | a |≤ m kh| x |i ∂ r ˜Γ a g ( t, x ) k L ( | x |≥h t i / . E m +1 ( t ) + X m +1 ( t ) + W m ( t )+ X | b | + | c |≤ m kh| x |i Q bc k L ( | x |≥h t i / . (3.5)In addition, it follows from (2.2) that G m ( t ) . X | a |≤ m kh| x |i ∂ r ˜Γ a g k L ( | x |≥h t i / + E m +1 ( t ) . (3.6)Collecting (3.5) and (3.6) together leads to (3.1). Lemma 3.2.
For | a | ≤ N − , | b | ≤ N − and | x | ≥ h t i / , it holds that h| x |i| ˜Γ a g ( t, x ) | . G | a | +1 ( t ) + E | a | +1 ( t ) , h| x |i |∇ ˜Γ b g ( t, x ) | . G | b | +2 ( t ) . (3.7) Proof.
Applying the Sobolev embedding on the unit circle and the Newton-Leibnitz formula in the radialdirection derive that h| x |i ℓ | U ( t, x ) | . h| x |i ℓ Z S | ˜Ω ≤ U ( t, | x | ω ) | dω . h| x |i ℓ Z ∞| x | Z S | ˜Ω ≤ U ( t, rω ) ∂ r ˜Ω ≤ U ( t, rω ) | rdωdr. Choosing U ( t, x ) = ˜Γ a g ( t, x ) , ∇ ˜Γ b g ( t, x ) in the above equality with ℓ = 1 , , respectively, we then getthat for | x | ≥ h t i / , h| x |i | ˜Γ a g ( t, x ) | . k ˜Ω ≤ ˜Γ a g ( t, y ) k L ( | y |≥h t i / + kh| y |i∇ ˜Ω ≤ ˜Γ a g ( t, y ) k L ( | y |≥h t i / , h| x |i |∇ ˜Γ b g ( t, x ) | . kh| y |i∇∇ ≤ ˜Ω ≤ ˜Γ b g ( t, y ) k L ( | y |≥h t i / . This completes the proof of Lemma 3.2.Based on Lemma 3.1 and 3.2, we have the following estimates.
Lemma 3.3.
Under bootstrap assumptions (2.8) , it holds that for m ≤ N − , X | b | + | c |≤ m kh| x |i ( | Q bc | + | Q bc | ) k L ( | x |≥h t i / . M ε G m ( t ) + E m +1 ( t )[1 + G N − ( t )] . (3.8) Proof.
According to (1.20) and equalities (3.3), one easily gets ˜Γ b u i = ˜Γ b g i + ω i Γ b σ + 1 | x | X b + b ≤ b f b,b i ( x )Γ b σ,∂ j ˜Γ c u i = ∂ j ˜Γ c g i + ω i ∂ j Γ c σ + 1 | x | X c + c ≤ c h f c,c i ( x ) ∂ j Γ c σ + f c,c ij ( x )Γ c σ i . (3.9)4Substituting (2.2) and (3.9) into (2.5) yields Q bc = Γ b σ n ∂ i ˜Γ c g i + 1 | x | X c + c ≤ c h f c,c i ( x ) ∂ i Γ c σ + f c,c ii ( x )Γ c σ io − ∂ i Γ c σ n ˜Γ b g i + 1 | x | X b + b ≤ b f b,b i ( x )Γ b σ o , (3.10)and Q bc i = 1 | x | Γ b σ ΩΓ c σ − ω i ∂ j Γ c σ n ˜Γ b g j + 1 | x | X b + b ≤ b f b j ( x )Γ b σ o − ˜Γ b u j n ∂ j ˜Γ c g i + 1 | x | X c + c ≤ c h f c i ( x ) ∂ j Γ c σ + f c ij ( x )Γ c σ io . (3.11)By applying the pointwise estimates (2.26) to the terms that containing the factor | x | in (3.10) and (3.11)directly, we obtain that kh| x |i ( | Q bc | + | Q bc | ) k L ( | x |≥h t i / . E m +1 ( t ) + kh| x |i|∇ ˜Γ c g | ( | Γ b σ | + | ˜Γ b u | ) k L ( | x |≥h t i / + kh| x |i ˜Γ b g ∇ Γ c σ k L ( | x |≥h t i / . (3.12)At last, by the virtue of the estimates in Lemma 3.1 and 3.2, we will deal with the remaining terms in(3.10) and (3.11).Due to | b | + | c | ≤ m ≤ N − ≤ N − , then | b | ≤ N − or | c | ≤ N − holds. For | b | ≤ N − ,it follows from (2.8) and (2.26) that (cid:13)(cid:13)(cid:13) | Γ b σ | + | ˜Γ b u | (cid:13)(cid:13)(cid:13) L ∞ ( | x |≥h t i / . h t i − (cid:8) E | b | +2 ( t ) + X | b | +2 ( t ) + W | b | +1 ( t ) (cid:9) . M ε.
This together with (3.7) implies that kh| x |i ˜Γ b g ∇ Γ c σ k L ( | x |≥h t i / + kh| x |i|∇ ˜Γ c g | ( | Γ b σ | + | ˜Γ b u | ) k L ( | x |≥h t i / . E | c | +1 ( t )[ G | b | +1 ( t ) + E | b | +1 ( t )] + M ε G | c | ( t ) . M ε G m ( t ) + E m +1 ( t )[1 + G N − ( t )] . (3.13)For | c | ≤ N − , by (2.19), we have k∇ Γ c σ k L ∞ ( | x |≥h t i / . h| x |i − h| x | − t i − (cid:8) E | c | +3 ( t ) + X | c | +3 ( t ) (cid:9) . M ε h| x |i − h| x | − t i − . Therefore, kh| x |i ˜Γ b g ∇ Γ c σ k L ( | x |≥h t i / + kh| x |i|∇ ˜Γ c g | ( | Γ b σ | + | ˜Γ b u | ) k L ( | x |≥h t i / . E | b | ( t )[ G | c | +2 ( t ) + E | c | +2 ( t )] + E | b | ( t ) + M ε kh| x |i h| x | − t i − ˜Γ b g k L ( | x |≥ h t i / . (3.14)For the last term in (3.14), performing the integration by parts for the radial direction yields kh| x |i h| x | − t i − ˜Γ b g k L ( | x |≥ h t i / . Z ∞ Z S | ˜Γ b g ( t, rω ) | h r i r h − χ (cid:0) r h t i (cid:1)i dωd arctan( r − t ) . G | b | ( t ) + E | b | ( t ) , (3.15)5where the cutoff function χ is defined by (2.21).Substituting (3.13)–(3.15) into (3.12) derives (3.8). This completes the proof of Lemma 3.3.Combining Lemma 3.1–3.3, we obtain the following result. Corollary 3.4.
Under bootstrap assumptions (2.8) , for | a | ≤ N − , | b | ≤ N − and | x | ≥ h t i / , itholds that h| x |i| ˜Γ a g ( t, x ) | . E | a | +1 ( t ) + X | a | +1 ( t ) + W | a | ( t ) , h| x |i |∇ ˜Γ b g ( t, x ) | . E | b | +2 ( t ) + X | b | +2 ( t ) + W | b | +1 ( t ) . (3.16) Moreover, for any integer m with ≤ m ≤ N − , it holds that X | b | + | c |≤ m kh| x |i ( | Q bc | + | Q bc | ) k L ( | x |≥h t i / . E m +1 ( t ) + M ε [ X m +1 ( t ) + W m ( t )] . (3.17) Proof.
It concludes from (3.1) and (3.8) with m = N − , (2.8) and the smallness of M ε that G N − ( t ) . E N − ( t )[1 + G N − ( t )] + X N − ( t ) + W N − ( t ) , which implies G N − ( t ) . M ε . Together with (3.1) and (3.8) again, this yields G m ( t ) . E m +1 ( t ) + X m +1 ( t ) + W m ( t ) . (3.18)Substituting (3.18) into (3.7) and (3.8) completes the proof of Corollary 3.4. X m ( t ) Lemma 3.5 (Weighted ˙ H x ) . Under bootstrap assumptions (2.8) , for any integer m with ≤ m ≤ N , itholds that X m ( t ) . E m ( t ) + W m − ( t ) . (3.19) Proof.
For | a | ≤ m − , it follows from direct computations and equations (2.4) that ( | x | − t ) ∂ t ˜Γ a u i = | x | ( Q a i − ∂ i Γ a σ ) − t S ˜Γ a u i + tx j ∂ j ˜Γ a u i = | x | Q a i − x j ( x j ∂ i − x i ∂ j )Γ a σ − x i S Γ a σ + tx i ∂ t Γ a σ − t S ˜Γ a u i + tx j ( ∂ j ˜Γ a u i − ∂ i ˜Γ a u j ) + t ( x j ∂ i − x i ∂ j )˜Γ a u j + tx i div ˜Γ a u = | x | Q a i − x j ǫ ji ΩΓ a σ − x i S Γ a σ + tx i Q a − t S ˜Γ a u i + tx j ǫ ji Γ a curl u + tǫ ji Ω(˜Γ a u j ) . (3.20)Here we point out that the main difference between (3.20) and the analogous equality of ∂ t P ˜Γ a u i in [27]lies in the presence of the vorticity Γ a curl u in (3.20).On the other hand, we can obtain ( | x | − t ) ∂ t Γ a σ = | x | Q a − x j ǫ ji Ω(˜Γ a u i ) − x i S ˜Γ a u i − t S Γ a σ + tx i Q a i , ( | x | − t ) ∂ i Γ a σ = x j ǫ ji ΩΓ a σ + x i S Γ a σ − tx i Q a − t Q a i + t S ˜Γ a u i + tx j ǫ ij Γ a curl u + tǫ ij Ω(˜Γ a u j ) , ( | x | − t ) div ˜Γ a u = x j ǫ ji Ω(˜Γ a u i ) + x i S ˜Γ a u i − tx i Q a i − t Q a + t S Γ a σ. (3.21)6In view of h| x | − t i . || x | − t | , by dividing | x | + t and then taking L x norm on the both sides of(3.20) and (3.21), we arrive at X m ( t ) . E m ( t ) + W m − ( t ) + X | b | + | c |≤ m − kh| x | + t i ( | Q bc | + | Q bc | ) k L x , (3.22)where Q bc , Q bc are defined in (2.5).We next investigate the L x norms of Q bc and Q bc , which are divided into the two parts of | x | ≥ h t i / and | x | ≤ h t i / .It is easy to deduce from (3.17) that X | b | + | c |≤ m − kh| x | + t i ( | Q bc | + | Q bc | ) k L ( | x |≥h t i / . E m ( t ) + W m − ( t ) + M ε X m ( t ) . (3.23)We now deal with kh| x | − t i ( | Q bc | + | Q bc | ) k L ( | x |≤h t i / . In fact, only ˜Γ b u · ∇ ˜Γ c u requires to be treatedsince the treatments on the other terms Γ b σ div ˜Γ c u , ˜Γ b u ·∇ Γ c σ , Γ b σ ∇ Γ c σ in Q bc and Q bc are analogous.Similarly to Lemma 3.3, it always holds that | b | ≤ N − or | c | ≤ N − . For the case of | c | ≤ N − , applying (2.26) to ∇ ˜Γ c u leads to kh| x | − t i ˜Γ b u · ∇ ˜Γ c u k L ( | x |≤h t i / . E | b | ( t )[ h t i W | c | +1 ( t ) + E | c | +3 ( t ) + X | c | +3 ( t )] . E m ( t ) , (3.24)where we have used assumptions (2.8).For the case of | b | ≤ min { N − , m − } , by utilizing (2.28) to ˜Γ b u , we can see that kh| x | − t i ˜Γ b u · ∇ ˜Γ c u k L ( | x |≤h t i / . k ˜Γ b u k L ∞ ( | x |≤h t i / (cid:13)(cid:13)(cid:13) h| x | − t i χ (cid:0) | x |h t i (cid:1) ∇ Γ c u (cid:13)(cid:13)(cid:13) L . (cid:13)(cid:13)(cid:13) h| x | − t i χ (cid:0) | x |h t i (cid:1) ∇ Γ c u (cid:13)(cid:13)(cid:13) L n W | b | +1 ( t ) + h t i − ln (2 + t )[ E | b | +2 ( t ) + X | b | +2 ( t )] o . (cid:13)(cid:13)(cid:13) h| x | − t i χ (cid:0) | x |h t i (cid:1) ∇ Γ c u (cid:13)(cid:13)(cid:13) L n W N − ( t ) + h t i − ln (2 + t )[ E m ( t ) + X m ( t )] o . (3.25)In addition, it follows from (2.11) that (cid:13)(cid:13)(cid:13) h| x | − t i χ (cid:0) | x |h t i (cid:1) ∇ Γ c u (cid:13)(cid:13)(cid:13) L . E | c | ( t ) + (cid:13)(cid:13)(cid:13) h| x | − t i∇ (cid:16) χ (cid:0) | x |h t i (cid:1) Γ c u (cid:17)(cid:13)(cid:13)(cid:13) L . E | c | ( t ) + kh| x | − t i div Γ c u k L + h t ikh| x |i curl Γ c u k L . E | c | ( t ) + X | c | +1 ( t ) + h t iW | c | ( t ) . E m ( t ) + X m ( t ) + h t iW m − ( t ) . (3.26)Substituting (3.26) into (3.25) derives kh| x | − t i ˜Γ b u · ∇ ˜Γ c u k L ( | x |≤h t i / . (cid:8) E m ( t ) + X m ( t ) + h t iW m − ( t ) (cid:9)(cid:8) M δ + h t i − ln (2 + t )[ E m ( t ) + X m ( t )] (cid:9) . E m ( t ) + M ε X m ( t ) + W m − ( t ) , (3.27)7where we have also used assumptions (2.8).Collecting (3.24) and (3.27) yields that X | b | + | c |≤ m − kh| x | + t i ( | Q bc | + | Q bc | ) k L ( | x |≤h t i / . E m ( t ) + M ε X m ( t ) + W m − ( t ) . (3.28)By combining (3.23) and (3.28), we eventually achieve X | b | + | c |≤ m − kh| x | + t i ( | Q bc | + | Q bc | ) k L x . E m ( t ) + M ε X m ( t ) + W m − ( t ) . (3.29)Plugging (3.29) into (3.22) with the smallness of M ε , then (3.19) is proved.
Note that the decay rate of the irrotational part P u of the velocity u is merely ε h t i − away from thelight cone (see Lemma 2.4). This is far to achieve the desired lifespan T δ = O ( δ ) , for examples, when δ = e − ε or δ = e − e ε are chosen, whose reason has been explained in Section 1. It is required toimprove the related pointwise estimates in Section 3. ( σ, u ) In this subsection, by the virtue of the weighted identities (3.4), (3.20) and (3.21), the pointwise estimatesof ∇ ˜Γ a u , ∇ Γ a σ in (2.26), (2.28) and ∇ ˜Γ a g in (3.16) can be improved as follows. Lemma 4.1.
Under bootstrap assumptions (2.8) , if | a | ≤ N − , then for | x | ≥ h t i / , it holds that | ∂ t ˜Γ a u ( t, x ) | + | ∂ t Γ a σ ( t, x ) | + | div ˜Γ a u ( t, x ) | + |∇ Γ a σ ( t, x ) | . M ε h| x |i M ′ ε − h| x | − t i − , |∇ ˜Γ a u ( t, x ) | . M ε h| x |i M ′ ε − h| x | − t i − , (4.1) and |∇ ˜Γ a g ( t, x ) | . M ε h| x |i M ′ ε − h| x | − t i − . (4.2) On the other hand, for | x | ≤ h t i / , we have | ∂ t ˜Γ a u ( t, x ) | + | ∂ t Γ a σ ( t, x ) | . M δ h t i M ′ ε − + M ε h t i M ′ ε − ln(2 + t ) , | div ˜Γ a u ( t, x ) | + |∇ Γ a σ ( t, x ) | . M δ h t i M ′ ε − + M ε h t i M ′ ε − ln(2 + t ) , h| x |i|∇ ˜Γ a u ( t, x ) | . M δ h t i M ′ ε + M ε h t i M ′ ε − ln(2 + t ) . (4.3) Proof.
For | x | ≥ h t i / , it follows from (2.23), (2.26), (3.20) and (3.21) that h| x | − t i ( | ∂ t ˜Γ a u ( t, x ) | + | ∂ t Γ a σ ( t, x ) | + | div ˜Γ a u ( t, x ) | + |∇ Γ a σ ( t, x ) | ) . X | b |≤| a | +1 ( | Γ b σ | + | ˜Γ b u | ) + h| x |i| Γ a curl u | + h| x |i X b + c ≤ a ( | Q bc | + | Q bc | ) . W | a | +1 ( t ) h| x |i − + h| x |i − h| x | − t i − n E | a | +3 ( t ) + X | a | +3 ( t ) o + W | a | +1 ( t ) h| x |i − + h| x |i X b + c ≤ a ( | Q bc | + | Q bc | ) . (4.4)8Applying (2.26) and (3.16) to (3.10) and (3.11) yields | Q bc | + | Q bc | . h| x |i − (cid:16) M δ h| x |i M ′ ε − + M ε h| x |i M ′ ε − h| x | − t i − (cid:17) + | ˜Γ b g ∇ Γ c σ | + |∇ ˜Γ c g | ( | Γ b σ | + | ˜Γ b u | ) . M δ h| x |i M ′ ε − + M ε h| x |i M ′ ε − h| x | − t i − , (4.5)where we have used the bootstrap assumptions (2.8). Substituting (4.5) into (4.4) infers h| x | − t i ( | ∂ t ˜Γ a u ( t, x ) | + | ∂ t Γ a σ ( t, x ) | + | div ˜Γ a u ( t, x ) | + |∇ Γ a σ ( t, x ) | ) . M δ h| x |i M ′ ε − + M ε h| x |i M ′ ε − h| x | − t i − + M ε h| x |i M ′ ε − h| x | − t i − . M ε h| x |i M ′ ε − h| x | − t i − . This leads to the first inequality in (4.1).Next, we turn to the proof of the second inequality in (4.1). By (2.2), we have h| x |i|∇ ˜Γ a u ( t, x ) | . | Γ ≤ ˜Γ a u ( t, x ) | + | r∂ r ˜Γ a u ( t, x ) | . | Γ ≤ ˜Γ a u ( t, x ) | + | t∂ t ˜Γ a u ( t, x ) | . (4.6)This, together with the estimate of ∂ t ˜Γ a u ( t, x ) in (4.1), yields the second inequality in (4.1).It is not hard to conclude from (3.4) that for | x | ≥ h t i / , h| x |i|∇ ˜Γ a g ( t, x ) | . h| x |i| Γ a curl u | + X | b |≤| a | +1 ( | ˜Γ b g | + | Γ b σ | + | ˜Γ b u | ) + h| x |i|Q a | + h| x | − t i| ∂ t Γ a σ | . h| x |i − W | a | +1 ( t ) + h| x |i − h| x | − t i − n E | a | +3 ( t ) + X | a | +3 ( t ) o + h| x |i|Q a | + h| x | − t i| ∂ t Γ a σ | , (4.7)where we have used (2.23) and (2.26).Combining (4.7) with (4.1) and (4.5) yields (4.2).Finally, we turn to the proof of (4.3). For | x | ≤ h t i / , by using (2.23) and (2.28) to (3.20) and(3.21) directly, we arrive at h t i ( | ∂ t ˜Γ a u ( t, x ) | + | ∂ t Γ a σ ( t, x ) | + | div ˜Γ a u ( t, x ) | + |∇ Γ a σ ( t, x ) | ) . X | b |≤| a | +1 ( | Γ b σ | + | ˜Γ b u | ) + h| x |i| Γ a curl u | + h t i X b + c ≤ a ( | Q bc | + | Q bc | ) . W | a | +1 ( t ) + h t i − ln (2 + t ) n E | a | +3 ( t ) + X | a | +3 ( t ) o + h t i − ln(2 + t ) (cid:16) E | a | +3 ( t ) + X | a | +3 ( t ) (cid:17) . M δ h t i M ′ ε + M ε h t i M ′ ε − ln(2 + t ) . Then we can achieve the first two inequalities in (4.3). Combining (4.6) with the estimates ∂ t ˜Γ a u ( t, x ) in (4.3), we get the third inequality in (4.3).9 L ∞ - L ∞ estimates for the linear wave equation In this subsection, we will establish some weighted L ∞ - L ∞ estimates for the solutions to the linear waveequations. Consider the following Cauchy problem (cid:3) ϕ := ∂ t ϕ − ∆ ϕ = F , ( ϕ, ∂ t ϕ ) | t =0 = ( ϕ , ϕ ) . (4.8)Then ϕ = ϕ hom + ϕ inh , where (cid:3) ϕ hom = 0 , ( ϕ hom , ∂ t ϕ hom ) | t =0 = ( ϕ , ϕ ) , (4.9)and (cid:3) ϕ inh = F , ( ϕ inh , ∂ t ϕ inh ) | t =0 = (0 , . Lemma 4.2. [Proposition 4.1 and 4.2 of [16]] Let < ν < and µ > , then it holds that h| x | + t i h| x | − t i ν | ϕ inh ( t, x ) | . ˜ M µ + ν ( F )( t ) , (4.10) h| x |i h| x | − t i ν |∇ ϕ inh ( t, x ) | . X | a | + j ≤ ˜ M µ + ν ( ∇ a Ω j F )( t ) , (4.11) where ˜ M µ + ν ( F )( t ) = sup ( s,y ) ∈ Λ ( t ) {h| y |i h| y | + s i µ + ν |F ( s, y ) |} + sup ( s,y ) ∈ Λ ( t ) {h s i + µ + ν h| y | − s i|F ( s, y ) |} , (4.12) and Λ ( t ) = { ( s, y ) ∈ [0 , t ] × R : || y | − s | ≤ s/ , | y | ≥ } , Λ ( t ) = [0 , t ] × R \ Λ ( t ) = { ( s, y ) ∈ [0 , t ] × R : || y | − s | ≥ s/ , or | y | ≤ } . (4.13) Remark 4.1.
The notation ˜ M µ + ν ( F )( t ) on the right hand side of (4.10) and (4.11) is slightly differentfrom that in [16], in which is M ν ( F )( t ) with M ν ( F )( t ) = sup ( s,y ) ∈ Λ ( t ) {h| y |i h| y | + s i µ + ν |F ( s, y ) |} + sup ( s,y ) ∈ Λ ( t ) {h s i + µ + ν h| y | − s i|F ( s, y ) |} . Unfortunately, Lemma 4.2 can not be applied directly for our problem. We next give a modifiedversion as follows.
Lemma 4.3.
Let < µ , ν < and µ > , then for | x | ≤ h t i , it holds that h| x | + t i − µ h| x | − t i ν | ϕ inh ( t, x ) | . M µ + ν − µ ( F )( t ) , (4.14) h| x |i h| x | − t i ν |∇ ϕ inh ( t, x ) | . X | a | + j ≤ M µ + ν ( ∇ a Ω j F )( t ) , (4.15) where M µ + ν ( F )( t ) = sup ( s,y ) ∈ Λ ( t ) , | y |≤ h t i {h| y |i h| y | + s i µ + ν |F ( s, y ) |} + sup ( s,y ) ∈ Λ ( t ) {h s i + µ + ν h| y | − s i|F ( s, y ) |} . (4.16)0 Proof.
Recall the Poisson formula ϕ inh ( t, x ) = 12 π Z t Z | y − x |≤ t − s F ( s, y ) dyds p ( t − s ) − | y − x | . In the domain { ( y, s ) : | y − x | ≤ t − s } , one has h| y | + s i . h| x | + t i and | y | ≤ | x | + t ≤ h t i .Therefore, we obtain h| x | + t i − µ | ϕ inh ( t, x ) | . Z t Z | y − x |≤ t − s h| y | + s i − µ |F ( s, y ) | dyds p ( t − s ) − | y − x | . Applying (4.10) to the above integration yields (4.14). The proof of (4.15) is analogous.Next, we study the pointwise estimates of ϕ hom . Lemma 4.4 (Estimates of ϕ hom ) . Let ϕ hom be defined by (4.9) . It holds that h| x | + t i h| x | − t i | ϕ hom ( t, x ) | . kh| y |i ϕ ( y ) k W , y + kh| y |i ϕ ( y ) k W , y , (4.17) h| x | + t i h| x | − t i |∇ ϕ hom ( t, x ) | . kh| y |i ϕ ( y ) k W , y + kh| y |i ϕ ( y ) k W , y . (4.18) Proof.
The inequality (4.17) is just Lemma 3.2 of [19].Next, we derive (4.18) by (4.17). In fact, for any vector field ˆ Z ∈ { ∂, S , Ω , t∂ i + x i ∂ t , i = 1 , } , onehas (cid:3) ˆ Zϕ hom = 0 . Applying (4.17) to ˆ Zϕ hom yields h| x | + t i h| x | − t i | ˆ Zϕ hom ( t, x ) | . kh| y |i ˆ Zϕ hom (0 , y ) k W , y + kh| y |i ∂ t ˆ Zϕ hom (0 , y ) k W , y . kh| y |i ϕ ( y ) k W , y + kh| y |i ϕ ( y ) k W , y . This, together with h| x | − t i|∇ ϕ hom ( t, x ) | . X ˆ Z ∈{ ∂, S , Ω ,t∂ i + x i ∂ t ,i =1 , } | ˆ Zϕ hom ( t, x ) | , derives Lemma 4.4. u This subsection is devoted to improve the pointwise estimates of u by the weighted L ∞ - L ∞ estimatesin subsection 4.2. For this purpose, we need to find the related wave equation hidden in the equations(1.19).By the Helmholtz decomposition in subsection 2.2, there exists a potential function φ ( t, x ) such that u = P u + P u = ∇ φ + P u. (4.19)The inherent wave equation of φ can be directly deduced from (1.19), see appendix A for details. (cid:3) φ = F := ∂ t A − (2 − σ ) Q − u · Q , (4.20)where the nonlocal term A is defined by A := − ( − ∆) − curl( u curl u ) , lim | x |→∞ A ( t, x ) = 0 . (4.21)1In addition, σ = − ∂ t φ + A −
12 ( | u | − σ ) , (4.22)which is achieved in appendix A.Acting ( S + 2) a s Z a z on (4.20), we can get the equation of Γ a φ : (cid:3) Γ a φ = F a := X b ≤ a C ab Γ b ∂ t A + X b + c ≤ a C abc Q bc + X b + c + d ≤ a C abcd n Q bc Γ d σ − Q bc i ˜Γ d u i o , (4.23)where C a ∗∗∗ are some suitable constants.At first, we deal with the pointwise estimates of the nonlocal term A . Lemma 4.5 (Estimates of A ) . Under bootstrap assumptions (2.8) , for | a ′ | ≤ N − , it holds that | Γ a ′ A ( t, x ) | . h| x |i − M δ h t i M ′ ε (cid:8) M δ + M ε h t i − ln(2 + t ) (cid:9) . (4.24) Proof.
In view of (2.16), (4.24) can be directly achieved by the following L p estimates X | a |≤| a ′ | +1 n kh| x |i ∇ Γ a A ( t, x ) k L + X p = , k∇ Γ a A ( t, x ) k L p o . M δ h t i M ′ ε (cid:8) M δ + M ε h t i − ln(2 + t ) (cid:9) . (4.25)Next we focus on the proof of (4.25). Note that Γ a A = − X b + c ≤ a C abc ( − ∆) − curl(˜Γ b u Γ c curl u ) , (4.26)which can be achieved by applying the following equality repeatedly ∆Γ A = curl( u Γ curl u + ˜Γ u curl u ) . Indeed, for
Γ = Ω (the other case of Γ is analogous), direct computation yields ∆Ω A = Ω∆ A = Ω curl( u curl u ) = curl ˜Ω( u curl u )= curl { Ω( u curl u ) − u ⊥ curl u } = curl { (Ω u ) curl u + u Ω curl u − u ⊥ curl u } = curl { ( ˜Ω u ) curl u + u Ω curl u } . It concludes from the L p boundedness of the Riesz operators and (4.26) that X p = , k∇ Γ a A ( t, x ) k L px . X p = , X b + c ≤ a k ˜Γ b u Γ c curl u k L px . X p = , X b + c ≤ a kh| x |i − Γ b u k L ∞ x kh| x |i Γ c curl u k L px . W | a | ( t ) (cid:8) W | a | +1 ( t ) + h t i − ln(2 + t )[ E | a | +2 ( t ) + X | a | +2 ( t )] (cid:9) . M δ h t i M ′ ε (cid:8) M δ + M ε h t i − ln(2 + t ) (cid:9) , h| x |i belongs to A class. Thereafter, we have kh| x |i ∇ Γ a A ( t, x ) k L x . X b + c ≤ a kh| x |i ˜Γ b u Γ c curl u k L x . X b + c ≤ a kh| x |i − ˜Γ b u k L ∞ x kh| x |i Γ c curl u k L x . M δ h t i M ′ ε (cid:8) M δ + M ε h t i − ln(2 + t ) (cid:9) . Thus, we have proved (4.25).Secondly, we focus on the pointwise estimates of the potential function Γ a φ , where φ is defined in(4.19). Lemma 4.6 (Estimates of Γ a φ ) . Under bootstrap assumptions (2.8) , for | a | ≤ N − and | x | ≤ h t i , itholds that h| x | + t i h| x | − t i | Γ a φ ( t, x ) | . M δ h t i + M ε. (4.27)
Proof.
By applying the weighted L ∞ - L ∞ estimates (4.14) and (4.17) with µ = and ν = µ = to Γ a φ in (4.23), we obtain h| x | + t i h| x | − t i | Γ a φ ( t, x ) | . kh| y |i Γ a φ (0 , y ) k W , y + kh| y |i ∂ t Γ a φ (0 , y ) k W , y + M − ( F a )( t ) . (4.28)According to the definition of F a in (4.23), we arrive at M − ( F a )( t ) . X b ≤ a M − (Γ b ∂ t A )( t ) + X b + c ≤ a M − ( Q bc )( t )+ X b + c + d ≤ a M − ( | Q bc Γ d σ | + | Q bc i ˜Γ d u i | )( t ) . (4.29)Note that it only suffices to deal with the two terms on the right hand side of the first line in (4.29), sincethe treatment on the cubic nonlinearities in the second line are much easier.It follows from the definition (4.16) of M − ( F )( t ) and (4.24) that X b ≤ a M − (Γ b ∂ t A )( t ) . X b ≤ a sup s ≤ t,y h| y |i h| y | + s i | Γ b ∂ t A ( s, y ) | . M δ sup s ≤ t,y h| y | + s i h| y |i − h s i M ′ ε (cid:8) M δ + M ε h s i − ln(2 + s ) (cid:9) . M δ. (4.30)The control of M − ( Q bc )( t ) will be divided into three parts corresponding to the domains Λ ( t ) , D and D , where D := Λ ( t ) ∩ { ( s, y ) : | y | ≤ h s i / } , D := Λ ( t ) ∩ { ( s, y ) : 4 h s i / ≤ | y | ≤ h t i} , and the definitions of Λ ( t ) and Λ ( t ) see (4.13).3In D , applying (2.26) to (2.5) yields that X b + c ≤ a sup ( s,y ) ∈D h| y |i h| y | + s i | Q bc ( s, y ) | . sup s ≤ t h s i +2 M ′ ε (cid:16) M δ + M ε h s i − (cid:17)(cid:16) M δ + M ε h s i − (cid:17) . M δ h t i + M ε. (4.31)In D , by using (2.26) again to (2.5), we deduce that X b + c ≤ a sup ( s,y ) ∈D h| y |i h| y | + s i | Q bc ( s, y ) | . sup | y |≤ h t i ,s ≤ t h| y |i +2 M ′ ε (cid:16) M δ + M ε h| y |i − (cid:17)(cid:16) M δ + M ε h| y |i − (cid:17) . M δ h t i + M ε. (4.32)In Λ ( t ) , the null condition structure in (3.10) will play a crucial rule (see Section 5 below). Applying(2.26) to the terms that containing the factor | x | in (3.10), we then see that X b + c ≤ a sup ( s,y ) ∈ Λ ( t ) h s i h| y | − s i| Q bc ( s, y ) | . sup | y |≤ s/ ≤ t/ h s i (cid:16) M δ h s i M ′ ε − h| y | − s i + M ε h s i M ′ ε − (cid:17) + X b + c ≤ a sup ( s,y ) ∈ Λ ( t ) h s i h| y | − s i ( |∇ Γ b σ ˜Γ c g | + | Γ b σ ∇ ˜Γ c g | ) . (4.33)By employing (2.26) and (3.16) to ∇ Γ b σ ˜Γ c g and Γ b σ ∇ ˜Γ c g , one has X b + c ≤ a sup ( s,y ) ∈ Λ ( t ) h s i h| y | − s i ( |∇ Γ b σ ˜Γ c g | + | Γ b σ ∇ ˜Γ c g | ) . M δ + M ε. (4.34)Collecting (4.31)–(4.34) together, we eventually arrive at X b + c ≤ a M − ( Q bc )( t ) . M δ h t i + M ε. (4.35)At last, we turn to estimate the initial data on the right hand side of (4.28). It is deduced from (1.14) and(4.22) that kh| y |i Γ a φ (0 , y ) k W , y + kh| y |i ∂ t Γ a φ (0 , y ) k W , y . ε . M ε. (4.36)Substituting (4.29), (4.30), (4.35) and (4.36) into (4.28) yields (4.27).Next, the pointwise estimates of the good unknown g can be improved. Lemma 4.7 (Improved estimates of ˜Γ a g ) . Under bootstrap assumptions (2.8) , for | a | ≤ N − and h t i / ≤ | x | ≤ h t i , it holds that | ˜Γ a g ( t, x ) | . M δ h t i − + M ε h t i − + M ε h t i M ′ ε − h| x | − t i . (4.37)4 Proof.
The key idea to achieve (4.37) is to make full use of the potential function φ with the relations u = ∇ φ + P u and (4.22). It follows from the definition (1.20), (3.9) and tedious computation that t ˜Γ a g i = ( t − | x | )˜Γ a u i + | x | ˜Γ a u i − tω i ˜Γ a σ − t | x | X b + c ≤ a f a,bi ( x )Γ c σ = ( t − | x | )˜Γ a u i + | x | P ˜Γ a u i + | x | ˜Γ a ∂ i φ + tω i ˜Γ a ∂ t φ − tω i Γ a A− t | x | X b + c ≤ a f a,bi ( x )Γ c σ + 12 tω i Γ a ( | u | − σ )= ( t − | x | )˜Γ a u i + | x | P ˜Γ a u i + X b ≤ a C ab ( | x | ∂ i + tω i ∂ t )Γ b φ − tω i Γ a A− t | x | X b + c ≤ a f a,bi ( x )Γ c σ + tω i X b + c ≤ a C abc (˜Γ b u j ˜Γ c u j − Γ b σ Γ c σ ) . (4.38)By utilizing (2.2) and the first line of (3.9) to the last summation in (4.38), we arrive at t ˜Γ a g i = ( t − | x | )˜Γ a u i + | x | P ˜Γ a u i + X b ≤ a C ab ( ω i S Γ b φ + ΩΓ b φ ) − tω i Γ a A− t | x | X b + c ≤ a f a,bi ( x )Γ c σ + tω i X b + c ≤ a C abc (˜Γ b g j ˜Γ c u j + ω j Γ b σ ˜Γ c g j )+ tω i | x | X b + b + c ≤ a f b,b j ( x )Γ b σ ˜Γ c u j + tω i | x | X b + c + c ≤ b f c,c j ( x ) ω j Γ b σ Γ c σ. (4.39)Applying (2.22), (2.26), (4.24) and (4.27) to (4.39) leads to h t i| ˜Γ a g i | . M δ h t i M ′ ε − + M ε h t i M ′ ε − h| x | − t i + h t i − (cid:8) M δ h t i + M ε (cid:9) + M δ h t i M ′ ε − (cid:8) M δ + M ε h t i − ln(2 + t ) (cid:9) + M ε h t i X b ≤ a | ˜Γ b g | . (4.40)Therefore, combining (4.40) with the smallness of M ε derives (4.37).Finally, we turn to the estimates of the velocity u = ∇ φ + P u . Lemma 4.8 (Estimates of ∇ Γ a φ ) . Under bootstrap assumptions (2.8) , for | a | ≤ N − and | x | ≤ h t i ,it holds that h| x | − t i |∇ Γ a φ ( t, x ) | . M δ h t i + M ε. (4.41)
Proof.
Applying (4.15) and (4.18) to (4.23) with ν = and µ = yields h| x |i h| x | − t i | Γ a φ ( t, x ) | . kh| y |i Γ a φ (0 , y ) k W , y + kh| y |i ∂ t Γ a φ (0 , y ) k W , y + X | b |≤| a | +1 M ( F b )( t ) . (4.42)Similarly to Lemma 4.6, we can obtain X | b |≤| a | +1 M ( F b )( t ) . X | b |≤| a | +1 M (Γ b ∂ t A )( t ) + X | b | + | c |≤| a | +1 M ( Q bc )( t )+ X | b | + | c | + | d |≤| a | +1 M ( | Q bc Γ d σ | + | Q bc i ˜Γ d u i | )( t ) . (4.43)5It follows from the definition (4.16) of M ( F )( t ) and (4.24) that X | b |≤| a | +1 M (Γ b ∂ t A )( t ) . X | b |≤| a | +1 sup s ≤ t,y h| y |i h| y | + s i | Γ b ∂ t A ( s, y ) | . M δ sup s ≤ t,y h| y | + s i h| y |i − h s i M ′ ε (cid:8) M δ + M ε h s i − ln(2 + s ) (cid:9) . M δ h t i +2 M ′ ε ln(2 + t ) . M δ h t i . (4.44)In D , applying (2.28) and (4.3) to (2.5) implies X b + c ≤ a sup ( s,y ) ∈D h| y |i h| y | + s i | Q bc ( s, y ) | . sup s ≤ t h s i (cid:16) M δ h s i M ′ ε + M ε h s i M ′ ε − ln(2 + s ) (cid:17) . M δ h t i + M ε. (4.45)In D , by using (2.26), (4.1) to (2.5), we get X b + c ≤ a sup ( s,y ) ∈D h| y |i h| y | + s i | Q bc ( s, y ) | . sup | y |≤ h t i ,s ≤ t h| y |i (cid:16) M δ h s i M ′ ε + M ε h| y |i M ′ ε − (cid:17) . M δ h t i + M ε. (4.46)Similarly to Lemma 4.6, the treatment in Λ ( t ) will also need to make full use of the null conditionstructure in (3.10). Indeed, applying (2.26) to the terms that containing the factor | x | in (3.10), we thenhave X b + c ≤ a sup ( s,y ) ∈ Λ ( t ) h s i h| y | − s i| Q bc ( s, y ) | . sup | y |≤ s/ ≤ t/ h s i (cid:16) M δ h s i M ′ ε h| y | − s i + M ε h s i M ′ ε − (cid:17) + X b + c ≤ a sup ( s,y ) ∈ Λ ( t ) h s i h| y | − s i n |∇ Γ b σ ˜Γ c g | + | Γ b σ ∇ ˜Γ c g | o . (4.47)On the other hand, by using (4.1) and (4.37) to ∇ Γ b σ ˜Γ c g , one has X b + c ≤ a sup ( s,y ) ∈ Λ ( t ) h s i h| y | − s i|∇ Γ b σ ˜Γ c g | . sup s ≤ t n M δ h s i M ′ ε + − − + M ε h s i M ′ ε + − − o . M δ h t i + M ε. (4.48)In addition, employing (2.19), (4.2) to Γ b σ ∇ ˜Γ c g yields that X b + c ≤ a sup ( s,y ) ∈ Λ ( t ) h s i h| y | − s i| Γ b σ ∇ ˜Γ c g | . M ε. (4.49)6Collecting (4.45)–(4.49) together, we eventually achieve X b + c ≤ a M ( Q bc )( t ) . M δ h t i + M ε. (4.50)Note that similarly to (4.36), we have kh| y |i Γ a φ (0 , y ) k W , y + kh| y |i ∂ t Γ a φ (0 , y ) k W , y . M ε. (4.51)Therefore, substituting (4.43), (4.44), (4.50) and (4.51) into (4.42) derives (4.41).Combining (4.41) and (2.22) with the decomposition ˜Γ a u = ∇ Γ a φ + P ˜Γ a u , we arrive at Corollary 4.9.
Under bootstrap assumptions (2.8) , for | a | ≤ N − and | x | ≤ h t i , it holds that | ˜Γ a u ( t, x ) | . M δ h| x |i − + h| x | − t i − (cid:8) M δ h t i + M ε (cid:9) . (4.52) This subsection is aimed to establish the elementary energy estimates for E m ( t ) , which is defined by(2.7). Lemma 5.1.
Under bootstrap assumptions (2.8) , we have that E N ( t ′ ) . E N (0) + Z t ′ E N ( t ) (cid:8) M δ + M ε h t i − (cid:9) dt + Z t ′ E N ( t ) M δ h t i M ′ ε (cid:8) M δ + M ε h t i − (cid:9) dt, (5.1) E N − ( t ′ ) . E N − (0) + Z t ′ E N − ( t ) (cid:8) M δ + M ε h t i − (cid:9) dt + Z t ′ E N − ( t ) M δ h t i M ′ ε (cid:8) M δ + M ε h t i − (cid:9) dt. (5.2) Proof.
For the multi-index a with | a | = m ≤ N , multiplying (2.4) by e q Γ a σ and e q ˜Γ a u , respectively,yields the following equality ∂ t { e q ( | Γ a σ | + | ˜Γ a u | ) } + 2 div { e q (1 − σ )Γ a σ ˜Γ a u } + div { e q u ( | Γ a σ | + | ˜Γ a u | ) } + e q h| x | − t i X i =1 n | ˜Γ a u i − ω i Γ a σ | − u i ω i ( | Γ a σ | + | ˜Γ a u | ) + 2 σω i Γ a σ ˜Γ a u i o = e q ( | Γ a σ | + | ˜Γ a u | ) div u − e q Γ a σ ˜Γ a u · ∇ σ + X b + c = a,c
M δ + M ε h t i − (cid:9) + E N − ( t ) (cid:8) M δ + M ε h t i − (cid:9) . (5.20)For all m ≤ N − , substituting (5.5), (5.6), (5.14), (5.17), (5.18) and (5.20) into (5.3) yields (5.2).0 Before taking the estimates of the vorticity, we will establish some useful lemmas. Recalling the defini-tion of the specific vorticity (2.6), then it is easy to check that ( ∂ t + u · ∇ ) w = 0 . (5.21) Lemma 5.2. Under bootstrap assumptions (2.8) , for m ≤ N − and k ≤ N − , it holds that X | a |≤ m h| x |i|∇ Γ a w ( t, x ) | . X | a ′ |≤ m +1 | Γ a ′ w ( t, x ) | + t h| x |i − W N − ( t ) X | b |≤ m | ˜Γ b u ( t, x ) | , (5.22) and X | a |≤ k h| x |i|∇ Γ a w ( t, x ) | . X | a ′ |≤ k +1 | Γ a ′ w ( t, x ) | . (5.23) Furthermore, for | x | ≥ h t i / , it holds that X | a |≤ m h| x |i|∇ Γ a w ( t, x ) | . X | a ′ |≤ m +1 | Γ a ′ w ( t, x ) | . (5.24) Proof. Similarly to the derivation of (2.4), acting ( S + 1) a s Z a z on (5.21) derives ( ∂ t + u · ∇ )Γ a w = X b + c = a,c
Lemma 5.3. Under bootstrap assumptions (2.8) , we have W N − ( t ) . W N − ( t ) , W N − ( t ) . W N − ( t ) . (5.29) Proof. Note that Γ a curl u = Γ a w + X b + c = a C abc Γ b σ Γ c curl u. Multiplying the above equality by h| x |i and taking L norm lead to kh| x |i Γ a curl u k L . kh| x |i Γ a w k L + X b + c = a, | b |≤ N − k Γ b σ k L ∞ kh| x |i Γ c curl u k L + X b + c = a, | b |≥ N − kh| x |i Γ b σ Γ c curl u k L . (5.30)In addition, it follows from (2.19) that X b + c = a, | b |≤ N − k Γ b σ k L ∞ kh| x |i Γ c curl u k L . X c ≤ a kh| x |i Γ c curl u k L . (5.31)For all | a | ≤ N − , substituting (5.31) into (5.30) yields the first inequality of (5.29).Next, we deal the second line of (5.30). It is easy to check that kh| x |i Γ b σ Γ c curl u k L . h t i − k Γ b σ k L ( | x |≥ h t i ) kh| x |i Γ c curl u k L ∞ ( | x |≥ h t i ) + h t i − (cid:13)(cid:13)(cid:13) h| x | − t i Γ b σ h| x |i (cid:13)(cid:13)(cid:13) L ( | x |≤ h t i ) kh| x |i Γ c curl u k L ∞ ( | x |≤ h t i ) . (5.32)Applying the Hardy inequality infers (cid:13)(cid:13)(cid:13) h| x | − t i Γ b σ h| x |i (cid:13)(cid:13)(cid:13) L . (cid:13)(cid:13)(cid:13) h| x | − t i Γ b σ | x | ln | x | (cid:13)(cid:13)(cid:13) L . E | b | ( t ) + X | b | +1 ( t ) . M ε h t i M ′ ε . (5.33)From | b | + | c | ≤ N − and | b | ≥ N − , then | c | ≤ N − holds. By plugging (2.18) and (5.33) into(5.32), we derive X b + c = a, | c |≤ N − kh| x |i Γ b σ Γ c curl u k L . M ε X | c |≤ N − W | c | +2 ( t ) . M ε W N − ( t ) . (5.34)For all | a | ≤ N − , combining (5.30), (5.31), (5.34) with the smallness of M ε yields the secondinequality of (5.29).With these lemmas, we begin to take the estimates of the vorticity. Lemma 5.4 ( L estimates) . Under bootstrap assumptions (2.8) , for W m ( t ) defined by (2.7) , it holds that W N − ( t ′ ) . W N − (0) + Z t ′ W N − ( t ) (cid:8) M δ + M ε h t i − (cid:9) dt, (5.35)2 W N − ( t ′ ) . W N − (0) + Z t ′ W N − ( t ) (cid:8) M δ + M ε h t i − (cid:9) dt + Z t ′ M εδ h t i M ′ ε − W N − ( t ) dt. (5.36) Proof. Multiplying (5.25) by h| x |i e q ( | x |− t ) Γ a w infers ∂ t (cid:16) e q (cid:12)(cid:12) h| x |i Γ a w (cid:12)(cid:12) (cid:17) + e q (cid:12)(cid:12) h| x |i Γ a w (cid:12)(cid:12) h| x | − t i + div (cid:16) e q u (cid:12)(cid:12) h| x |i Γ a w (cid:12)(cid:12) (cid:17) = e q (cid:12)(cid:12) h| x |i Γ a w (cid:12)(cid:12) (cid:16) div u + u · ∇ q (cid:17) + u · ∇ (cid:16) h| x |i (cid:17) e q | Γ a w | + X b + c = a,c