Global solutions for the critical Burgers equation in the Besov spaces and the large time behavior

ScienceDirect

Ann. I. H. Poincaré – AN 32 (2015) 687–713

www.elsevier.com/locate/anihpc

Global solutions for the critical Burgers equation in the Besov spaces and the large time behavior

Tsukasa Iwabuchi

Department of Mathematics, Chuo University, Kasuga, Bunkyoku, Tokyo, 115-8551, Japan Received 2 October 2013; received in revised form 20 March 2014; accepted 22 March 2014

Available online 2 April 2014

Abstract

We consider the Cauchy problem for the critical Burgers equation. The existence and the uniqueness of global solutions for small initial data are studied in the Besov spaceB˙⁰_∞,1(Rⁿ)and it is shown that the global solutions are bounded in time. We also study the large time behavior of the solutions with the initial datau₀∈L¹(Rⁿ)∩ ˙B_∞⁰_,1(Rⁿ)to show that the solution behaves like the Poisson kernel.

©2014 Elsevier Masson SAS. All rights reserved.

Keywords:Burgers equation; Besov spaces; Large time behavior; Poisson kernel

1. Introduction

We consider then-dimensional Burgers equation

⎧⎪

⎨

⎪⎩

∂_tu+ n j=1

u∂_xju+Λ^αu=0 fort >0, x∈Rⁿ, u(0, x)=u₀(x) forx∈Rⁿ,

(1.1)

whereΛ=(−)¹². The Burgers equation withα=0 andα=2 has received an extensive amount of attention since the studies by Burgers in the 1940s. Ifα=0, the equation is the basic example of a PDE evolution leading to shocks.

Ifα=2, it provides an accessible model for studying the interaction between nonlinear and dissipative phenomena.

The valueα=1 is a threshold for the occurrence of singularity in finite time or the global regularity (see[3,14,15,20]).

The aim of this paper is to study the existence and the uniqueness of global solutions to(1.1)withα=1, which is bounded in time, and to show the solutions behaving like the Poisson kernel in the large time. There is also another purpose, namely, establishing a method to deal with such problems for a wider class of equations including the quasi-geostrophic equation which derivative orders are balanced on the linear part and the nonlinear part.

For the Cauchy problem(1.1), Kiselev, Nazarov and Shterenberg[20]considered in the periodic setting S¹ to show the finite time blow up for the supercritical case 0< α <1 and the global well-posedness inH¹²(S¹)for the

http://dx.doi.org/10.1016/j.anihpc.2014.03.002

0294-1449/©2014 Elsevier Masson SAS. All rights reserved.

critical caseα=1 and inH^s(S¹) (s >3/2−α)for the subcritical case 1< α <2. Dong, Du and Li[14]considered both of spacesS¹andRto show the finite time blow up for the supercritical case and the global well-posedness in H¹²(S¹), H¹²(R)for the critical case and inL^1/(α−1)(S¹), L^1/(α−1)(R)(1< α <2) for the subcritical case. Finite time blow up for the supercritical case is also shown by Alibaud, Droniou and Vovelle[3]. Miao and Wu[23]showed the global well-posedness in the critical caseα=1 for the initial data in the Besov spacesB˙

1 p

p,1(R)with 1p <∞.

The notion of entropy solution has been introduced by Alibaud[1]to show the global well-posedness inL^∞(Rⁿ).

On the study of weak solutions, Alibaud and Andreianov[2]showed that the uniqueness of a weak solution fails for 0< α <1. Chan and Czubak[13]establish global regularizing effects for then-dimensional Burgers equation with sufficiently integrable initial datau₀in the critical caseα=1. We also refer to the results on the global regularizing effects by Droniou, Gallouet and Vovelle[15]in the subcritical caseα >1 and Silvestre[27]for the Hamilton–Jacobi equations. Our goal on global solutions is considering the critical caseα=1 to show the existence of global solutions, which are bounded in time, for small initial data u0∈ ˙B_∞⁰_,1(Rⁿ) in all space dimensions. We note the following embeddings

B˙

n p

p,1

Rⁿ

→ ˙B_∞⁰_,1 Rⁿ

→L^∞ Rⁿ

for 1p∞,

and that the boundedness in time of solutions are needed to deal with the large time behavior. We also mention the scaling invariance to (1.1)in the critical caseα=1. For the solution u to(1.1)with α=1, letu_λ be defined by u_λ(t, x):=u(λt, λx)forλ >0. Thenu_λmaintains Eq.(1.1)and we have the following norm invariance:

sup

t >0

u_λ(t)

L^∞=sup

t >0

u(t )

L^∞ for anyλ >0.

ThenL^∞(Rⁿ)satisfies the above invariant property, and the spacesH˙¹²(R),L^∞(Rⁿ)andB˙

n p

p,1(Rⁿ) (1p∞)also satisfy such norm invariance.

On the large time behavior, Biler, Karch and Woyczynski[6]considered the equation with the semigroup generated by(−)^α2 − (0< α <2)to study the asymptotic expansion of solutions. Biler, Karch and Woyczynski[7–9]also studied with a Lévy semigroup, which includes the semigroups generated by(−)^α2 −ε(1< α <2,ε >0), to show the large time behavior like the self-similar solutions. For Eq.(1.1), Karch, Miao and Xu[19]considered the subcritical case 1< α <2 in one space dimension to show that the large time asymptotic is described by the rarefaction waves.

Alibaud, Imbert and Karch[4]considered the critical caseα=1 and the supercritical case 0< α <1 to consider the initial data satisfying

u₀(x)=c+ x

−∞

m(dy), u₀(·)−c, u₀(·)−

c+ ∞

−∞

m(dy)

∈L¹(R),

wherec∈Randmis a finite signed measure onRwith_∞

−∞m >0. They[4] showed that forα=1 the solutions converge to the self-similar solution, and for α <1 the nonlinearity is negligible in the asymptotic expansion of solutions. Our goal on the large time behavior is considering the case α=1 to show that the solutions for initial data u₀∈L¹(Rⁿ)∩ ˙B_∞⁰_,1(Rⁿ)behave like the Poisson kernel as t→ ∞. We also show a higher order asymptotic expansion, imposing the additional decay in the distance on the initial data.

Throughout this paper,P is defined by P (x):=F⁻¹

e^−|ξ|

(x)= Γ (ⁿ⁺₂¹) πⁿ⁺²¹(1+ |x|²)ⁿ⁺²¹

forx∈Rⁿ, whereΓ (·)is the Gamma function, and letP_t be the Poisson kernel:

Pt(x):=t⁻ⁿP t⁻¹x

fort >0, x∈Rⁿ.

To study Eq.(1.1)withα=1, we consider the following integral equation u(t )=P_t∗u₀−1

2 t

0

P_t−τ∗ _n

j=1

∂_xju(τ )²

dτ fort0. (1.2)

The following are our results on the global solutions and the large time behavior.

Theorem 1.1.Letn1. There exist positive constantsδandCsuch that for anyu₀∈ ˙B_∞⁰_,1(Rⁿ)withu_0B˙⁰

∞,1δ, a unique global solutionuto(1.2)exists in the spaceC([0,∞); ˙B_∞⁰_,1(Rⁿ))∩L¹(0,∞; ˙B_∞¹_,1(Rⁿ)). Furthermore, it holds that fort0

u(t ) _˙

B_∞⁰_,1Cu0_B˙⁰

∞,1exp

C t 0

∇u(τ )

L^∞dτ

. (1.3)

Theorem 1.2. Let n 1. There exists δ > 0 such that for any initial data u₀∈ L¹(Rⁿ)∩ ˙B_∞⁰_,1(Rⁿ) with u_0B˙⁰

∞,1 δ, a unique global solutionuto(1.2)exists in the spaceC([0,∞);L¹(Rⁿ))∩C([0,∞); ˙B_∞⁰_,1(Rⁿ))∩ L¹(0,∞; ˙B_∞¹_,1(Rⁿ)). Furthermore, the solutionusatisfies the following.

(i) For1p∞, it holds that

tlim→∞t^n(1−p1) u(t )−MPt

L^p=0, (1.4)

whereM:=

Rⁿu0(y) dy.

(ii) Letu₀satisfy| · |u₀(·)∈L¹(Rⁿ)and1p∞. Then it holds that fort >0 u(t )−MP_t

L^p

⎧⎨

⎩

Ct^{−(1−p1)−1}log(e+t ) ifn=1, Ct^{−n(1−p1)−1} ifn2,

(1.5)

tlim→∞t^1+n(1−1p)

u(t )−MPt+ ∇Pt·

Rⁿ

yu0(y) dy+1 2

n j=1

(∂x_jPt) t 0

Rⁿ

u(τ, y)²dy dτ

+1 2

n j=1

t 0

(∂_xjP_t−τ)∗(MP_τ+1)²dτ−1 2

n j=1

(∂_xjP_t) t 0

Rⁿ

MP_τ+1(y)2

dy dτ

L^p

=0. (1.6)

Remark 1.3.On the boundedness(1.5)in the case of one space dimension, the order is optimal and we cannot remove log(e+t ). Indeed, it is possible to show that

u(t )−MP_t

L²cM²t⁻³²log(e+t ) (1.7)

for larget. The above estimate will be proved in Section7.

The methods of the proof ofTheorem 1.1are applying contraction argument and making use of a priori estimate. To see a key idea, we explain importance on introducing the spaceL¹(0,∞; ˙B_∞¹_,1(Rⁿ)), and how to prove the uniqueness of the solution briefly. The spaceL¹(0,∞; ˙B_∞¹_,1(Rⁿ))plays an essential role in the proof of the above theorems. If we try to show the estimate inL^∞(Rⁿ), we hope to show the following estimate

∞ 0

∇P_t∗fL^∞dtCfL^∞.

However the above estimate never holds since for the functionf defined byf (x)=e^ik·x(k∈Rⁿ) it holds that ∇Pt∗f (x)

L^∞= e^ik·xxe^−t|x|

L^∞=Ct⁻¹

andt⁻¹ is not inL¹(0,∞). On the other hand, it follows from the embedding B˙_∞¹_,1(Rⁿ) → ˙W^1,∞(Rⁿ)that for Littlewood–Paley’s dyadic decomposition{φ_k}k∈Zwith suppφ_k⊂ {ξ∈Rⁿ|2^k−1|ξ|2^k+1}

∞ 0

∇P_t∗fL^∞dtC ∞ 0

P_t∗f_B˙¹

∞,1dtC

k∈Z

2^k ∞

0

P_t∗φ_k∗fL^∞dt

C

k∈Z

2^k ∞ 0

e^−ct2kdtφ_k∗fL^∞

=Cf_B˙⁰

∞,1.

Then∇Pt∗fL^∞ is integrable and it is one of the sufficient methods to study the Cauchy problem in the space L¹(0,∞; ˙B_∞¹_,1(Rⁿ))for the initial datau0∈ ˙B_∞⁰_,1(Rⁿ). On the proof of the uniqueness of the solution, we consider the following equation of the divergence form

⎧⎪

⎨

⎪⎩

∂_tu+ n j=1

∂_xj(vu)+Λu=0, u(0, x)=u₀(x),

(1.8)

for any given functionv∈L¹(0,∞; ˙B_∞¹_,1(Rⁿ))and to show a priori estimate u(t ) _˙

B_∞⁰_,1Cu₀B˙_∞⁰_,1exp

C t 0

v(τ ) _˙

B_∞¹_,1dτ

, (1.9)

modifying the result by Miao and Wu[23], where the termv∂_xjuis considered instead of∂_xj(vu)of(1.8). We will consider a modified version of Eq.(1.8)to show a priori estimate. We should note that(1.9)is not enough for(1.3) but we will show that it is possible to replacev_B˙¹

∞,1 with∇vL^∞ forv=uwith certain adjustment.

The method of the proof ofTheorem 1.2is to show the decay estimates of the solutionuast→ ∞and to estimate the integral equation(1.2)with the decay estimates. It is easy to deal with the linear partP_t ∗u₀ by the analogous methods for the heat kernel. On the nonlinear part, it is difficult to show the decay estimates ast→ ∞by the analogous argument for heat equations. Indeed, when we consider∂tu−u=∂xu²to treat the derivative, we use the estimate

∇e^t f

L^pCt⁻¹²fL^p

andt⁻¹² is integrable locally. One can deal with the solution in Lebesgue spacesL^p(Rⁿ)as a solution satisfying a integral equation by the integrability in time. On the other hand, it holds for the Poisson kernel that

∇Pt∗fL^pCt⁻¹fL^p,

and t⁻¹ is not integrable. Then, we need to impose the regularity of one order derivative on the solution, and L¹(0,∞; ˙B_∞¹_,1(Rⁿ)) is important to obtain the decay estimate. Considering the solution in L¹(0,∞; ˙B_∞¹_,1(Rⁿ)), we will apply a modified version of Gronwall’s inequality to show decay estimates of the solutions. Once we obtain the decay estimates, we can show the large time behavior of the solution in the analogous way to that for the heat equations (see[16–18,24,25]). We also refer on the asymptotic expansion of solutions to the dissipative equation with the fractional Laplacian(−)^α2 (1< α2)to the result[29].

This paper is organized as follows. In Section2, we introduce the definition of Besov spaces, its properties and some propositions on the Poisson kernel. We show a priori estimate related to(1.3) and(1.9) in Section 3, and Theorem 1.1is proved in Section4. We show the decay estimate of the solution in Section5, andTheorem 1.2is proved in Section6. The estimate(1.7)is proved in Section7.

Throughout this paper, we use the following notations.{φ_k}k∈Zdenotes Littlewood–Paley’s dyadic decomposition, i.e., letφ∈S(Rⁿ)satisfy suppφ⊂ {ξ∈Rⁿ|2⁻¹|ξ|2},

k∈Zφ(2^−kξ )≡1 for anyξ∈Rⁿ\ {0}, whereφis the Fourier transform ofφ, and let{φ_k}k∈Zbe defined byφ_k(ξ ):=φ(2 ^−kξ ). LetS_kbe defined byS_kf :=

kkφ_k∗f.

2. Preliminary

We introduce the definition of Besov spaces and propositions on Besov spaces, the linear estimates and the Poisson kernel. The asymptotic behavior of the solution to∂tu+Λu=0 is also shown. The idea of proof in this section is known for the heat kernel but we give the proof for the paper being self-contained.

Definition.Let{φ_k}k∈Zbe Littlewood Paley’s dyadic decomposition. Fors∈Rand 1p, q∞, the homogeneous Besov spaceB˙_p,q^s (Rⁿ)is defined by

B˙_p,q^s Rⁿ

:=

u∈S

Rⁿ u_B˙^s

p,q<∞,

k∈Z

φ_k∗u=uinS Rⁿ

, uB˙_p,q^s := 2^skφ_k∗uL^p

k∈Z

^q(Z).

Remark 2.1.By the argument in Kozono and Yamazaki[22], the Besov spaces of the above definition are complete ifs < n/p and 1q∞, ors=n/p andq=1. The Besov space B˙_∞⁰_,1(Rⁿ), in which we consider the Cauchy problem, is complete.

Proposition 2.2.(See[12,28].) Lets∈Rand1p, q, r, p_j∞(j=1,2,3,4).

(i) B˙_p,r^s (Rⁿ) → ˙B_p,q^s (Rⁿ)ifrq. (ii) B˙^s+n(

1 r−_p¹)

r,q (Rⁿ) → ˙B_p,q^s (Rⁿ)ifrp.

(iii) B˙_p,1⁰ (Rⁿ) →L^p(Rⁿ) → ˙B_p,⁰_∞(Rⁿ).

(iv) Lets >0and1/p=1/p1+1/p2=1/p3+1/p4. Then it holds that f g_B˙^s

p,1C

f_B˙^s

p1,1gL^p2+ fL^p3g_B˙^s

p4,1

. (2.1)

Proposition 2.3.Lets >0 and letp,p_j (j=1,2,3,4)satisfy1p, p_j∞ (j=1,2,3,4)and1/p=1/p1+ 1/p2=1/p3+1/p4. Then it holds that

f g_B˙^s

p,1C

f_B˙^s

p1,1g_B˙⁰

p2,1+ f_B˙⁰

p3,1g_B˙^s

p4,1

. (2.2)

Proof. (2.2)is obtained by(2.1)and the embeddingB˙_p,1⁰ (Rⁿ) →L^p(Rⁿ). 2 Proposition 2.4.Lets,s∈R,1p, q∞andα∈(N∪ {0})ⁿ.

(i) There existC, c >0such that

φk∗Pt∗fL^pCe^−ct2kφk∗fL^p, (2.3)

for anyk∈Zandf ∈L^p(Rⁿ).

(ii) Ifs >s, it holds that

Pt∗fB˙_p,1^s Ct^−(s−˜s)f_B˙s˜

p,∞. (2.4)

(iii) Ifqp, it holds that ∇^αP_t∗f

L^pCt^{−|α|−n(1q−p1)}fL^q, (2.5)

where|α| =α₁+α₂+ · · · +α_n.

Remark 2.5.We note on the estimate(2.4)that this is a smoothing effect on not only the derivative indicess,s but also the interpolation indices 1,∞and the power of t depends only ons,s. Such estimate fore^t is known in the result by Kozono, Ogawa and Taniuchi[21].

Proof of Proposition 2.4. LetΨ_k be defined byΨ_k=φ_k−1+φ_k+φ_k+1. We have fromφ_k=Ψ_kφ_k and Hausdorff Young’s inequality

φk∗Pt∗f (x)

L^p= F⁻¹

e^−t|ξ|Ψkφkf

L^pC F⁻¹

e^−t|ξ|Ψk

L¹φk∗fL^p.

On the estimate ofF⁻¹[e^−t|ξ|Ψ_k]L¹, lets > n/2 and we apply the change of variableξ →2^kξ,x→2^−kx and Hölder’s inequality, and estimate the norm ofH^s(Rⁿ)directly to obtain

F⁻¹

e^−t|ξ|Ψ_k(ξ )

L¹= F⁻¹

e^−t2k|ξ|Ψ₀(ξ ) (x)

L¹_x

C 1+ |x|s

F⁻¹

e^−t|ξ|Ψ₀(ξ )(x)

L²_x

C e^−t2k|ξ|Ψ₀(ξ )

H_ξ^s

Ce^−ct2k. Then(2.3)is obtained.

To prove(2.4), we have from(2.3) Pt∗fB˙_p,1^s C

k∈Z

e^−ct2k2^skφk∗fL^p

=Ct^−(s−˜s)

k∈Z

e^−ct2k t2^ks−˜s

2^sk˜ φ_k∗fL^p

Ct^−(s−˜s)f_B˙s˜ p,∞

k∈Z

e^−ct2k t2^ks−˜s

.

It is sufficient to show that the supremum of the last sum with respect tot >0 is finite. Lett be in[2^k0,2^k0+1]for somek₀∈Z, then it holds that 2^k+k0t2^k2^k+k0+1and

k∈Z

e^−ct2k t2^ks−˜s

k∈Z

e^−c2k+k0

2^k+k0+1s−˜s

=

k∈Z

e^−c2k

2^k+1s−˜s

<∞. This completes the proof of(2.4).

For the proof of(2.5)in the caseq=pandα=0, it is sufficient to apply Hausdorff–Young’s inequality and the factPtL¹ =1. In the caseq < porα >0, it follows from Hausdorff–Young’s inequality with 1/p=1/r+1/q−1 and∇P_tL^r=Ct^−n(1−1r)=Ct^{−n(q1−p1)}that

∇^αP_t∗f

L^p ∇^αP_t

L^rfL^q =Ct^{−n(1q−p1)}fL^q. Therefore, we complete the proof of(2.5). 2

Proposition 2.6.Lets∈Rand1p∞. (i) It holds that

P_t∗f_L¹

t(0,∞; ˙B^s+_p,1¹)CfB˙_p,1^s (2.6)

for allf ∈ ˙B_p,1^s (Rⁿ).

(ii) It holds that ^t

0

P_t−τ∗f (τ ) dτ

L¹_t(0,∞; ˙B_p,1^s+1)

Cf_L¹_{(0,∞; ˙B}^s

p,1) (2.7)

for allf ∈L¹(0,∞; ˙B_p,1^s (Rⁿ)).

Proof. By(2.3), it holds that P_t∗φ_k∗f_L¹

t(0,∞;L^p)C e^−ct2kφ_k∗fL^p

L¹_t(0,∞)C2^−kφ_k∗fL^p. Therefore, we have from the above estimate

P_t∗f_L¹

t(0,∞; ˙B_p,1^s+1)=

k∈Z

2^(s+1)kP_t∗φ_k∗f_L¹_t(0,∞;L^p₎C

k∈Z

2^skφ_k∗fL^p= fB˙_p,1^s .

This completes the proof of(2.6). In order to prove(2.7), we apply Minkowski’s inequality and(2.6)to obtain ^t

0

P_t−τ∗f (τ ) dτ

L¹_t(0,∞; ˙B_p,1^s+1)

∞ 0

P_t−τ∗f (τ )

L¹_t(τ,∞; ˙B_p,1^s+1)dτ

C

∞ 0

f (τ ) _˙

B_p,1^s dτ

=CfL¹(0,∞; ˙B_p,1^s ). This completes the proof of(2.7). 2

Proposition 2.7.Let1p∞.

(i) It holds that

P_t+1(·)−P_t(·)

L^pCt^{−n(1−p1)−1} (2.8)

for allt >0.

(ii) Forf∈L¹(Rⁿ), it holds that

tlim→∞t^n(1−p1)

P_t∗f−P_t

Rⁿ

f (y) dy L^p

=0. (2.9)

(iii) Forf∈L¹(Rⁿ)with| · |f (·)L¹(Rⁿ), it holds that sup

t >0

t^n(1−p1)+1

P_t∗f−P_t

Rⁿ

f (y) dy L^p

<∞, (2.10)

tlim→∞t^n(1−p1)+1

P_t∗f −P_t

Rⁿ

f (x) dx+ ∇P_t·

Rⁿ

yf (y) dy L^p

=0. (2.11)

Proof. On the proof of(2.8), since

P_t+1(x)−P_t(x)= 1 0

∂_θP_t+θ(x) dθ

= 1 0

(−n)(t+θ )⁻ⁿ⁻¹P

(t+θ )⁻¹x

+(t+θ )⁻ⁿ(∇P )

(t+θ )⁻¹x

· −x (t+θ )²

dθ, we have from Minkowski’s inequality and the change of variable

P_t+1−P_tL^pC 1 0

(t+θ )⁻ⁿ⁻¹ P

(t+θ )⁻¹·

L^p+(t+θ )⁻ⁿ⁻² | · |(∇P )

(t+θ )⁻¹·

L^p

dθ

C

1 0

(t+θ )^{−n(1−1p)−1}dθ

PL^p+ | · |∇P

L^p

Ct⁻ⁿ⁽¹⁻

1 p)−1

. Therefore(2.8)is obtained.

For the proof of(2.9), we considerf ∈C₀^∞(Rⁿ)firstly. Iff ∈C₀^∞(Rⁿ), there existsR >0 such that suppf ⊂ {x∈Rⁿ| |x|R}. It holds that

(P_t∗f )(x)−P_t(x)

Rⁿ

f (y) dy=

Rⁿ

P_t(x−y)−P_t(x) f (y) dy

=

Rⁿ

1 0

∂_θP_t(x−θy) dθ f (y) dy

=

|y|R

1 0

t⁻ⁿ(∇P )

t⁻¹(x−θy)

·−y

t dθ f (y) dy. (2.12)

Then, we takeL^p(Rⁿ)norm and apply Minkowski’s inequality to obtain t^n(1−1p)

(P_t∗f )(x)−P_t(x)

Rⁿ

f (y) dy L^p

Ct^n(1−p1) t⁻ⁿ∇P t⁻¹·

L^p

|y|R

|y| t f (y) dy Ct⁻¹RfL¹

→0 ast→ ∞. Forf ∈L¹(Rⁿ), sinceC₀^∞(Rⁿ)is dense inL¹(Rⁿ)and

t^n(1−1p)

(Pt∗f )−Pt

Rⁿ

f (y) dy L^p

CfL¹,

we also obtain(2.9)by density argument.

On the proof of(2.10), we have from(2.12)and Minkowski’s inequality t^n(1−1p)+1

(P_t∗f )−P_t

Rⁿ

f (y) dy L^p

Ct^n(1−p1)+1 t⁻ⁿ∇P t⁻¹·

L^p

Rⁿ

|y|

t f (y)dy C | · |f

L¹<∞. (2.13)

Therefore(2.10)is obtained.

To prove(2.11), we assumef ∈C₀^∞(Rⁿ)before dealing with the casef ∈L¹(Rⁿ)with| · |f (·)∈L¹(Rⁿ). Let R >0 satisfy suppf ⊂ {ξ∈Rⁿ| |x|R}. It follows from(2.12)that

(P_t∗f )(x)−P_t(x)

Rⁿ

f (y) dy+ ∇P_t·

Rⁿ

yf (y) dy

= −t⁻ⁿ n j=1

Rⁿ

1 0

(∂_xjP )

t⁻¹(x−θy)

−(∂_xjP ) t⁻¹x

dθ·y_j

t f (y) dy

= −t⁻ⁿ n j=1

Rⁿ

1 0

1 0

∂_μ (∂_xjP )

t⁻¹(x−μθy) dθ y_j

t f (y) dy

=t⁻ⁿ n j=1

Rⁿ

1 0

1 0

(∇∂_xjP )

t⁻¹(x−μθy) dμ·θy

t dθy_j

t f (y) dy.

Then, we have from the above identity t^n(1−pn)+1

(Pt∗f )(·)−Pt(·)

Rⁿ

f (y) dy+ ∇Pt(·)·

Rⁿ

yf (y) dy L^p

Ct^n(1−p1)+1t⁻ⁿ n j=1

|y|R

1 0

1 0

∇∂_xjP t⁻¹·

L^pθ dθ dμ t⁻²|y|²f (y)dy Ct⁻¹R²

|y|R

f (y)dy

→0 ast→ ∞.

For the proof in the casef ∈L¹(Rⁿ)with| · |f (·)∈L¹(Rⁿ), we use the estimate(2.13)and the density argument.

Then the proof of(2.11)is completed. 2 3. A priori estimate

We consider a priori estimate of the following equation:

⎧⎪

⎨

⎪⎩

∂_tu+ n j=1

∂_xjF (v_j, u)+(−)¹²u=G(u) fort >0, x∈Rⁿ,

u(0, x)=u₀(x) forx∈Rⁿ,

(3.1) wherev=(v₁, v₂, . . . , v_n)is the given vector field, andF (·,·)be defined by

F (v, u):=

k2∈Z

(Sk₂−3v)(φk₂∗u)=

k2∈Z

k1k2−3

(φk₁∗v)(φk₂∗u), (3.2)

andGis a smooth function. F (v, u)is one factor of the decomposition due to J.-M. Bony[10]on the interaction of low and high frequency ofvu, andG(u)can be regarded as the remainder term. We show a lemma related to the commutator estimates to show a priori estimate.

Lemma 3.1.Letuk:=φk∗ufor simplicity and lets0and1p∞. Then it holds that

k∈Z

2^sk φ_k∗divF (v, u)−(S_k−8v)· ∇u_k

L^pC∇vL^∞u_B˙^s

p,1. (3.3)

Remark 3.2.For the Euler equations, such estimate is known and we refer to[11,26]on the commutator estimates for divergence free vector fieldv.

Proof of Lemma 3.1. To prove(3.3), we show the following φk∗divF (Sk−8v, u)−(Sk−8v)· ∇uk

L^p

C∇vL∞

3 μ=−3

uk+μL^p+CdivvL∞

3 μ=−3

uk+μL^p, (3.4)

φ_k∗divF

k₁k−7

φ_k1∗v, u L^p

Cv_B˙1

∞,∞

3 μ=−3

u_k+μL^p. (3.5)