On The Radius Of Analyticity Of Solutions To Semi-Linear Parabolic Systems

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On The Radius Of Analyticity Of Solutions To Semi-Linear Parabolic Systems

Jean-Yves Chemin, Isabelle Gallagher, Ping Zhang

To cite this version:

Jean-Yves Chemin, Isabelle Gallagher, Ping Zhang. On The Radius Of Analyticity Of Solutions To Semi-Linear Parabolic Systems. 2020. �hal-02537907�

ON THE RADIUS OF ANALYTICITY OF SOLUTIONS TO SEMI-LINEAR PARABOLIC SYSTEMS

JEAN-YVES CHEMIN, ISABELLE GALLAGHER, AND PING ZHANG

Abstract. We study the radius of analyticity R(t) in space, of strong solutions to sys- tems of scale-invariant semi-linear parabolic equations. It is well-known that near the ini- tial time,R(t)t⁻¹² is bounded from below by a positive constant. In this paper we prove that lim inf

t→0 R(t)t⁻¹² =∞, and assuming higher regularity for the initial data, we obtain an improved lower bound near time zero. As an application, we prove that for any global solu- tionu∈C([0,∞);H¹²(R³)) of the Navier-Stokes equations, there holds lim inf

t→∞ R(t)t⁻¹² =∞.

Keywords:Semi-linear parabolic systems, Radius of analyticity, Navier-Stokes equations AMS Subject Classification (2000):35K55

1. Introduction

We consider the following system ofN equations onR⁺×R^d: (SP)

∂_tU −∆U =P(U)

U|t=0=U0, with P_j(U)^def= X

`∈N^N

|`|=k

A<sub>j,`</sub>(D)(U<sup>`</sup>) for j in{1, . . . , N}

whereA<sub>j,`</sub>(D) are homogeneous Fourier multipliers of degree β ∈[0,2[, and U = (Uj)1≤j≤N. The order of the nonlinearity is k ≥ 2 and we have written U<sup>`</sup> =

N

Y

j=1

U_j^`j. An important property of a such a system is its scaling invariance: if a functionU satisfies (SP) on a time interval [0, T] with the initial dataU₀, then the function U_λ defined by

Uλ(t, x)^def= λ^αU(λ²t, λx)

satisfies (SP) on the time interval [0, λ⁻²T] with the initial dataU_0,λ^def= λ^αU0(λ·) for α^def= 2−β

k−1·

Note that α is positive, and in the following we shall assume that α≤d/k. For example for the Navier-Stokes equations there holds β = 1 andk = 2, while for the cubic heat equation there holdsβ= 0 andk= 3. In both casesα = 1. The scaling invariant Sobolev space for the initial data isH^scrit(R^d), withscrit def

= ^d₂−α, recalling thatH^s(R^d) is defined by the following norm, fors < d/2:

kfk_Hs(R^d) def= Z

R^d

|ξ|^2s|fb(ξ)|²dξ¹₂ , wherefb=Ff is the Fourier transform off.

The question of solving the Cauchy problem for systems such as (SP) in scale invariant spaces has been widely studied. We shall make no attempt at listing all the results on the subject but simply recall the typical so-called Kato-type theorem, which may be proved by a Banach fixed point argument (see for instance [1, 7, 9, 13, 14] among others)

1

Theorem 1.1. Assume α ∈]1/k, d/k] and define scrit def

= d/2−α. Then, for any δ ∈ [0, α[, for any initial data U₀ belonging to H^scrit+δ(R^d), a positive time T exists such that the system (SP) has a unique solutionU in the Kato space K_T^p such that

(1.1) kUk_K^p

T

def= sup

t≤T

t^1pkU(t)k

H^scrit+2p <∞. Moreover, ifδ is positve, a constantc exists such thatT ≥ckU₀k⁻

2 δ

H^scrit+δ.

The goal of this article is to analyze the instantaneous smoothing effect of (SP): let us recall that in [5], the analyticity of smooth periodic solutions to the Navier-Stokes equations (NS) is proved, in the sense that ifvsolves (NS) thene^σ

√−t∆v(t) is a smooth function for someσ >0.

This result was extended in [3, 8, 11, 12] where it is proved for instance that Z

R³

|ξ| sup

t≤T

e

√t|ξ||bv(t, ξ)|2

dξ+ Z T

0

Z

R³

|ξ|³ e

√t|ξ||v(t, ξ)|b 2

dξdt <∞, which shows that the radius of analyticityR(t) ofv(t) is bounded from below by√

t. Note that the above condition is equivalent to the fact that e

√−t∆v(t) belongs to E_T^∞∩E_T², where E_T^q denotes the space of vector fields V such that

kVk_E^q

T

def= 2^j

1 2+²_q

k∆_jVk_Lq([0,T];L²(R³))

`²(Z).

This type of result is also known to hold in the more general context of (SP) (see [4, 10] for instance) and may be stated as follows.

Theorem 1.2. The solution constructed in Theorem1.1is analytic for positivetwith radius of analyticityR(t)greater than √

t.

The purpose of this work is the proof of the following improved theorem.

Theorem 1.3. (a) Ifδ is positive, the solution constructed in Theorem1.1satisfies lim inf

t→0

R(t) t¹²

r

−log tkU₀k

2 δ

H^scrit+δ

≥√ 2δ .

(b) If δ= 0, then, for any positive εsmall enough, we have lim inf

t→0

R(t) t¹²q

−log ke^ετ∆U0k_K^p

t

≥2√ 1−ε .

In particular lim

t→0

R(t) t¹²

=∞.

We remark that in the case of three-dimensional incompressible Navier-Stokes system (NS)







∂tu+u· ∇u−∆u+∇p= 0, (t, x)∈R⁺×R³, divu= 0,

u|_t=0 =u₀,

where u = (u¹, u², u³) denotes the velocity of the fluid and p the scalar pressure function, part (a) of Theorem 1.3 coincides with Theorem 1.3 of [8]. Moreover, the main idea used to prove Theorem 1.3 can be applied to investigate the radius of analyticity of any global solution of (NS). More precisely we can prove the following result.

Corollary 1.1. Let u∈C([0,∞);H¹²(R³)) be a global solution of (NS). Then one has

(1.2) lim inf

t→∞

R(t) t¹² =∞.

2

2. Proof of Theorem 1.3

We shall perform all our computations on the approximated system (SP_n)

∂tU −∆U =Pn(U)

U_|t=0=U₀, with P_n,j(U)^def= X

`∈N^N

|`|=k

1_B(0,n)(D)A<sub>j,`</sub>(D)(U<sup>`</sup>)

forj in {1, . . . , N},and where we have written 1_B(0,n) for the characteristic function of the ball B(0, n)^def=

ξ∈R^d; |ξ| ≤n . The system (SP_n) is an ordinary differential equation in all Sobolev spaces. All the quantities we shall write are defined in this case, and we neglect the index n in all that follows. We also skip the final stage of passing to the limit when n tends to infinity.

Let us consider three positive real numbersT,λand εwhich will be chosen later on in the proof. Motivated by [8], we define

(2.1) U_a(t, x)^def= F⁻¹ e^{− λ}

2 4(1−ε)

t T+λ^√t

T|ξ|

|Ub(t, ξ)|

.

The main point is that the function Ua behaves like a solution of a modified system (SP) where the viscosity isε instead of 1 and the non-linear term has a factore⁻

λ2(k−1)

4(1−ε) . We shall make this idea more precise in what follows.

The key ingredient used to prove Theorem 1.3 will be the following lemma.

Lemma 2.1. Let U_a be defined by (2.1). Then for anyp in]max(2/α, k),∞[,there exists a positive constantCk,ε such that

(2.2) kU_ak_K^p

T ≤ ke^εt∆U₀k_K^p

T +C_k,ε e ^λ

2

4(1−ε)kU_ak_K^p

T

k−1

kU_ak_K^p

T. Proof. A solution of (SP_n) satisfies

(2.3) |Ub(t, ξ)| ≤e^−t|ξ|2|Ub0(ξ)|+C Z t

0

e^{−(t−t0)|ξ|2}|ξ|^β |Ub(t⁰)|?· · ·?|Ub(t⁰)|

| {z }

k times

(ξ)dt⁰. Let us observe that

− λ² 4(1−ε)

1 T + λ

√

T|ξ| − |ξ|² ≤ −ε|ξ|². Thus by definition (2.1), we infer from (2.3) that

Uba(t, ξ)≤e^−εt|ξ|2|Ub0(ξ)|+C Z t

0

e^{−ε(t−t0)|ξ|2}e⁻

λ2 4(1−ε)

t0 T+λ^√t0

T|ξ|

|ξ|^β |Ub(t⁰)|?· · ·?|Ub(t⁰)|

| {z }

ktimes

(ξ)dt⁰. Notice that

|Ub(t⁰)|?· · ·?|Ub(t⁰)|

| {z }

ktimes

(ξ) = Z

Pk

`=1ξ`=ξ

Y^k

`=1

|Ub(t⁰, ξ_`)|

dξ₁. . . dξ_k,

and using that, for any (ξj)1≤j≤k in (R^d)^k such that

k

X

j=1

ξj =ξ, there holdse^|ξ|≤

k

Y

j=1

e^|ξj|,we infer that

(2.4) Uba(t, ξ)≤e^−εt|ξ|2|Ub0(ξ)|+Ce

λ2(k−1) 4(1−ε)

Z t 0

e^{−ε(t−t0)|ξ|2}|ξ|^β Uba(t⁰)?· · ·?Uba(t⁰)

| {z }

k times

(ξ)dt⁰.

3

Let us recall the following result on products in Sobolev spaces: for any positive real numbers, smaller thand/2 and greater thand/2−d/k, there holds

(2.5)

k

Y

`=1

ak

H^{ks−(k−1)d2} ≤Ck k

Y

`=1

ka_kk_H^s.

Now let us choose pin [1,∞] such that

(2.6) 0< 2

p < α and set s_p ^def= s_crit+2 p = d

2+ 2 p −α . Notice that

Uba(t⁰)?· · ·?Uba(t⁰)

| {z }

ktimes

= (2π)^kdF(U_a^k(t⁰)).

The assumption thatα≤ ^d_k implies thatα < ^d_k+²_p and s_p > ^d₂ −_k^d,so (2.5) ensures that Uba(t⁰)?· · ·?Uba(t⁰)

| {z }

k times

(ξ)≤C^k|ξ|⁻

sp+(k−1)

2 p−α

f(t⁰, ξ)kU_a(t⁰)k^k_Hsp withkf(t⁰)k_L2(R^d)= 1.

Asα(k−1) = 2−β, plugging the above inequality in (2.4) gives Uba(t, ξ)≤e^−εt|ξ|2|Ub0(ξ)|+C^ke

λ2(k−1) 4(1−ε)

Z t 0

e^{−ε(t−t0)|ξ|2}|ξ|^{−sp+2−(k−1)2p}f(t⁰, ξ)kU_a(t⁰)k^k_Hspdt⁰. By multiplication of this inequality byt^1p|ξ|^sp and by definition of the normk · k_K^p

T in (1.1), we get, for any tin the interval [0, T],

t^1p|ξ|^spUb_a(t, ξ)≤t^1p|ξ|^spe^−εt|ξ|2|Ub₀(ξ)|

+C^ke

λ2(k−1)

4(1−ε) kU_ak^k_Kp Tt^1p

Z t 0

e^{−ε(t−t0)|ξ|2} 1 (t⁰)^kp

|ξ|²

1−^k−1

p

f(t⁰, ξ)dt⁰.

If we assume that p is greater than k, the function y∈[0,∞]7→y¹⁻

k−1

p e^−εy is bounded, we infer that for anytin the interval [0, T],

t^1p|ξ|^spUba(t, ξ)≤t^1p|ξ|^spe^−εt|ξ|2|Ub0(ξ)|

+C_k,εe

λ2(k−1)

4(1−ε) kU_ak^k_Kp Tt^1p

Z t 0

1 (t−t⁰)¹⁻

k−1 p

1 t^0kp

f(t⁰, ξ)dt⁰.

Taking theL² norm with respect to the variable ξ gives, for any tin the interval [0, T], t^1pkU_a(t)k_H^sp ≤t^1pke^εt∆U₀k_H^sp+C_k,εe

λ2(k−1)

4(1−ε) kU_ak^k_K^p

T.

Taking the supremum with respect tot in the interval [0, T] gives (2.2). 2 Let us now turn to the proof of Theorem 1.3.

Proof of Theorem 1.3. LetU andU_abe determined respectively by (SP_n) and (2.1). We make the following induction hypothesis

(2.7) kU_ak_K^p

T ≤c_k,εe^{− λ}

2

4(1−ε) with c_k,ε ^def= 1

(4C_k,ε)^k−11

with Ck,ε being determined by Lemma 2.1. As long as this induction hypothesis is satisfied, Inequality (2.2) becomes

(2.8) kU_ak_K^p

T ≤ 4

3ke^εt∆U₀k_K^p

T.

4

Now let us distinguish the case whenU₀ belongs to the space H^scrit+δ from the case whenU₀ belongs only to the critical spaceH^scrit.

(a) The case whenU0 belongs to the space H^scrit+δ. We first observe that

(2.9) ke^εt∆U₀k_K^p

T ≤C_δ,pT^δ2kU₀k_Hscrit+δ

Let us define

T_ε(U₀)^def= η_εkU₀k⁻

2 δ

H^scrit+δ with η_ε ^def= c_k,ε

2C_δ,p ²_δ

· By definition ofT_ε(U₀), we have that for anyT ≤T_ε(U₀),

2Cδ,pT^δ2kU₀k_Hscrit+δ ≤ck,ε. Now let us define

λT def= p

2δ(1−ε) log¹²

ηε

TkU₀k

2 δ

H^scrit+δ

·

Then for T ≤T_ε(U₀),we deduce from the Ineqalities (2.8) and (2.9) that kU_ak_K^p

T ≤ 4

3Cδ,pT^δ2kU₀k_Hscrit+δ <2Cδ,pT^δ2kU₀k_Hscrit+δ =ck,εe⁻

λ2 T 4(1−ε).

This in turn shows that (2.7) holds forT ≤T_ε(U₀).Furthermore, according to (1.1) and (2.1), there holds

T^p1 e^λT

√

T|D|U(T)

H^scrit+

2

p ≤c_k,ε. Asδ is less than α, takingp= 2

δ ensures that

∀T ≤Tε(U0), R(T)≥p

2δ(1−ε)T¹²log¹²

ηε

TkU₀k

2 δ

H^scrit+δ

·

This inequality means exactly that lim inf

T→0

R(T) T¹²

r

−log TkU₀k

2 δ

H^scrit+δ

≥p

2δ(1−ε).

Due the fact thatεis arbitrary, we conclude the proof of part (a) of Theorem 1.3.

(b) The case whenU₀ belongs to the critical spaceH^scrit. Let us use the fact that in this case

(2.10) lim

T→0ke^εt∆U0k_K^p

T = 0. Then we considerTε(U0) such that

(2.11) ke^εt∆U0k_K^p

Tε(U0)

≤c_k,ε.

For anyT ≤Tε(U0), let us define λT def

= 2(1−ε)¹²log¹²

c_k,ε 2ke^εt∆U₀k_K^p

T

.

Then it follows from Inequality (2.8) that kU_ak_K^p

T ≤ 4

3ke^εt∆U₀k_K^p

T <2ke^εt∆U₀k_K^p

T =c_k,εe⁻

λ2 T 4(1−ε).

5

This shows that (2.7) indeed holds forT ≤T_ε(U₀).Furthermore, according to (1.1) and (2.1), there holds

T^1p e^λT

√

T|D|U(T)

H^scrit+2p ≤c_k,ε. By definition ofλ_T this means in particular that

∀T ≤T_ε(U₀), R(T)≥2(1−ε)¹²T¹² log¹²

c_k,ε 2ke^εt∆U0k_K^p

T

.

This inequality means exactly that for any small strictly positiveε, we have lim inf

T→0

R(T) T¹²

q

−log¹² ke^εt∆U₀k_K^p

T

≥ lim

T→02(1−ε)¹²

1− logc_k,ε log 2ke^εt∆U0k_K^p

T

¹₂

,

which together with (2.10) ensures part (b) of of Theorem 1.3. This completes the proof of

the theorem. 2

3. Proof of Corollary 1.1

Let u ∈ C([0,∞);H¹²(R³)) be a global solution of the Navier-Stokes system (NS) with initial datau0.Then it follows from [2] that this solution is unique, so that applying Theorem 2.1 of [6] yields

(3.1) lim

t→∞ku(t)k

H¹² = 0. Moreover, for anyt0 >0, u verifies

(N St0)







∂tu+u· ∇u−∆u+∇p= 0, (t, x)∈]t₀,∞[×R³, divu= 0,

u|_t=t0 =u(t₀).

Similar to (2.1) we denote

(3.2) ua,t0(t, x)^def= F⁻¹ e⁻

λ2 4(1−ε)

t−t0 T +λ^t−t√0

T |ξ|

|u(t, ξ)|b , and

kuk_K^p

t0,T

def= sup

t∈[t0,t0+T]

(t−t₀)^p1ku(t)k

H^{12+ 2p}

.

Then along the same line to proof of Lemma 2.1, we deduce that (3.3) ku_a,t0k_K^p

t0,T ≤ ke^{ε(t−t0)∆}u(t₀)k_K^p

t0,T +C_εe ^λ

2

4(1−ε)ku_a,t0k²_Kp t0,T . By (3.1), we can chooset₀ so large that

ku(t₀)k

H¹² ≤ cε

2K_ε withKε being determined bykε^εt∆u0k_K^p

∞ ≤Kεku₀k

H¹². Let us make the induction assumption that

(3.4) ku_a,t0k_K^p

t0,T ≤cεe⁻

λ2

4(1−ε) with cε def= 1

4Cε

·

Then as long as the induction assumption is satisfied, we infer from (3.3) that ku_a,t0k_K^p

t0,T ≤ 4

3ke^{ε(t−t0)∆}u(t0)k_K^p

t0,T <2ke^{ε(t−t0)∆}u(t0)k_K^p

t0,T ≤2Kεku(t₀)k

H¹² ≤cε. So that for anyT >0,we define

λ_T ^def= 2(1−ε)¹²log¹²

cε

2K_εku(t₀)k

H¹²

,

6

we have

ku_a,t0k_K^p

t0,T <2Kεku(t₀)k

H¹² ≤cεe⁻

λ2 4(1−ε)T .

This in turn shows that (3.4) holds for anyT >0.(3.4) in particular implies that T^1p

e^λT

√ T|D|

u(t0+T)

H^{12+ 2p} ≤cε. As a result, it comes out

∀T , R(t₀+T)≥2(1−ε)¹²T¹² log¹²

cε

2K_εku(t₀)k

H¹²

,

from which we infer that lim inf

T→∞

R(t0+T)

√t₀+T = lim inf

T→∞

R(t0+T)

√T ≥2(1−ε)¹² log¹²

cε

2K_εku(t₀)k

H¹²

.

This together with (3.1) ensures (1.2). This finishes the proof of Corollary 1.1.

Acknowledgments. Part of the work was done when P. Zhang was visiting Laboratoire J.

L. Lions of Sorbonne Universit´e in the fall of 2018. He would like to thank the hospitality of the Laboratory. P. Zhang is partially supported by NSF of China under Grants 11371347 and 11688101, and innovation grant from National Center for Mathematics and Interdisciplinary Sciences.

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[7] Y. Giga, Solutions for semilinear parabolic equations inL^pand regularity of weak solutions of the Navier- Stokes system,J. Differential Equations,62, 1986, pages 186–212.

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(J.-Y. Chemin)Laboratoire Jacques Louis Lions - UMR 7598, Sorbonne Universit´e, Boˆıte cour- rier 187, 4 place Jussieu, 75252 Paris Cedex 05, France

E-mail address:[email protected]

(I. Gallagher)DMA, ´Ecole normale sup´erieure, CNRS, PSL Research University, 75005 Paris, and UFR de math´ematiques, Universit´e Paris-Diderot, Sorbonne Paris-Cit´e, 75013 Paris, France.

E-mail address:[email protected]

(P. Zhang)Academy of Mathematics&Systems Science and Hua Loo-Keng Key Laboratory of Mathematics, The Chinese Academy of Sciences, Beijing 100190, China, and School of Mathe- matical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China.

E-mail address:[email protected]

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