On global existence of strong and classical solutions of Navier-Stokes equations

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On global existence of strong and classical solutions of Navier-Stokes equations

Andrey Polyakov

To cite this version:

Andrey Polyakov. On global existence of strong and classical solutions of Navier-Stokes equations.

2019. �hal-02096215v3�

Noname manuscript No.

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On global existence of strong and classical solutions of Navier-Stokes equations

Andrey Polyakov

the date of receipt and acceptance should be inserted later

Abstract This short note studies the problem of a global expansion of local results on existence of strong and classical solutions of Navier-Stokes equations in R

.

Keywords Navier-Stokes-Equations · Strong Solutions · Dilation Symetry

1 Introduction

In this short note we exploit a very simple idea of global expansion of regularity by means of dilation symmetry, which is well known for systems in R

.

Let a vector field f : R

→ R

be symmetric with respect to a dilation of the argument, i.e. ∃α ∈ R such that

f (e

u) = e

f (u), ∀u ∈ R

, ∀s, ∈ R If u(·) : [0, +∞) → R

is a classical solution of

du

dt = f (u), t > 0 with the initial condition u(0) = u

then

u

(t) := e

u(e

t) is defined on [0, +∞) and, due to symmetry, we derive

du

dt = e

f (u(e

t)) = f(u

(t)), t > 0,

A. Polyakov

Inria Lille, Univ. Lille, CNRS, UMR 9189 - CRIStAL, (F-59000 Lille, France), Tel.: +33-359577802

E-mail: andrey.polykov@inria.fr

i.e. u

is a classical solution of the same differential equation with the initial condition u

(0) = e

x

, where s ∈ R .

Let ∃ε > 0 such that a classical solution of the differential equation exists on [0, +∞) for any initial value u(0) = u

∈ B

:= {u ∈ R

: |u| < ε}. To construct a solution for u

∈ / B

we first need to scale u

→ e

u

, where s

∈ R is such that

|e

u

| < ε.

If u(, e

u

) is a solution with the initial condition u(0) = λ

u

then ˜ u(t) = e

u(e

, e

u

) is also a solution of the considered system and, obviously,

˜

u(0) = e

u(0, e

u

) = e

e

u

= u

.

In this paper we use the dilation symmetry (see, [1, formula (1.5)]) of the Navier-Stokes equations in R

∂

u = ν∆u − (u · ∇)u − ∇p, 0 = divu

where u denotes the velocity of a fluid, p denotes the scalar pressure and ν > 0 denotes viscosity of the fluid, in order to understand when a global-in- time existence of strong or classical solutions of the Navier-Stokes equations for small initial data implies the existence of global-in-time strong or classical solutions for large initial data. We refer the reader, for example, to [2] and [4], for more details about global-in-time existence of strong solutions for small initial data.

Mainly, the standard notation is utilized through the paper, e.g. R is the field of real numbers; L

((0, T )× R

, R ) denotes the space of locally integrable functions (0, T ) × R

→ R ; L

( R

, R

), 1 ≤ p ≤ +∞ is a Lebesgue space of function R

→ R

with the norm k · k

; C

((0, T ) × R

, R

) is a space of smooth functions (0, T ) × R

→ R

with compact support and C

([0, T ) × R

, R

) is a space of smooth functions which vanish at infinity, where 0 <

T ≤ ∞. For composition of operators A, B we also use the notation A ◦ B.

Let L

( R

, R

) denotes the following normed vector space of functions R

→ R

L

( R

, R

) := {u : kuk

< +∞} , µ ∈ R

kuk

:=

Z

Rⁿ

|x|

|u(x)|

dx

, 0 < p < ∞

kuk

:= ess sup(|x|

u(x)), p = ∞,

which can be treated as a wighted L

.

On global existence of strong and classical solutions of Navier-Stokes equations 3

2 Preliminaries: Dilations in functional spaces

The following lemmas deal with the most common dilation groups in functional spaces.

Lemma 1 The operator d(s) given by

(d(s)z)(x) = e

z(e

t, e

x), (1) where s ∈ R , z is a function (0, T ) × R

→ R

, 0 < T ≤ +∞, x ∈ R

and α, β, γ ∈ R are constant parameters,

– maps C

((0, T ) × R

, R

) onto C

((0, e

T ) × R

, R

);

– maps C

([0, T ) × R

, R

) onto C

([0, e

T ) × R

, R

).

The inverse operator is given by [d(s)]

= d(−s).

Proof 1) Since the linear function (t, x) → (e

t, e

x) maps a compact in (0, T ) × R

to a compact in (0, e

T ) × R

then d(s) defined on whole C

((0, T ), R

) and if z ∈ C

((0, T ), R

) (i.e. z is smooth and has a com- pact support in (0, T ) × R

) then d(s)z ∈ C

((0, e

T ), R

) (i.e. d(s)z is also smooth, but it has a compact support in (0, e

T) × R

). Obviously, (d(s) ◦ d(−s))z = (d(−s) ◦ d(s))z = z for any z ∈ C

((0, T ), R

) and any s ∈ R .

Let us show that d(s) maps C

((0, T ), R

) onto C

((0, e

), R

). Sup- pose the opposite: ∃z

∈ C

((0, e

T ), R

) such that z

6= d(s)y, ∀y ∈ C

((0, T ), R

). This is impossible, since d(−s)z

∈ C

((0, T ), R

) and z

= (d(s) ◦ d(−s))z

∈ d(s)C

((0, T ), R

).

2) The proof for C

is almost identical. Obviously, if z ∈ C

([0, T ), R

) (i.e. z is smooth and vanishing at infinity) then d(s)z ∈ C

([0, e

T ), R

) (i.e. d(s)z is also smooth and vanishing at infinity) and d(s) ◦ d(−s)z = d(−s) ◦ d(s)z = z for any z ∈ C

([0, T ), R

).

Let us show that d(s) maps C

([0, T ), R

) onto C

([0, e

T), R

). Sup- pose the opposite: there exists z

∈ C

([0, e

T ), R

) such that d(s)y 6= z

,

∀y ∈ C

([0, T ), R

). This is impossible since d(−s)z

∈ C

([0, T ), R

) and z

= d(s) ◦ d(−s)z

∈ d(s)C

([0, T ), R

).

Lemma 2 The operator d(s) given by

(d(s)z)(x) = e

z(e

x), (2)

where s ∈ R , z is a function R

→ R

, x ∈ R

and α, β ∈ R are constant

parameters, is

– a linear bounded invertible operator on L

( R

, R

),

kd(s)zk

= e

kzk

, z ∈ L

( R

, R

), s ∈ R , – a linear bounded invertible operator on L

( R

, R

),

kd(s)zk

= e

kzk

, z ∈ L

( R

, R

), s ∈ R , where 0 < p ≤ ∞. The inverse operator is given by [d(s)]

= d(−s).

Proof Notice that L

= L

.

Let 1 ≤ p < ∞. If z ∈ L

( R

, R

) then Z

Rⁿ

|x|

|z(x)|

dx < +∞

and

Z

Rⁿ

|x|

|z(x)|

dx = e

Z

Rⁿ

|e

x|

|z(e

x)|

dx =

e

Z

Rⁿ

|x|

(d(s)z)(x)|

dx < +∞.

Since e

> 0 for any α, β, p, s ∈ R then d(s)z ∈ L

( R

, R

) for any s ∈ R . Obviously, d(s) is a linear operator on L

, i.e. d(s)(µ

z

+ µz

) = µ

d(s)z

+ µ

d(s)z

, for any µ

, µ

∈ R and z

, z

∈ L

( R

, R

). Moreover, the latter identities imply that

kd(s)zk

=e

kzk

, kzk

:=

Z

Rⁿ

|x|

|z(x)|

dx

.

Hence, the operator d(s) : L

( R

, R

) → L

( R

, R

) is bounded for any s ∈ R .

Let p = ∞. If z ∈ L

( R

, R

) then

ess sup|z(x)| = ess sup(|e

x|

|z(e

x)|) < +∞

for any β, s, µ ∈ R and kd(s)zk

= e

kzk

for any s ∈ R . Therefore, d(s) is also a linear bounded operator on z ∈ L

( R

, R

).

Obviously, (d(s) ◦ d(−s))z = (d(−s) ◦ d(s))z for any z : R

→ R

and any

s ∈ R and we derive [d(s)]

= d(−s).

On global existence of strong and classical solutions of Navier-Stokes equations 5

3 Global Existence of Strong Solutions of Navier-Stokes Equations Below for shortness we omit R

in the notations of R

R³

, L

, C

and C

if the context is clear. Without loss of generality (see e.g. [4, page 4]) we also assume that ν = 1, where ν is a viscosity coefficient.

Let us consider the weak form of the Navier-Stokes equations Z

R³

u(0)·ξ(0)+

T

Z

0

Z

R³

u·(∂

ξ+∆ξ)+p divξ =

T

Z

0

Z

R³

u·(u·∇)ξ, ∀ξ ∈ C

([0, T )× R

, R

) (3) with u(t) ∈ V for t ∈ (0, T ), where V is a set of the so-called weakly divergence free velocity fields:

V :=

u ∈ L

: Z

R³

u · ∇φ = 0, ∀φ ∈ C

( R

, R )

.

The classical idea of analysis is to prove existence and regularity of weak solutions and next to show that any weak solution is smooth. For Navier- Stokes equations this analysis has been initiated by Jean Leray in 1938 (see [2]). We use the recent review [4] of his results.

Definition 1 ([4], Definition 3.7) A pair (u, p) is said to be a strong solution of the Navier-Stokes equations on [0, T ) if u(t) ∈ V for t ∈ (0, T ), p ∈ L

((0, T ) × R

, R ),

u ∈ C([0, T ), L

) ∩ C((0, T ), L

), ku(t)k

is bounded as t → 0

and (3) is satisfied.

A possible way for expansion of regularity to a larger set of initial condi- tions is to use the dilation symmetry as explained in the introduction. Several symmetries for Navier-Stokes equations are known (see e.g. [1] and references therein). Below we use just one of them (see [1, formula (1.5)]).

Lemma 3 If (u, p) is a strong solution of the Navier-Stokes equations with the initial data u(0) = u

∈ V ∩ L

defined on [0, T ) then for any s ∈ R the pair (u

, p

) given by

u

(t, x) = e

u(e

t, e

x), p

(t, x) = e

p(e

t, e

x), t ≥ 0, x ∈ R

, s ∈ R .

is a strong solution of the Navier-Stokes equations defined on [0, e

T )

with the initial value u

(0) = d(s)u

.

Proof 1) Let d

(s) be defined by the formula (2) with α = 1, β = 1.

Since u is a strong solution then the function t → u(t) is continuous in L

and in L

. According to Lemma 2, d

(s) is a linear bounded invertible oper- ator on L

and on L

. Hence, for any fixed s ∈ R the function t → d

(s)u(t) is also continuous in L

and L

, consequently, u

∈ C([0, e

T ), L

) and u

∈ C((0, e

T ), L

).

Notice kd

(s)zk

= e

kzk

for any z ∈ L

and any s ∈ R (see Lemma 2). Since ku(t)k

is bounded as t → 0

then ku

(t)k

= e

ku(e

t)k

is also bounded as t → 0

.

If u

∈ V then d

(s)u

∈ V for any s ∈ R . Indeed, Z

u

· ∇φ = 0, ∀φ ∈ C

and using the change-of-variable theorem in the Lebesgue integral we derive 0 =

Z

R³

u

(x)·∇φ(x)dx = e

Z

R³

u

(e

x)·(∇φ)(e

x)dx = e

Z

R³

(d

(s)u

)·∇ φ, ˜ where ˜ φ = d

(s)φ. Since d

(s) maps C

onto C

(see Lemma 1) then

0 = Z

(d

(s)u

) · ∇ φ, ˜ ∀ φ ˜ ∈ C

,

i.e. d

(s)u

∈ V for any s ∈ R . Hence, the inclusion u(t) ∈ V , t ∈ [0, T ) implies u

(t) ∈ V , t ∈ [0, e

T ).

2) Let d

(s) be given by the formula (1) with α = 2, β = 1 and γ = 2.

Since p ∈ L

((0, T ) × R

, R ) then Z

0

Z

R³

|p(t, x)ξ(t, x)|dxdt < +∞, ∀ξ ∈ C

((0, T ) × R

, R )

hence using the change-of-variable theorem in the Lebesgue integral for the functions t → e

t and x → e

x we derive

Z

0

Z

R³

|p(t, x)ξ(t, x)|dxdt = e

Z

0

Z

R³

|p(e

t, e

x)ξ(e

t, e

x)|dxdt =

e

Z

0

Z

R³

|p

· d

(s)ξ| < +∞, ∀ξ ∈ C

((0, T ) × R

, R ).

Since the operator d

(s) maps C

((0, T )× R

, R ) onto C

((0, e

T )× R

, R ) then

Z

0

Z

R³

|p

· ξ| ˜ < +∞, ∀ ξ ˜ ∈ C

((0, e

T ) × R

, R ).

Therefore, p

∈ L

((0, e

T ) × R

, R ).

On global existence of strong and classical solutions of Navier-Stokes equations 7

3) Let us show that (u

, p

) satisfies the equation (3). Since (u, p) is a solution defined on [0, +∞) then ∀ξ ∈ C

([0, T ) × R

, R

) we have

Z

R³

u(0, x)·ξ(0, x)dx+

T

Z

0

Z

R³

u(t, x)·(∂

ξ(t, x)+(∆ξ)(t, x))+p(t, x)(divξ)(t, x)dx dt =

T

Z

0

Z

R³

u(t, x) · (u(t, x) · ∇)ξ(t, x)dx dt

Using the change-of-variable theorem in the Lebesgue integral for the functions t → e

t and x → e

x we derive

e

Z

R³

u(0, e

x) · ξ(0, e

x)dx + e

e^−2sT

Z

0

Z

R³

u(e

t, e

x) · (∂

ξ(e

t, e

x)+

(∆ξ)(e

t, e

x)) + p(e

t, e

x)(divξ)(e

t, e

x)dx dt =

e

e^−2sT

Z

0

Z

R³

u(e

t, e

x) · (u(e

t, e

x) · ∇)ξ(e

t, e

x)dx dt or, equivalently,

e

Z

R³

u

(0, x)·ξ(0, e

x)dx+e

e^−2sT

Z

0

Z

R³

e

u

(t, x)· ∂

ξ(e

t, e

x)+(∆ξ)(e

t, e

x)

+e

p

(t, x)(divξ)(e

t, e

x)dx dt =

e

e^−2sT

Z

0

Z

R³

u

(t, x) · (u

(t, x) · ∇)ξ(e

t, e

x)dx dt Let us denote ξ

(t, x) = ξ(e

, e

x). Hence,

(∆ξ

)(t, x) = e

(∆ξ)(e

, e

x), (∇ξ

)(t, x) = e

(∇ξ)(e

t, e

x), (∂

ξ

)(t, x) = e

(∂

ξ)(e

t, e

x), (divξ

)(t, x) = e

(divξ)(e

t, e

x), and

e

Z

R

u

(0)·ξ

(0)+e

e^−2sT

Z

0

Z

u

· (∂

ξ

+∆ξ

+p

(divξ

) = e

e^−2sT

Z

0

Z

R³

u

·(u

·∇)ξ

,

where ξ

∈ C

([0, e

T ), × R

, R

). Since for any s > 0 the operator d

(s) defined as

(d

(s)ξ)(t, x) = ξ(e

t, e

x), t > 0, x ∈ R

maps C

([0, T ) × R

, R

) onto C

([0, e

T ), × R

, R

) (see Lemma 1), then Z

R³

u

(0)·ξ

(0)+

e^−2sT

Z

0

Z

R³

u

(t, x)·(∂

ξ

+∆ξ

+p

(divξ

) =

e^−2sT

Z

0

Z

R³

u

(t, x)·(u

(t, x)·∇)ξ

.

holds for all ξ

∈ C

([0, e

T ) × R

, R

). Therefore, (u

, p

) is a strong solution of the Navier-Stokes equations on [0, e

T ) and u

(0) = d(s)u

.

The proven lemma implies the following result, which describes the cases when global-in-time existence of strong solutions for small initial data is equiv- alent to global-in-time existence of strong solutions for large initial data.

Corollary 1 Let q

, q

∈ [1, ∞] and r

, r

, µ

, µ

∈ R and r

(1 − µ

− 3/q

) + r

(1 − µ

− 3/q

) 6= 0.

A strong solution of the Navier-Stokes equations with arbitrary the initial data u(0) = u

∈ V ∩ L

∩ L

exists on [0, +∞) if and only if there exist ε > 0 such that for any

u

∈ {u ∈ V ∩ L

1

∩ L

2

: kuk

1,µ₁

kuk

2,µ₂

< ε}

a strong solution (u, p) with the initial data u(0) = u

exists on [0, +∞).

Proof Let d

be defined as in the proof of Lemma 3. Assume that for any u

∈ {u ∈ V ∩ L

1

∩ L

2

: kuk

1,µ₁

kuk

2,µ₂

< ε} there exists a global in time strong solution with u(0) = u

and let us show that for any u

∈ V ∩L

∩L

: ku

k

ku

k

≥ ε then the Navier-Stokes equations also have a strong solution on [0, +∞).

In the proof of Lemma 3 we have shown that d

(s)u

∈ V for any s ∈ R , by Lemma 2 and d

(s)u

∈ L

1

and for d

(s)u

∈ L

1

any s ∈ R . By Lemma 2 we also derive

kd

(s)u

k

= e

ku

k

, and

kd

(s)u

k

= e

ku

k

and

kd

(s)u

k

kd

(s)u

k

= e

ku

k

ku

k

.

On global existence of strong and classical solutions of Navier-Stokes equations 9

Since by assumption r

(1 − µ

− 3/q

) + r

(1 − µ

− 3/q

) 6= 0 then for any u

∈ V ∩ L

∩ L

there exists s

∈ R such that

kd

(s)u

k

kd

(s)u

k

< ε.

Hence, if a strong solution (u, p) with u(0) = d

(s

)u

exists on [0, +∞) then by Lemma 3 the pair (˜ u, p) given by ˜

˜

u(t, x) = e

u(e

t, e

x), p(t, x) =e ˜

p(e

t, e

x), t ∈ [0, +∞), x ∈ R

is also a strong solution of the Navier-Stokes equation. Since u(0) = d

(s

)u

means that

u(0, x) = e

u

(e

x), x ∈ R

then

˜

u(0, x) = e

u(0, e

x) = u

(x), i.e. ˜ u(0) = u

∈ V ∩ L

1

∩ L

2

and the proof is complete.

Notice that taking µ

= µ

= 0 we derive the usual L

and L

spaces in the latter corollary.

Let us mention the following properties of the strong solutions (see Defini- tion 1) proven before:

– Global-in-time existence for small initial data [4, Corollary 3.13 and Lemma 3.10]

There exist ε > 0 and C > 0 such that for any

u

∈ {u ∈ V ∩ L

∩ L

: kuk

kuk

< ε} (4) or

u

∈ {u ∈ V ∩ L

∩ L

: kuk

k∇uk

< ε} (5) or

u

∈ {u ∈ V ∩ L

∩ L

: kuk

kuk

< ε}, q > 3 (6) a strong solution (u, p) with the initial data u(0) = u

exists on [0, +∞) and ku(t)k

≤ Cku

k

.

– Uniqueness of strong solutions [4, Theorem 3.9]

For any u

∈ V ∩ L

a strong solution with u(0) = u

is unique.

– Smothness [4, Corollary 3.3].

If (u, p) is a strong solution of the Navier-Stokes equation then

∂

∇

u, ∂

∇

p ∈ C((0, T ), L

) ∩ C((0, T ), L

), ∀m, k ≥ 0 and, in particular, u, p ∈ C

( R

× (0, T )) constitute a classical solution of the Navier-Stokes equations on (0, T ) × R

.

None of conditions (4), (5), (6) satisfy Corollary 1.

To expand globally the regularity of the Navier-Stokes equation a global-

in-time existence of strong solutions for small initial data has to be proven for

the norms satisfying Corollary 1.

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3. Netuka, I.: The change-of-variables theorem for the lebesgue integral. ACTA UNIVER- SITATIS MATTHIAE BELII, series MATHEMATICS19, 37–42 (2011)

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