Olsens inequality and its application to Navier Stokes equations - - PowerPoint PPT Presentation

Olsen’s inequality and its application to Navier Stokes equations

Yoshihiro Sawano and Hitoshi Tanaka March 2011, Waseda University Joint work with Professor Sadek Gala (Mostaganem University Algeria)

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 1 / 15

Outline

1

Motivation

2

Main result

3

Applications to PDEs

4

References

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 2 / 15

Motivation and orientation

Sharp estimate of the factor g · Iαf , ∇g · f and its applications to PDEs. Here Iαf is given by Iαf (x) =

• Rd

f (y) |x − y|n−α dy. Here we take up the Navier equation as a special case of the MHD equation when b = 0.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 3 / 15

Morrey spaces

We define f Q,L logP L = inf

• λ > 0 :

1 |Q|

• Q

|f (x)| λ logP

• 2 + |f (x)|

λ

• dx ≤ 1
• for P > 1. We define

f Mp

L logP L = sup

Q∈Q

|Q|1/pf Q,L logP L. This is a very complicated definition and it might be more helpful when we replace t logP(2 + t) with tP.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 4 / 15

Morrey spaces

The above is an extention of the Morrey space Mp

q whose norm is given by

f Mp

q =

sup

Q : cube

|Q|

1 p − 1 q

• Q

|f (y)|q dy 1

q

, where 0 < q ≤ p < ∞.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 5 / 15

Fractional integral operators

Our fundamental result reads as follows:

Theorem

Let 0 < α < n/2. The fractional integral operator Iα is defined by Iαf (x) =

• Rn

f (y) |x − y|n−α dy. (1) If P > 1, then we have g · Iαf L2 ≤ C gMn/α

L2 logP L

f L2. (2) This inequality is named Olsen’s inequality after Olsen who investigated first this type of inequality. Originally Olsen applied to Schr¨

• dinger equations and we also sharped and

applied Olsen’s inequality. However, today we do not allude to this more.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 6 / 15

Remarks

1 The next result supports that our result is new.

Lemma

Let P > 0 and 0 < u < ˜ u < v. Then Lv ֒ → Lv,∞ ֒ → Mv

˜ u ֒

→ Mv

Lu logP L ֒

→ Mv

u

(3) in the sense of continuous embedding and the inclusion is proper.

2 It is well known that

Lemma

Let 1 < q ≤ p < ∞, 1 < t ≤ s < ∞ and q p = t s , 1 s = 1 p − α n . Then Iα : Mp

q → Ms t is bounded. Since we also have the H¨

• lder inequality, we

can obtain a “partial ” result. But this result is weaker than our main result.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 7 / 15

Example

As an example we considered the following 3D incompressible magneto-hydrodynamic (MHD in short) equations :        ∂tu − ∆u + (u · ∇)u + ∇p + 1

2∇ |b|2 − (b · ∇)b = 0

∂tb − ∆b + (u · ∇)b − (b · ∇)u = 0 div(u) = div(b) = 0 u (·, 0) = u0, b (·, 0) = b0 , (4) where u = u(x, t) is the velocity field, b ∈ R3 is the magnetic field, p = p(x, t) is the scalar pressure while u0 and b0 are given initial velocity and initial magnetic field with div(u0) = div(b0) = 0 in the sense of

• distribution. For simplicity, we assume that the external force has a scalar

potential and is included into the pressure gradient. When b = 0, this is just the Navier-Stokes equation.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 8 / 15

Definition of the weak solution

A measurable vector pair (u, b) is called a weak solution to MHD equations, if (u, b) satisfies the following properties (i) u ∈ L∞([0, T); L2) ∩ L2([0, T); H1), b ∈ L∞([0, T); L2) ∩ L2([0, T); H1) (ii) (u, b) verifies the equation in the sense of distribution; that is T

• R3

∂φ ∂t + ((u · ∇))φ

• udxdt

+

• R3 u0φ(x, 0)dx =

T

• R3 (u∇φ + (b · ∇)φ · b) dxdt

T

• R3

∂φ ∂t + ((u · ∇))φ

• bdxdt

+

• R3 b0φ(x, 0)dx =

T

• R3 (b∇φ + (b · ∇)φ · u) dxdt

for all φ ∈ C ∞

0 (R3 × [0, T)) with div(φ) = 0,

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 9 / 15

Definition of the weak solution continued

and T

• R3(u · ∇)φdxdt = 0,

T

• R3(b · ∇)φdxdt = 0

for every φ ∈ C ∞

0 (R3 × [0, T)). (iii) The energy inequality; that is

u(t)2

L2 + 2

t ∇u(s)2

L2ds ≤

u02

L2,

b(t)2

L2 + 2

t ∇b(s)2

L2ds ≤

b02

L2,

for 0 ≤ t ≤ T.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 10 / 15

Application to existence

Let (u0, b0) ∈ L2 R3 with div(u0) = div(b0) = 0. Assume that there exists a solution (u, b) of the magneto-hydrodynamic equations on (0, T) × R3 (for some T ∈ (0, +∞] with some initial data (u0, b0) such that (u, b) ∈ L∞ (0, T); L2

σ

• R3

∩ L2

• (0, T);

.

H

1 σ

• R3

, and for some r ∈ (0, 1), (u, b) ∈ L

2 1−r

• (0, T); M3/r

L2 logP L(R3)

• Then, (u, b) is the unique Leray solution associated with (u0, b0) on (0, T).

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 11 / 15

Related works

We considered the existence of the equation            ∂tu + (u · ∇)u − (µ + χ)∆u − (b · ∇)b + ∇(p + b2) − χ∇ × ω = 0, ∂tω − γ∆ω − κ∇divω + 2κω + (u · ∇)ω − χ∇ × u = 0, ∂tb − ν∆b + (u · ∇)b − (b · ∇)u = 0, div u = div b = 0, u(x, 0) = u0(x), b(x, 0) = b0(x), ω(x, 0) = ω0(x), (5) where u0, ω0 and b0 are the prescribed initial data for the velocity and angular velocity and magnetic field such that u0 and b0 are divergence free; div u0 = 0 and div b0 = 0. The constant µ is the kinematic viscosity, χ denotes the vortex viscosity, κ and γ are spin viscosities, and µ is the reciprocal of the magnetic Reynold. If the magnetic field b = 0, (5) reduces to the micropolar fluid system.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 12 / 15

References

• S. Gala, Y. Sawano and H. Tanaka, On the uniqueness of weak

solutions of the 3D MHD equations in the Orlicz-Morrey space, submtted.

• S. Gala, Y. Sawano and H. Tanaka, A new Beale-Kato-Majda criteria

for the 3D magneto-micropolar fluid equations in the Orlicz-Morrey space, submitted.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 13 / 15

References

• Y. Sawano, S. Sugano and H. Tanaka, Generalized fractional integral
• perators and fractional maximal operators in the framework of

Morrey spaces, to appear in Trans. Amer. Math. Soc..

• Y. Sawano, S. Sugano and H. Tanaka, A Note on Generalized

Fractional Integral Operators on Generalized Morrey Space, Boundary Value Problems, vol.2009 (2009), Article ID 835865, 18pp.

• Y. Sawano, S. Sugano and H. Tanaka, Orlicz-Morrey spaces and

fractional operators, in preparation.

• Y. Sawano, S. Sugano and H. Tanaka, Olsen’s inequality and its

applications to Schr¨

• dinger equations, to appear in RIMS Kˆ
• kyˆ

uroku Bessatsu.

• H. Tanaka, Morrey spaces and fractional operators, J. Aust. Math.

Soc., 88 (2010), 247-259.

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 14 / 15

Last slide... Thank you for your attention!

Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality March 2011, Waseda University 15 / 15