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 Motivation 1 Main result 2 Applications to PDEs 3 References 4 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 · I α f , ∇ g · f and its applications to PDEs. Here I α f is given by � f ( y ) I α f ( x ) = | x − y | n − α dy . R d 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 � 1 � | f ( x ) | � 2 + | f ( x ) | � � log P � f � Q , L log P L = inf λ > 0 : dx ≤ 1 | Q | λ λ Q for P > 1. We define | Q | 1 / p � f � Q , L log P L . � f � M p L log P L = sup Q ∈Q This is a very complicated definition and it might be more helpful when we replace t log P (2 + t ) with t P . 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 M p q whose norm is given by � 1 �� q 1 p − 1 | f ( y ) | q dy � f � M p q = sup | Q | , q Q : cube 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 I α is defined by � f ( y ) I α f ( x ) = | x − y | n − α dy . (1) R n If P > 1 , then we have � g · I α f � L 2 ≤ C � g � M n /α � f � L 2 . (2) L2 log P L This inequality is named Olsen’s inequality after Olsen who investigated first this type of inequality. Originally Olsen applied to Schr¨ odinger 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 L v ֒ → L v , ∞ ֒ → M v → M v → M v u ֒ L u log P L ֒ (3) ˜ u in the sense of continuous embedding and the inclusion is proper. 2 It is well known that Lemma s , 1 s = 1 p − α Let 1 < q ≤ p < ∞ , 1 < t ≤ s < ∞ and q p = t n . Then I α : M p q → M s t is bounded. Since we also have the H¨ older 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 3 D incompressible magneto-hydrodynamic (MHD in short) equations : 2 ∇ | b | 2 − ( b · ∇ ) b = 0 ∂ t u − ∆ u + ( u · ∇ ) u + ∇ p + 1 ∂ t b − ∆ b + ( u · ∇ ) b − ( b · ∇ ) u = 0 , (4) div ( u ) = div ( b ) = 0 u ( · , 0) = u 0 , b ( · , 0) = b 0 where u = u ( x , t ) is the velocity field, b ∈ R 3 is the magnetic field, p = p ( x , t ) is the scalar pressure while u 0 and b 0 are given initial velocity and initial magnetic field with div ( u 0 ) = div ( b 0 ) = 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 ); L 2 ) ∩ L 2 ([0 , T ); H 1 ), b ∈ L ∞ ([0 , T ); L 2 ) ∩ L 2 ([0 , T ); H 1 ) (ii) ( u , b ) verifies the equation in the sense of distribution; that is � T � � ∂φ � ∂ t + (( u · ∇ )) φ udxdt R 3 0 � T � � + R 3 u 0 φ ( x , 0) dx = R 3 ( u ∇ φ + ( b · ∇ ) φ · b ) dxdt 0 � T � � ∂φ � ∂ t + (( u · ∇ )) φ bdxdt R 3 0 � T � � + R 3 b 0 φ ( x , 0) dx = R 3 ( b ∇ φ + ( b · ∇ ) φ · u ) dxdt 0 0 ( R 3 × [0 , T )) with div ( φ ) = 0, for all φ ∈ C ∞ 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 � T � � R 3 ( u · ∇ ) φ dxdt = 0 , R 3 ( b · ∇ ) φ dxdt = 0 0 0 0 ( R 3 × [0 , T )). (iii) The energy inequality; that is for every φ ∈ C ∞ � t �∇ u ( s ) � 2 � u 0 � 2 � u ( t ) � 2 L 2 + 2 L 2 ds ≤ L 2 , 0 � t �∇ b ( s ) � 2 � b 0 � 2 � b ( t ) � 2 L 2 + 2 L 2 ds ≤ L 2 , 0 for 0 ≤ t ≤ T . March 2011, Waseda University 10 / Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality 15
Application to existence Let ( u 0 , b 0 ) ∈ L 2 � R 3 � with div ( u 0 ) = div ( b 0 ) = 0. Assume that there exists a solution ( u , b ) of the magneto-hydrodynamic equations on (0 , T ) × R 3 (for some T ∈ (0 , + ∞ ] with some initial data ( u 0 , b 0 ) such that � . R 3 �� 1 (0 , T ); L 2 R 3 �� ∩ L 2 ( u , b ) ∈ L ∞ � � � (0 , T ); H , σ σ and for some r ∈ (0 , 1) , 2 � (0 , T ); M 3 / r � L 2 log P L ( R 3 ) ( u , b ) ∈ L 1 − r Then, ( u , b ) is the unique Leray solution associated with ( u 0 , b 0 ) on (0 , T ). March 2011, Waseda University 11 / Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality 15
Related works We considered the existence of the equation ∂ t u + ( u · ∇ ) u − ( µ + χ )∆ u − ( b · ∇ ) b + ∇ ( p + b 2 ) − χ ∇ × ω = 0 , ∂ t ω − γ ∆ ω − κ ∇ div ω + 2 κω + ( u · ∇ ) ω − χ ∇ × u = 0 , ∂ t b − ν ∆ b + ( u · ∇ ) b − ( b · ∇ ) u = 0 , div u = div b = 0 , u ( x , 0) = u 0 ( x ) , b ( x , 0) = b 0 ( x ) , ω ( x , 0) = ω 0 ( x ) , (5) where u 0 , ω 0 and b 0 are the prescribed initial data for the velocity and angular velocity and magnetic field such that u 0 and b 0 are divergence free; div u 0 = 0 and div b 0 = 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. March 2011, Waseda University 12 / Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality 15
References S. Gala, Y. Sawano and H. Tanaka, On the uniqueness of weak solutions of the 3 D 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. March 2011, Waseda University 13 / Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality 15
References Y. Sawano, S. Sugano and H. Tanaka, Generalized fractional integral operators 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¨ odinger equations, to appear in RIMS Kˆ okyˆ uroku Bessatsu. H. Tanaka, Morrey spaces and fractional operators, J. Aust. Math. Soc., 88 (2010), 247-259. March 2011, Waseda University 14 / Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality 15
Last slide... Thank you for your attention! March 2011, Waseda University 15 / Yoshihiro Sawano and Hitoshi Tanaka (Kyoto University, Tokyo University) Olsen’s inequality 15
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