Optimal decay estimates on the framework of Besov spaces for - PowerPoint PPT Presentation

Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Optimal decay estimates on the framework of Besov spaces for hyperbolic systems with degenerate dissipation Jiang Xu Nanjing University of Aeronautics and Astronautics, China Joint work with Shuichi Kawashima Mathflows 2015 September 13-18th, 2015, Porquerolles Jiang Xu hyperbolic dissipative systems

Hyperbolic systems with symmetric dissipation Hyperbolic systems with Non-symmetric dissipation References Outline Hyperbolic systems with symmetric dissipation 1 Introduction New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Littlewood-Paley pointwise estimates Nonlinear applications Hyperbolic systems with Non-symmetric dissipation 2 Examples Frequency-localization time-decay inequality An open problem References 3 Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Outline Hyperbolic systems with symmetric dissipation 1 Introduction New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Littlewood-Paley pointwise estimates Nonlinear applications Hyperbolic systems with Non-symmetric dissipation 2 Examples Frequency-localization time-decay inequality An open problem References 3 Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Hyperbolic systems with dissipation Consider the following hyperbolic system � A 0 ∂ t w + � n j = 1 A j ∂ x j w + Lw = 0 , (1) w ( 0 , x ) = w 0 , for ( t , x ) ∈ [ 0 , + ∞ ) × R n . w ( t , x ) : R N -valued function; A j ( j = 0 , 1 , · · · , n ) and L are constant matrices of order N ; Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Hyperbolic systems with dissipation Assume that the equation of (1) is “symmetric hyperbolic" in the following sense: (A1) Matrices A j ( j = 0 , · · · , n ) are real symmetric and, in addition, A 0 is positive definite. L is real symmetric and nonnegative definite. Also, assume (1) satisfies the “Shizuta-Kawashima" condition ( � [Shizuta & Kawashima, Hokkaido Math. J., 1985)]) (A2) Let φ ∈ R N and ( λ, ω ) ∈ R × S n − 1 . If L φ = 0 and λ A 0 φ + A ( ω ) φ = 0, then φ = 0. Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Decay framework L 2 ( R n ) ∩ L p ( R n ) � [Umeda, Kawashima & Shizuta (Japan J. Appl. Math., 1984)] The dissipative structure of (1) satisfies | ξ | 2 Re λ ( i ξ ) ≤ − c η 1 ( ξ ) with η 1 ( ξ ) = 1 + | ξ | 2 for c > 0, which leads to the optimal decay estimate � w � L 2 ( R n ) � � w 0 � L 2 ( R n ) ∩ L p ( R n ) ( 1 + t ) − n 2 ( 1 p − 1 2 ) (2) for 1 ≤ p < 2. Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Outline Hyperbolic systems with symmetric dissipation 1 Introduction New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Littlewood-Paley pointwise estimates Nonlinear applications Hyperbolic systems with Non-symmetric dissipation 2 Examples Frequency-localization time-decay inequality An open problem References 3 Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications New decay framework (motivation) In [X.-Kawashima (ARMA, 2015)], we give a new decay framework for (1): L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n )( 0 < s ≤ n / 2 ) which can be regarded as the natural generalization, since → ˙ L p ( R n ) ֒ B − s 2 , ∞ ( R n ) with s + n / 2 = n / p . In particular, → ˙ → ˙ B − n / 2 L 1 ( R n ) ֒ B 0 1 , ∞ ( R n ) ֒ 2 , ∞ ( R n ) . Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Decay property Theorem 1.1 Let the assumptions (A1)-(A2) hold. Suppose w 0 ∈ L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) for s > 0, then the solution of (1) has the decay estimate 2 , ∞ ( R n ) ( 1 + t ) − s / 2 . � w � L 2 ( R n ) � � w 0 � L 2 ( R n ) ∩ ˙ (3) B − s In particular, suppose w 0 ∈ L 2 ( R n ) ∩ L p ( R n )( 1 ≤ p < 2), one further has � w � L 2 ( R n ) � � w 0 � L 2 ( R n ) ∩ L p ( R n ) ( 1 + t ) − n 2 ( 1 p − 1 2 ) . (4) Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Sketch of Proof of Theorem 1.1 The proof is divided into several steps. Step 1. Pointwise energy estimte gives w | 2 + c 1 κ | ξ | 2 d w | 2 ≤ 0 dt E [ ˆ w ] + ( c 0 − κ C ) | ( I − P ) ˆ 1 + | ξ | 2 | ˆ (5) with � � w ] = 1 w ) + κ | ξ | 2 ( A 0 ˆ 1 + | ξ | 2 K ( ω ) A 0 ˆ E [ ˆ w , ˆ w , ˆ 2 Im w , where we chosen κ > 0 so small that c 0 − κ C ≥ 0 and w | 2 , since A 0 is positive definite. E [ ˆ w ] ≈ | ˆ Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Sketch of Proof of Theorem 1.1 Step 2. The low- and high-frequency decompositions (the unit decomposition): w = w L + w H with w L = F − 1 [ φ ( ξ ) ˆ w ( ξ )] and w H = F − 1 [ ϕ ( ξ ) ˆ w ( ξ )] , where 1 ≡ φ ( ξ ) + ϕ ( ξ ) , and φ, ϕ ∈ C ∞ c ( R n ) (0 ≤ φ ( ξ ) , ϕ ( ξ ) ≤ 1) satisfy φ ( ξ ) ≡ 1 , if | ξ | ≤ R ; φ ( ξ ) ≡ 0 , if | ξ | ≥ 2 R with R > 0. Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Sketch of Proof of Theorem 1.1 Step 3. The high-frequency estimate w ]) + c 1 R 2 d w | 2 ≤ 0 , dt ( ϕ 2 E [ ˆ 1 + R 2 | ϕ ˆ (6) which implies that � w H � L 2 ≤ Ce − c 2 t � w 0 � L 2 , (7) for c 2 > 0 depending on R . Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Sketch of Proof of Theorem 1.1 Step 4. The low-frequency estimate d c 1 w L | 2 ≤ 0 , dt (˜ w ] 2 ) + 1 + R 2 | ξ | 2 | ˆ E [ ˆ (8) w ] } 1 / 2 and ˜ where ˜ w ] := { φ 2 E [ ˆ E [ ˆ E [ ˆ w ] ≈ | ˆ w L | . Furthermore, Plancherel’s theorem gives d E 2 ˜ L + c 3 �∇ w L � 2 L 2 ≤ 0 , (9) dt where ˜ E L ≈ � w L � L 2 and the constant c 3 > 0 depends on R . Jiang Xu hyperbolic dissipative systems

Introduction Hyperbolic systems with symmetric dissipation New decay framework L 2 ( R n ) ∩ ˙ B − s 2 , ∞ ( R n ) Hyperbolic systems with Non-symmetric dissipation Littlewood-Paley pointwise estimates References Nonlinear applications Sketch of Proof of Theorem 1.1 Step 5. Applying the interpolation inequality � � s � f � L 2 � �∇ f � θ L 2 � f � 1 − θ θ = . (10) B − s ˙ 1 + s 2 , ∞ to the low-frequency part w L , we obtain the differential equality from (9): d 1 + C � w 0 � − 2 / s 2 , ∞ � w L � 2 ( 1 + 1 / s ) E 2 ˜ ≤ 0 , (11) B − s ˙ L 2 dt which implies that 2 , ∞ ( 1 + t ) − s / 2 . � w L � L 2 � � w 0 � ˙ (12) B − s Jiang Xu hyperbolic dissipative systems