Functional inequalities and applications B. Al TAKI Sorbonne Universit´ e and INRIA-Paris July 31, 2019 CEMRACS’19 Geophysical Fluids, Gravity Flows B. Al Taki Functional inequalities and applications CEMRACS’19 1 / 22
Outline ◮ Part I: Weighted Sobolev inequalities ◮ Applications to geophysics models: Lake equations ◮ Difficulties... ◮ Part II: Logarithmic Sobolev inequalities ◮ Application to gas dynamic system. ◮ Diffusive capillary models of Korteweg type. B. Al Taki Functional inequalities and applications CEMRACS’19 2 / 22
Importance of weighted spaces e-Wirtinger’s inequality: let f : Ω ⊂ R n − Poincar´ → R smooth, compactly supported 1 � � � u | p dx ≤ C |∇ u | p dx | u − ¯ u = ¯ | u | dx . | Ω | Ω Ω Ω For p = 2, this inequality is the key tool of solving � − ∆ u = f in Ω , ( P 1 ) = g on ∂ Ω . u for Ω smooth domain and f , g such that f ∈ H − 1 (Ω) , g ∈ H 1 / 2 ( ∂ Ω) . ∈ H − 1 (Ω)?? But: What happens when f / B. Al Taki Functional inequalities and applications CEMRACS’19 3 / 22
Importance of weighted spaces e-Wirtinger’s inequality: let f : Ω ⊂ R n − Poincar´ → R smooth, compactly supported 1 � � � u | p dx ≤ C |∇ u | p dx | u − ¯ u = ¯ | u | dx . | Ω | Ω Ω Ω For p = 2, this inequality is the key tool of solving � − ∆ u = f in Ω , ( P 1 ) = g on ∂ Ω . u for Ω smooth domain and f , g such that f ∈ H − 1 (Ω) , g ∈ H 1 / 2 ( ∂ Ω) . ∈ H − 1 (Ω)?? But: What happens when f / B. Al Taki Functional inequalities and applications CEMRACS’19 3 / 22
Example 1: � − ∆ u = − div f in Ω , ( P 2 ) u = 0 , on ∂ Ω . For x ∈ Ω, we define f and d M as follows � f ( x ) = | x | − N / p ′ x 2 1 + x 2 2 + . . . + x 2 x ∈ Ω d M := dist( x , M ) = | x | = N . Notice that � 1 � | f ( x ) | p ′ dx = � � f � p ′ | x | − N dx = const r − 1 dr = ∞ , L p ′ (Ω) = Ω Ω 0 However � 1 � � � f � p ′ | f ( x ) | p ′ | x | ε dx = | x | − N + ε dx = const r − 1+ ε dr < ∞ . M ) = L p ′ (Ω , d ε Ω Ω 0 ∈ L p ′ (Ω) but f ∈ L p ′ (Ω , d ε ⇒ possibility of solution in W 1 , p (Ω , d ǫ f / M ) = m )? Remark Details about Laplacian equation in weighted spaces can be found in the works of Farwig, Sohr,. B. Al Taki Functional inequalities and applications CEMRACS’19 4 / 22
For Ω smooth domain, consider � − div( b ∇ u ) = b f in Ω , u = 0 on ∂ Ω . Variational problem: find u such that � � v ∈ H 1 a ( u , v ) := ∇ u ∇ vb dx = f · vb dx := L ( v ) 0 (Ω) Ω Ω ◮ If 0 < c 1 ≤ b ( x ) ≤ c 2 < ∞ � |∇ u | 2 b dx ≥ c 1 � u � H 1 a ( u , u ) = ⇒ Coercivity on Sob. space 0 (Ω) Ω � L ( v ) = f · vb dx ≤ c 2 � f � L 2 (Ω) � v � L 2 (Ω) ⇒ Continuity on Sob. spaces Ω Lax Milgram’s Theo. ⇒ existence. of sol. in H 1 0 (Ω) . ◮ When b ( x ) → 0 or b ( x ) → ∞ when x → ∂ Ω, We lose the coercivity and continuity in Sobolev spaces What about existence of sol. in weighted Sob. spaces.?? B. Al Taki Functional inequalities and applications CEMRACS’19 5 / 22
If we define ”formally” V b as follows � � � |∇ v | 2 b dx < ∞ V b = v measurable s . t Ω and suppose that � |∇ · | 2 b dx �∇ · � L 2 b (Ω) := Ω define a norm. Then we can verify that • a ( · , · ) is bilinear and coercive,i.e., � |∇ u | 2 b dx ≥ � u � 2 a ( u , u ) = V b Ω • L ( · ) is linear and continuous, i.e., � � � � ≤ C � b 1 / 2 f � L 2 (Ω) � b 1 / 2 u � L 2 (Ω) | L ( u ) | = f · v b dx � � � Ω B. Al Taki Functional inequalities and applications CEMRACS’19 6 / 22
◮ To finish the proof, it remains to verify that • �∇ · � L 2 b (Ω) is a norm • V b endowed with the ”norm” �∇ · � L 2 b (Ω) is a Hilbert space In other words, the question is: What conditions on the weight function b guarantee the validity of these two statements Among the most known weights, we found the Muckenhoupt weight. That means � 1 �� 1 � � � q − 1 b − 1 / q − 1 dx < ∞ , b ∈ L 1 b ∈ A q ⇐ ⇒ sup b dx loc (Ω) | Q | | Q | Q Q Q This family of weights ensure that W n , p (Ω) admits similar properties as W n , p (Ω) . b ◮ Some examples in A q : b ( x ) = ρ ( x ) α := dist( x , ∂ Ω) α , − ( n − 1) < α < ( n − 1)( q − 1) b ( x ) = | x − x 0 | α or Ref. B. O. Turessson , Nonlinear Potential Theory and Weighted Sobolev Spaces (2000). B. Al Taki Functional inequalities and applications CEMRACS’19 7 / 22
Importance of Muckenhoupt weight Ω =]0 , 1[ and u smooth s.t. u | ∂ Ω = 0 . Then if b ∈ A q , then we have � 1 � 1 | u | q b dx ≤ | u ′ | q b dx . 0 0 Proof. Since u | ∂ Ω = 0 , then we can write � x � x q � � | u | q b = b u ′ ( y ) dy � u ′ ( y ) dy � u ( x ) = . hence � � 0 � 0 � Hence � 1 � 1 � x q � � � | u | q b dx = � u ′ ( y ) dy � � b dx � � � � 0 0 0 � 1 � 1 q � � � � | u ′ ( y ) | dy � � ≤ C b dx � � � � 0 0 � � 1 � 1 q � � � | u ′ ( y ) | b 1 / q b − 1 / q dy � � ≤ C b dx � � � � 0 0 � � 1 �� � 1 � q − 1 � � 1 b − 1 / q − 1 dx | u ′ ( y ) | q b dx � ≤ C b dx . 0 0 0 B. Al Taki Functional inequalities and applications CEMRACS’19 8 / 22
Difficulties ♣ Trace: Characterization when b = ρ α ( x ) = dist α ( x , ∂ Ω) , − 1 < α < q − 1 ( ⇒ b ∈ A q ) → W 1 − 1+ α , q ( ∂ Ω) bounded linear operator T 1 , q : W 1 , q (Ω) − q b b ♣ Regular solution: Ω =]0 , 1[, b = ρ α , − 1 < α < 1 and ρ ( x ) = dist( x , ∂ Ω) � − ∂ x ( b ( x ) ∂ x u ) = f , in Ω , u = 0 , on ∂ Ω , Straightforward computation yields to � x x u = − b ′ ( x ) b ′ ( x ) 1 ∂ 2 f ( z ) dz + b ( x ) f ( x ) − ( b ( x )) 2 c . ( b ( x )) 2 0 However � 1 1 � 2 ⇒ α < 1 � ρ∂ 2 x u ∈ L 2 (Ω) ⇐ ⇒ dx < ∞ = 2 . ρ α ( x ) 0 ♣ Higher dimension? Big issue! Tangential regularity = ⇒ Normal regularity? B. Al Taki Functional inequalities and applications CEMRACS’19 9 / 22
Lake equations Theo. � ∂ t ( bu µ ) + div( bu µ ⊗ u µ ) − 2 µ div( bD ( u µ ) + b div u µ I ) + b ∇ p µ = 0 div( bu µ ) = 0 Navier boundary condition bu µ · n = 0 ( bD ( u µ ) · n + b div u µ I · n ) · τ = η u · τ. ◮ If b ∈ W 1 , ∞ b = ρ α 0 < α < 1 ( ⇒ b ∈ A 2 ) loc (Ω) with ⇒ ∃ ! u µ sol. with u µ ∈ L ∞ (0 , T ; L 2 b (Ω)) ∩ L 2 (0 , T ; V b ) ∩ C ((0 , T ) , H b − weak ) = v ∈ H 1 � � V b = b (Ω) , bv · n = 0 , div( bv ) = 0 . ◮ Moreover if b = ρ ( x ) α , 0 < α < 1 / 2 , ( ⇒ b ∈ A 3 / 2 ) for x ∈ V ( ∂ Ω) , ⇒ we can find p µ with ∇ p µ ∈ W − 1 , ∞ (0 , T ; H − 1 = (Ω)) b Work in progress with C. Lacave (Institute of Fourier) ◮ Regular solutions? ◮ What happen when µ tends to zero? B. Al Taki Functional inequalities and applications CEMRACS’19 10 / 22
Logarithmic Sobolev inequalities Sobolev inequality: let f : R n − → R smooth, compactly supported � � � 2 / p � 2 n R n | f | p dx R n |∇ f | 2 dx , ≤ C p = n − 2 ( > 2) n ≥ 3 = ⇒ 2 � � � � R n | f | p dx � R n |∇ f | 2 dx � p log ≤ C log � R n f 2 dx = 1 , Jensen’s inequality for f 2 dx Assume � � � � � � R n | f | p dx � R n | f | p − 2 f 2 dx � f 2 dx � | f | p − 2 � log = log ≥ R n log � p − 2 � � � R b f 2 log f 2 dx ≤ log R n |∇ f | 2 dx � p Form of logarithmic sobolev inequality B. Al Taki Functional inequalities and applications CEMRACS’19 11 / 22
Different forms and applications (Euclidean) Logarithmic Sobolev inequality � 2 � R n f 2 log f 2 dx ≤ n � � R n |∇ f | 2 dx � R n f 2 dx = 1 2 log n π e dx → d µ ( x ) = e −| x | 2 / 2 dx − (2 π ) n / 2 µ standard Gaussian probability measure on R n change f 2 into f 2 e −| x | 2 / 2 � f : R n − R n f 2 d µ = 1 → R , smooth s.t. � � R n f 2 log f 2 d µ ≤ 2 R n |∇ f | 2 d µ (Gaussian) Logarithmic Sobolev inequality B. Al Taki Functional inequalities and applications CEMRACS’19 12 / 22
Different forms and applications (Euclidean) Logarithmic Sobolev inequality � 2 � R n f 2 log f 2 dx ≤ n � � R n |∇ f | 2 dx � R n f 2 dx = 1 2 log n π e dx → d µ ( x ) = e −| x | 2 / 2 dx − (2 π ) n / 2 µ standard Gaussian probability measure on R n change f 2 into f 2 e −| x | 2 / 2 � f : R n − R n f 2 d µ = 1 → R , smooth s.t. � � R n f |∇ log f | 2 d µ (entropy) R n f log f d µ ≤ 2 (Fisher information) (Gaussian) Logarithmic Sobolev inequality B. Al Taki Functional inequalities and applications CEMRACS’19 13 / 22
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