Variable Exponents Spaces and Their Applications to Fluid Dynamics

Transcription

1 Variable Exponents Spaces and Their Applications to Fluid Dynamics Martin Rapp TU Darmstadt November 7, 213 Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

2 Overview 1 Variable Exponent Spaces 2 Applications to Fluid Dynamics Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

3 Variable Exponent Lebesgue Spaces p : Ω R d [1, ] measurable is called exponent. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

4 Variable Exponent Lebesgue Spaces p : Ω R d [1, ] measurable is called exponent. p + := ess sup x Ω p(x) p := ess inf x Ω p(x) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

5 Variable Exponent Lebesgue Spaces p : Ω R d [1, ] measurable is called exponent. p + := ess sup x Ω p(x) p := ess inf x Ω p(x) For measurable f define the modular ϱ p( ) (f ) := f (x) p(x) χ {p } dx + f χ {p= } Ω Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

6 Variable Exponent Lebesgue Spaces p : Ω R d [1, ] measurable is called exponent. p + := ess sup x Ω p(x) p := ess inf x Ω p(x) For measurable f define the modular ϱ p( ) (f ) := f (x) p(x) χ {p } dx + f χ {p= } Ω L p( ) (Ω) := {f : ϱ p( ) (f ) < } with norm ( ) f f p( ) := inf{λ > : ϱ p( ) 1} λ Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

7 Variable Exponent Sobolev Spaces Define the modular ϱ k,p( ) (f ) := α k ϱ p( ) ( α f ) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

8 Variable Exponent Sobolev Spaces Define the modular ϱ k,p( ) (f ) := α k ϱ p( ) ( α f ) W k,p( ) (Ω) := {f : ϱ k,p( ) (f ) < } with norm ( ) f f k,p( ) := inf{λ > : ϱ k,p( ) 1} λ Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

9 Variable Exponent Sobolev Spaces Define the modular ϱ k,p( ) (f ) := α k ϱ p( ) ( α f ) W k,p( ) (Ω) := {f : ϱ k,p( ) (f ) < } with norm ( ) f f k,p( ) := inf{λ > : ϱ k,p( ) 1} λ This norm is equivalent to k m= m f p( ) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

10 Variable Exponent Sobolev Spaces Define the modular ϱ k,p( ) (f ) := α k ϱ p( ) ( α f ) W k,p( ) (Ω) := {f : ϱ k,p( ) (f ) < } with norm ( ) f f k,p( ) := inf{λ > : ϱ k,p( ) 1} λ This norm is equivalent to k m= m f p( ) Let W k,p( ) (Ω) be the closure of C (Ω) w.r.t. k,p( ) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

11 Properties of Variable Exponent Spaces Basic Properties of L p( ) (Ω), W k,p( ) (Ω), W k,p( ) (Ω) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

12 Properties of Variable Exponent Spaces Basic Properties of L p( ) (Ω), W k,p( ) (Ω), W k,p( ) (Ω) Banach spaces Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

13 Properties of Variable Exponent Spaces Basic Properties of L p( ) (Ω), W k,p( ) (Ω), W k,p( ) (Ω) Banach spaces p + < separable Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

14 Properties of Variable Exponent Spaces Basic Properties of L p( ) (Ω), W k,p( ) (Ω), W k,p( ) (Ω) Banach spaces p + < separable 1 < p p + < reflexive Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

15 Properties of Variable Exponent Spaces Basic Properties of L p( ) (Ω), W k,p( ) (Ω), W k,p( ) (Ω) Banach spaces p + < separable 1 < p p + < reflexive Further Properties of the Lebesgue Spaces Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

16 Properties of Variable Exponent Spaces Basic Properties of L p( ) (Ω), W k,p( ) (Ω), W k,p( ) (Ω) Banach spaces p + < separable 1 < p p + < reflexive Further Properties of the Lebesgue Spaces Hölder inequality Ω f (x)g(x) dx 2 f p( ) g p ( ), where 1 p + 1 p = 1 a.e. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

17 Properties of Variable Exponent Spaces Basic Properties of L p( ) (Ω), W k,p( ) (Ω), W k,p( ) (Ω) Banach spaces p + < separable 1 < p p + < reflexive Further Properties of the Lebesgue Spaces Hölder inequality Ω f (x)g(x) dx 2 f p( ) g p ( ), where 1 p + 1 p ( L p( ) (Ω) ) = L p ( ) (Ω) = 1 a.e. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

18 Properties of Variable Exponent Spaces Basic Properties of L p( ) (Ω), W k,p( ) (Ω), W k,p( ) (Ω) Banach spaces p + < separable 1 < p p + < reflexive Further Properties of the Lebesgue Spaces Hölder inequality Ω f (x)g(x) dx 2 f p( ) g p ( ), where 1 p + 1 p ( L p( ) (Ω) ) = L p ( ) (Ω) = 1 a.e. If Ω is bounded and p q a.e., then L p( ) (Ω) L q( ) (Ω) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

19 log-hölder Continuity An exponent p on a bounded domain Ω is called log-hölder continuous if p(x) p(y) c log(e + 1 x y ) x, y Ω, x y. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

20 log-hölder Continuity An exponent p on a bounded domain Ω is called log-hölder continuous if p(x) p(y) c log(e + 1 x y ) x, y Ω, x y. For d 2 and an exponent p define the Sobolev-conjugate exponent p (x) := { dp(x) d p(x) : p(x) < d : p(x) d. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

21 p log-hölder Continuity If Ω is bounded and p is log-hölder continuous, then the following theorems hold: Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

22 p log-hölder Continuity If Ω is bounded and p is log-hölder continuous, then the following theorems hold: Poincaré: u p( ) c u p( ) for all u W 1,p( ) (Ω) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

23 p log-hölder Continuity If Ω is bounded and p is log-hölder continuous, then the following theorems hold: Poincaré: u p( ) c u p( ) for all u W 1,p( ) (Ω) p + < d W 1,p( ) (Ω) L p ( ) (Ω) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

24 p log-hölder Continuity If Ω is bounded and p is log-hölder continuous, then the following theorems hold: Poincaré: u p( ) c u p( ) for all u W 1,p( ) (Ω) p + < d W 1,p( ) (Ω) L p ( ) (Ω) p + < d W 1,p( ) (Ω) L p ( ) ɛ (Ω) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

25 From now on we assume Ω R d bounded domain p is log-hölder continuous f L p ( ) (Ω) or f L p ( ) (Ω) d δ Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

26 Navier-Stokes equations Navier-Stokes equations div (δ + Du ) p( ) 2 Du + div (u u) + π = f in Ω div u = in Ω u = on Ω Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

27 Navier-Stokes equations Navier-Stokes equations div (δ + Du ) p( ) 2 Du + div (u u) + π = f in Ω div u = in Ω u = on Ω Solving Methods Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

28 Navier-Stokes equations Navier-Stokes equations div (δ + Du ) p( ) 2 Du + div (u u) + π = f in Ω div u = in Ω u = on Ω Solving Methods Classical theory of monotone operators: Existence of weak solution u W 1,p( ) (Ω) d for p > 3d d+2. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

29 Navier-Stokes equations Navier-Stokes equations div (δ + Du ) p( ) 2 Du + div (u u) + π = f in Ω div u = in Ω u = on Ω Solving Methods Classical theory of monotone operators: Existence of weak solution u W 1,p( ) (Ω) d for p > 3d d+2. Method of Lipschitz-Truncations: Existence of weak solution u W 1,p( ) (Ω) d for p > 2d d+2. (Diening, Málek, Steinhauer 28 and Diening, Růžička 21) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

30 convenction-diffusion equations convection-diffusion equations div (δ + u ) p( ) 2 u + a u = f in Ω u = on Ω where a L r( ) (Ω) with div a = is given. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

31 convenction-diffusion equations convection-diffusion equations div (δ + u ) p( ) 2 u + a u = f in Ω u = on Ω where a L r( ) (Ω) with div a = is given. Theorem For 1 < p p + < d or d < p p + <. Let a Lσ r( ) (Ω) d, where r is continuous and r > (p ) a.e. or r = 1 respectively. Then there is a weak solution u W 1,p( ) (Ω) of the equation. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

32 Sketch of the Proof Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

33 Sketch of the Proof Solve by classical theory of monotone operators for every n N (δ + u n ) p( ) 2 u n ϕ dx u n a n ϕ dx Ω Ω + 1 u n q 2 u n ϕ dx = (f, ϕ) n Ω in W 1,p( ) (Ω) L q (Ω) where the a n are smooth divergence free functions with a n a in L r( ) (Ω) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

34 Sketch of the Proof Solve by classical theory of monotone operators for every n N (δ + u n ) p( ) 2 u n ϕ dx u n a n ϕ dx Ω Ω + 1 u n q 2 u n ϕ dx = (f, ϕ) n Ω in W 1,p( ) (Ω) L q (Ω) where the a n are smooth divergence free functions with a n a in L r( ) (Ω) Get a sequence u n W 1,p( ) (Ω) L q (Ω) which is bounded in W 1,p( ) (Ω) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

35 Sketch of the Proof Solve by classical theory of monotone operators for every n N (δ + u n ) p( ) 2 u n ϕ dx u n a n ϕ dx Ω Ω + 1 u n q 2 u n ϕ dx = (f, ϕ) n Ω in W 1,p( ) (Ω) L q (Ω) where the a n are smooth divergence free functions with a n a in L r( ) (Ω) Get a sequence u n W 1,p( ) (Ω) L q (Ω) which is bounded in W 1,p( ) (Ω) There is a weakly convergent subsequence: u n u in W 1,p( ) (Ω) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

36 Sketch of the Proof Solve by classical theory of monotone operators for every n N (δ + u n ) p( ) 2 u n ϕ dx u n a n ϕ dx Ω Ω + 1 u n q 2 u n ϕ dx = (f, ϕ) n Ω in W 1,p( ) (Ω) L q (Ω) where the a n are smooth divergence free functions with a n a in L r( ) (Ω) Get a sequence u n W 1,p( ) (Ω) L q (Ω) which is bounded in W 1,p( ) (Ω) There is a weakly convergent subsequence: u n u in W 1,p( ) (Ω) For limit in nonlinear p( )-Laplacian use method of Lipschitz-Truncations Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

37 Sketch of the Proof u solves then (δ + u) p( ) 2 u ϕ dx Ω for all ϕ W 1, (Ω) Ω ua n ϕ dx = (f, ϕ) Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

38 References L. Diening, P. Harjulehto, P. Hästö, M. Růžička (211): Lebesgue and Sobolev Spaces with Variable Exponents Springer-Verlag, Berlin Heidelberg. L. Diening, J. Málek, M. Steinhauer (28): On Lipschitz Truncations of Sobolev Functions (With Variable Exponent) and Their Selected Applications ESAIM: Control, Optimisation and Calculus of Variations 14(2): L. Diening, M. Růžička (21): An existence result for non-newtonian fluids in non-regular domains Discrete and Continuous Dynamical Systems Series S 3(2): Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14

39 Thank you for your attention! Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, / 14