Variable Exponents Spaces and Their Applications to Fluid Dynamics

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, 213 1 / 14

Overview 1 Variable Exponent Spaces 2 Applications to Fluid Dynamics Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 2 / 14

Variable Exponent Lebesgue Spaces p : Ω R d [1, ] measurable is called exponent. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 3 / 14

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, 213 3 / 14

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, 213 3 / 14

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, 213 3 / 14

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, 213 4 / 14

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, 213 4 / 14

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, 213 4 / 14

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, 213 4 / 14

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, 213 5 / 14

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, 213 5 / 14

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, 213 5 / 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, 213 5 / 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 Further Properties of the Lebesgue Spaces Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 5 / 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 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, 213 5 / 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 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, 213 5 / 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 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, 213 5 / 14

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, 213 6 / 14

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, 213 6 / 14

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, 213 7 / 14

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, 213 7 / 14

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, 213 7 / 14

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, 213 7 / 14

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, 213 8 / 14

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, 213 9 / 14

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, 213 9 / 14

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, 213 9 / 14

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, 213 9 / 14

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, 213 1 / 14

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, 213 1 / 14

Sketch of the Proof Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 11 / 14

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, 213 11 / 14

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, 213 11 / 14

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, 213 11 / 14

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, 213 11 / 14

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, 213 12 / 14

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):211-232. 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):255-268. Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 13 / 14

Thank you for your attention! Martin Rapp (TU Darmstadt) Variable Exponent Spaces November 7, 213 14 / 14