On Solvability of a non-linear heat equation with non-integrable convective term and the right-hand side involving measures Josef M´ alek Mathematical Intitute of the Charles University Sokolovsk´ a 83, 186 75 Prague 8, Czech Republic June 19, 2008 M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 1 / 29
References [F1] M. Bul´ ıˇ cek , E. Feireisl , J. M´ alek: Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, to appear in Nonlinear Analysis and Real World Applications , 2008. [F2] M. Bul´ ıˇ cek , J. M´ alek, K. R. Rajagopal : Mathematical analysis of unsteady flows of fluids with pressure, shear-rate and temperature dependent material moduli, that slip at solid boundaries, preprint at http://ncmm.karlin.mff.cuni.cz [F3] M. Bul´ ıˇ cek , L. Consiglieri , J. M´ alek: Slip boundary effects on unsteady flows of incompressible viscous heat conducting fluids with a nonlinear internal energy-temperature relationship [Q1] M. Bul´ ıˇ cek , L. Consiglieri , J. M´ alek: On Solvability of a non-linear heat equation with a non-integrable convective term and the right-hand side involving measures M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 2 / 29
Problem formulation/1 e , t + div ( e v ) + div q ( · , e , ∇ e ) = f ≥ 0 in Q := (0 , T ) × Ω e (0 , x ) = e 0 ( x ) ≥ c > 0 in Ω (*) q ( t , x , e ( t , x ) , ∇ e ( t , x )) · n ( x ) = 0 (0 , T ) × ∂ Ω • for all ( e , u ) ∈ R × R d : q ( · , e , u ) is measurable, • for almost all ( t , x ) ∈ Q : q ( t , x , · , · ) is continuous in R × R d , • there are C 1 , C 2 > 0 such that for all ( e , u ) ∈ R × R d q ( · , e , u ) · u ≥ C 1 | u | q and | q ( · , e , u ) | ≤ C 2 | u | q − 1 , • for all e ∈ R and for all u 1 , u 2 ∈ R d , u 1 � = u 2 ( q ( · , e , u 1 ) − q ( · , e , u 2 )) · ( u 1 − u 2 ) > 0 . M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 3 / 29
Problem formulation/2 e , t + div ( e v ) + div q ( · , e , ∇ e ) = f ≥ 0 in Q := (0 , T ) × Ω e (0 , x ) = e 0 ( x ) > 0 in Ω ( P ) q ( t , x , e ( t , x ) , ∇ e ( t , x )) · n ( x ) = 0 (0 , T ) × ∂ Ω Data: Ω ⊂ R d with Lipschitz boundary, T ∈ (0 , ∞ ) e 0 ∈ L 1 (Ω) f ∈ L 1 ( Q ) M ( Q ) := ( C ( Q )) ∗ or v ∈ L r (0 , T ; L s (Ω)) (1 ≤ r , s ≤ ∞ ) div v = 0 in Q , v · n = 0 on (0 , T ) × ∂ Ω Task: Large data mathematical theory (notion of solution, its existence, uniqueness, ...) to Problem P , for any set of data and for largest class of constitutive relations M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 4 / 29
Approximations and apriori estimates/1 , t + div ( e n H n ( v )) + div q ( · , e n , ∇ e n ) = f n ≥ 0 e n e n (0 , · ) = e n 0 > 0 [ic] ( P n ) q ( · , e n , ∇ e n ) · n ( x ) = 0 [bc] where H n ( v ) := ( χ n v ) ∗ ω n − ∇ η n = ⇒ div H n ( v ) = 0 and H n ( v ) · n = 0 ⇒ H n ( v ) ∈ L ∞ (0 , T ; L k (Ω)) = ∀ k ∈ [1 , ∞ ) ⇒ H n ( v ) → v ∈ L r (0 , T ; L s (Ω)) = f n ∈ L ∞ ( Q ) f n → f in M ( Q ) or in L 1 ( Q ) 0 < e n 0 ∈ L ∞ (Ω) e n 0 → e 0 in L 1 (Ω) M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 5 / 29
Approximations and apriori estimates/2 , t + div ( e n H n ( v )) + div q ( · , e n , ∇ e n ) = f n ≥ 0 e n e n (0 , · ) = e n ( P n ) 0 > 0 [ic] q ( · , e n , ∇ e n ) · n ( x ) = 0 [bc] Truncation operators � z if | z | ≤ k , T k ( z ) := sign ( z ) k if | z | > k , � z if | z | ≤ k , T k ,δ ( z ) := if | z | > k + δ , sign ( z )( k + δ/ 2) such that T k ,δ ∈ C 2 ( R ), 0 ≤ T ′ k ,δ ≤ 1. � s � s Θ k ( s ) := T k ( t ) dt , Θ k ,δ ( s ) := T k ,δ ( t ) dt . 0 0 M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 6 / 29
Approximations and apriori estimates/3 , t + div ( e n H n ( v )) + div q ( · , e n , ∇ e n ) = f n ≥ 0 e n e n (0 , · ) = e n ( P n ) 0 > 0 [ic] q ( · , e n , ∇ e n ) · n ( x ) = 0 [bc] For any λ > 0 q − 1 − λ � � e ∈ L ∞ (0 , T ; L 1 (Ω)) , ∈ L q (0 , T ; L q (Ω) d ) E := e ≥ 0; ∇ (1+ e ) q if q > 2 d + 1 ⇒ � | e n | q − 1 � L 1 ( Q ) ≤ C � e n � E ≤ C = d + 1 �∇ T k ( e n ) � L q ( Q ) ≤ C . � T k ( e n ) , t � L 1 (0 , T ;( W 1 , z ) ∗ ) ≤ C , for sufficiently large z . Consequently, e n → e almost everywhere in Q M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 7 / 29
Weak Solution Let q > 2 d +1 d +1 and v ∈ L r (0 , T ; L s (Ω)) with r ′ s < q ( d + 1) − 2 d d ( q − 1) and s > q ( d + 1) − 2 d d We say that: e ∈ E is a weak solution to Problem ( P ) if for all ϕ ∈ D ( −∞ , T ; C ∞ (Ω)) − ( e , ϕ , t ) Q + ( q ( · , e , ∇ e ) , ∇ ϕ ) Q = � f , ϕ � + ( e v , ∇ ϕ ) Q + ( e 0 , ϕ (0)) Ω Theorem ( Bul´ ıˇ cek, Consiglieri , M´ alek) There exists a weak solution to Problem ( P ) . M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 8 / 29
Entropy solution Let q > 1 and v ∈ L 1 (0 , T ; L 1 (Ω)) and f ∈ L 1 ( Q ). We say that: e ∈ E is an entropy solution to Problem ( P ) if for a.a. t ∈ (0 , T ) � � ϕ , t , T k ( e − ϕ ) � Q t + Θ k ( e ( t ) − ϕ ( t )) + ( q ( · , e , ∇ e ) , ∇ T k ( e − ϕ )) Q t Ω � ≤ ( T k ( e − ϕ ) v , ∇ ϕ ) Q t + ( f , T k ( e − ϕ )) Q t + Θ k ( e (0) − ϕ (0)) dx Ω for all ϕ ∈ L ∞ (0 , T ; W 1 , ∞ (Ω)) with ϕ , t ∈ L q ′ (0 , T ; W − 1 , q ′ (Ω)) Theorem ( Bul´ ıˇ cek, Consiglieri , M´ alek) There exists an entropy solution to Problem ( P ) . This solution is unique in the class of entropy solutions provided that v ∈ L q ′ ( Q ) and q does not explicitly depends on e. M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 9 / 29
Results and their relation to earlier studies e , t + div ( e v ) + div q ( · , e , ∇ e ) = f ≥ 0 v given with div v = 0 Theorem W/a. ( Bocardo, Murat ’92 ) div ( v θ ) ∈ L 1 , f non-negative measure = ⇒ existence of weak solution. Theorem W/b. ( Diening, R˚ uˇ ziˇ cka, Wolf ’08 ) v θ ∈ L 1 , f ∈ L q ′ (0 , T ; W − 1 , q ′ ) = ⇒ existence of weak solution. Theorem W/c. ( Bul´ ıˇ cek, Consiglieri , M´ alek ’08 ) v θ ∈ L 1 , f non-negative measure = ⇒ existence of weak solution. Theorem E/a. ( Prignet ’97 ) v = 0 , f ∈ L 1 ( Q ) = ⇒ existence and uniqueness of entropy solution. Theorem E/b. ( Bul´ alek ’08 ) ıˇ cek, Consiglieri , M´ v ∈ L 1 ( Q ), f ∈ L 1 ( Q ) = ⇒ existence of entropy solution. v ∈ L q ′ ( Q ), q = q ( · , ∇ e ) and f ∈ L 1 ( Q ) = ⇒ uniqueness. M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 10 / 29
Key step: almost everywhere convergence of { e n } Theorem Let given q fulfil the assumptions with q > 1 and v ∈ L 1 ( Q ) . Assume that {| e n |} ∞ n =1 is bounded in E , { f n } ∞ n =1 is bounded in L 1 (0 , T ; L 1 (Ω)) , and � T k ,δ ( e n ) , t , ϕ � + ( q ( · , e n , ∇ e n ) , ∇ ( T ′ k ,δ ( e n ) ϕ )) Q = ( f n T ′ k ,δ ( e n ) , ϕ ) Q + ( e n H n ( v ) , ∇ ( T ′ k ,δ ( e n ) ϕ )) Q , for all ϕ ∈ L ∞ (0 , T ; W 1 , ∞ (Ω)) and all k , δ ∈ R + . 0 Then there exists a subsequence e n and e: | e | ∈ E and ∇ e n → ∇ e a.e. in Q M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 11 / 29
Key tool: Lipschitz approximations of Bochner functions/1 Lemma. Let for 1 < q < ∞ u ∈ L ∞ (0 , T ; L 2 (Ω)) ∩ L q (0 , T ; W 1 , q (Ω)) f ∈ L 1 ( Q ) q ∈ L q ′ (0 , T ; L q ′ (Ω)) fulfil in D ′ ( Q ) . u , t = div q + f Moreover, let E ⊂⊂ Q be an open set such that M α ( |∇ u | ) + α M α ( | q | ) + α M α ( | f | ) ≤ C < + ∞ , a.e. in Q \ E . (1) Then there holds ∇L α E u ∈ L ∞ (0 , T ; L ∞ (Ω)) ∂ t ( L α E u ) ( L α E u − u ) ∈ L 1 loc ( Q ) M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 12 / 29
Key tool: Lipschitz approximations of Bochner functions/2 and for all φ 1 ∈ C ∞ 0 (Ω) and all φ 2 ∈ C ∞ 0 (0 , T ) � T � � ∂ t u , T ε ( L α Θ ε ( L α E u ) φ 1 � φ 2 dt = − E u ) φ 1 ( ∂ t φ 2 ) dx dt 0 Q � ( u − L α E u ) ∂ t ( T ε ( L α − E u )) φ 1 φ 2 dx dt Q � ( u − L α E u ) T ε ( L α − E u ) φ 1 ( ∂ t φ 2 ) dx dt Q Proof is a minor (important) generalization (due to BCM) of the assertion due to Diening, R˚ uˇ ziˇ cka and Wolf (2008) . M´ alek (Charles University in Prague) On nonlinear convection-diffusion equation June 19, 2008 13 / 29
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