Higher integrability for doubly nonlinear parabolic equations Juha Kinnunen, Aalto University, Finland juha.k.kinnunen@aalto.fi http://math.aalto.fi/ ∼ jkkinnun/ August 26, 2019 Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
References V. B¨ ogelein, F. Duzaar, J. Kinnunen and C. Scheven, Higher integrability for doubly nonlinear parabolic systems , submitted (2018). V. B¨ ogelein, F. Duzaar, R. Korte and C. Scheven, The higher integrability of weak solutions of porous medium systems , Adv. Nonlinear Anal. 8 (2018) 1004–1034. V. B¨ ogelein, F. Duzaar and C. Scheven, Higher integrability for the singular porous medium system , submitted (2018). U. Gianazza and S. Schwarzacher, Self-improving property of degenerate parabolic equations of porous medium-type , Amer. J. Math. (to appear). U. Gianazza and S. Schwarzacher, Self-improving property of the fast diffusion equation , submitted (2018). Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
Nonlinear parabolic equations The porous medium equation/system u t − ∆( | u | m − 1 u ) = 0 , 0 < m < ∞ . Sometimes written in the form ( | u | m − 2 u ) t − ∆ u = 0 , 1 < m < ∞ . The parabolic p -Laplace equation/system u t − div( | Du | p − 2 Du ) = 0 , 1 < p < ∞ . The doubly nonlinear equation/system ( | u | p − 2 u ) t − div( | Du | p − 2 Du ) = 0 , 1 < p < ∞ . Sometimes called Trudinger’s equation. All equations above are special cases of a general doubly nonlinear equation/system ( | u | m − 2 u ) t − div( | Du | p − 2 Du ) = 0 , 1 < m < ∞ , 1 < p < ∞ . Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
Outline of the talk Goal To show that the gradient of a weak solution to ( | u | p − 2 u ) t − div( | Du | p − 2 Du ) = 0 , is locally integrable to a better power than assumed in the definition, with a reverse H¨ older inequality estimate for the gradient. Motivation To extend the recent breakthroughs by Gianazza–Schwarzacher and others to cover a wider class of equations and systems. To develop direct methods that only apply energy estimates, Sobolev–Poincar´ e inequalities and Calder´ on–Zygmund type covering arguments. To develop methods that apply to sign-changing solutions and systems. In particular, Harnack estimates and the expansion of positivity are not applied in the argument. Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
Review of the elliptic case Let u ∈ W 1 , p loc (Ω), with 1 < p < ∞ , be a weak solution of the stationary p -Laplace equation div( | Du | p − 2 Du ) = 0 in Ω ⊂ R n . Then there exists ε > 0 such that 1 � 1 �� � � � p + ε | Du | p + ε dy | Du | p dy p ≤ c B ( x , r ) B ( x , 2 r ) for every ball B ( x , 2 r ) ⊂ Ω. In particular, u ∈ W 1 , p + ε (Ω) . loc (Gehring 1973, Meyers and Elcrat 1975, Giaquinta and Modica 1979, Stredulinsky 1980) Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
A sketch of the proof Step 1: An energy (Caccioppoli) estimate � 1 � 1 �� �� ≤ c p p | Du | p dy | u − u B ( x , 2 r ) | p dy . r B ( x , r ) B ( x , 2 r ) Step 2: A Sobolev–Poincar´ e inequality � 1 � 1 �� �� p q | u − u B ( x , 2 r ) | p dy | Du | q dy ≤ cr B ( x , 2 r ) B ( x , 2 r ) for some q < p . Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
Step 3: A reverse H¨ older inequality � 1 � 1 �� �� p q | Du | p dy | Du | q dy ≤ c , B ( x , r ) B ( x , 2 r ) for every B ( x , 2 r ) ⊂ Ω with some q < p . Step 4: The Gehring–Meyers–Elrcat lemma: There exists ε > 0 such that 1 � 1 �� � �� p + ε p | Du | p + ε dy | Du | p dy ≤ c B ( x , r ) B ( x , 2 r ) for every B ( x , 2 r ) ⊂ Ω. Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
Higher integrability results for parabolic equations 1(2) Giaquinta–Struwe 1982 : Parabolic systems with a quadratic structure, i.e. p = 2. Kinnunen–Lewis 2000 : Systems of the parabolic p -Laplacian structure u t − div( | Du | p − 2 Du ) = div( | F | p − 2 F ) , 2 n p > n +2 Gianazza–Schwarzacher 2016 : Nonnegative solutions to the porous medium type equations u t − ∆ u m = f , m ≥ 1 . Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
Higher integrability results for parabolic equations 2(2) B¨ ogelein–Duzaar–Korte–Scheven 2017 : Systems of the porous medium type | u | m − 1 u � � u t − ∆ = div F , m ≥ 1 . Gianazza–Schwarzacher 2018 : Nonnegative solutions to the porous medium type equations u t − ∆ u m = f , ( n − 2) + < m < 1 . n +2 B¨ ogelein–Duzaar–Scheven 2018 : Systems of the porous medium type ( n − 2) + | u | m − 1 u � � u t − ∆ = div F , < m ≤ 1 . n +2 Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
Potential applications Partial regularity result for parabolic systems: Local H¨ older continuity outside a small set (Misawa 2002). Nonlinear Calder´ on-Zygmund theory (Acerbi and Mingione 2007). Estimates up to the boundary (Parviainen 2009, Moring-Scheven-Schwarzacher-Singer 2019). Higher order systems (Parviainen and B¨ ogelein 2010). Stability of solutions as p varies (Kinnunen and Parviainen 2010). Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
The parabolic Sobolev space Ω is an open subset of R n and 0 ≤ t 1 < t 2 ≤ T . The space-time cylinders are Ω T = Ω × (0 , T ) and D t 1 , t 2 = D × ( t 1 , t 2 ) , where D ⊂ Ω is an open set. The parabolic Sobolev space L p (0 , T ; W 1 , p (Ω)) consists of measurable functions u : Ω T → [ −∞ , ∞ ] such that x �→ u ( x , t ) belongs to W 1 , p (Ω) for almost all t ∈ (0 , T ), and �� ( | u | p + | Du | p ) dx dt < ∞ . Ω T u ∈ L p loc (0 , T ; W 1 , p loc (Ω)), if u belongs to the parabolic Sobolev space for every D t 1 , t 2 ⋐ Ω T . Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
The doubly nonlinear equation Let 1 < p < ∞ . A function u ∈ L p loc (0 , T ; W 1 , p loc (Ω)) is a weak solution to the doubly nonlinear equation ( | u | p − 2 u ) t − div( | Du | p − 2 Du ) = 0 in Ω T , if �� | Du | p − 2 Du · D ϕ − | u | p − 2 u ϕ t � � dx dt = 0 Ω T for every ϕ ∈ C ∞ 0 (Ω T ). Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
The Barenblatt solution Example The Barenblatt solution 1 n u ( x , t ) = t − � � | x | p � p − 1 � p ( p − 1) exp − p − 1 , p pt where x ∈ R n and t > 0, is a solution to the doubly nonlinear equation in the upper half space. Observe: The Barenblatt solution is strictly positive for every x ∈ R n and t > 0. This indicates that disturbances propagate with infinite speed. Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
More general doubly nonlinear parabolic systems We focus on the prototype equation, but it is possible to consider solutions u : Ω T → R N , N ≥ 1, to a system ( | u | p − 2 u ) t − div A ( x , t , u , Du ) = div( | F | p − 2 F ) in Ω T where F : Ω T → R N and A : Ω T × R N × R Nn → R Nn with A ( x , t , u , ξ ) · ξ ≥ α | ξ | p , � | A ( x , t , u , ξ ) | ≤ β | ξ | p − 1 , for almost every ( x , t ) ∈ Ω T and every u ∈ R N , ξ ∈ R Nn with 0 < α ≤ β < ∞ . Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
Structural properties The equation is nonlinear: The sum of two solutions is not a solution, in general. Solutions can be scaled. Constants cannot be added to solutions. Thus the boundary values cannot be perturbed in a standard way by adding an epsilon. In the natural geometry a scaling by r in the spatial variable corresponds to r p in the time direction. For p = 2, this works for the heat equation. Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
The Cole–Hopf transformation Consider a nonnegative solution of ( | u | p − 2 u ) t − div( | Du | p − 2 Du ) = 0 . The transformation v = log u leads to the diffusive Hamilton–Jacobi equation p − 1 div( | Dv | p − 2 Dv ) = −| Dv | p , 1 v t − with Dv = Du u . Note: In the elliptic case log u is a subsolution to the p -Laplace equation, but for the doubly nonlinear equation the equation changes. Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
A change of variables Consider a solution of ( | u | p − 2 u ) t − div( | Du | p − 2 Du ) = 0 . The transformation v = | u | p − 2 u leads to � p − 2 �� � | Dv | v t = div Dv . | v | Takeaway: The quotient | Du | | u | appears again. This indicates that it should play some role in the intrinsic geometry. Juha Kinnunen, Aalto University Higher integrability for doubly nonlinear parabolic equations
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