Analyzing a special case of the Hele-Shaw flow using - PowerPoint PPT Presentation

Analyzing a special case of the Hele-Shaw flow using integro-differential operators Russell Schwab (Michigan State University) Swedish Summer PDEs at KTH 26-28 August 2019

The Law of the Instrument “I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail.” – thanks, wikipedia

A theme to keep in mind There are a few instances where regularity is shown to occur for Hele-Shaw. Is it regular izing ? Where does it come from ?

Hele-Shaw Let us recall what the one phase Hele-Shaw problem looks like

Hele-Shaw A few references for existence, uniqueness, regularity: • Escher-Simonett 1997 • Kim 2003, 2006 • Jerison-Kim 2005 • Choi-Jerison-Kim 2007 • Chang Lara - Guillen 2016

Hele-Shaw, special case for the interface Then, U : R d +1 × R + → R is a non-negative function solving +  ∆ U = 0 in { U > 0 } ,  (HS) U = 1 on { y = 0 } , V = |∇ U | on ∂ { U > 0 } .  V denoting the normal velocity of the free boundary ∂ { U > 0 } .

Hele-Shaw, special case for the interface D f = { ( x , x d +1 ) ∈ R d +1 : 0 < x d +1 < f ( x ) } and Γ f = graph( f )

Hele-Shaw, special case for the interface These methods actually apply to a two-phase version of Hele-Shaw � 2 − � 2 ) V = G ( ∂ + n U + f , ∂ − n U − � ∂ + n U + � ∂ − n U − � � � � f ) (e.g. G = f f

Hele-Shaw, the goal So, the goal of studying the problem is to produce a function U and describe its properties. Thus, in our special case, this is equivalent to producing and describing the function, f .

Integro-Differential Equations ∂ t f − L ( f , x ) = g ( x , t ) in R d × (0 , T ] and f ( · , 0) = f 0 . Where L ( f , x ) has the form � L ( f , x ) = b ( x ) · ∇ f ( x ) + R d ( f ( x + h ) − f ( x ) − ✶ B 1 ( h ) ∇ f ( x ) · h ) µ ( x , dh ) , with b bounded, and µ ( x , · ) ≥ 0 is a measure, possibly singular at h = 0 . NOTATION: δ h f ( x ) := f ( x + h ) − f ( x ) − ✶ B 1 ( h ) ∇ f ( x ) · h

Integro-Differential Equations The (arguably) simplest and most canonical case is for some α ∈ (0 , 2) b = 0 and µ ( x , dh ) = C d ,α | h | − d − α dh , giving L ( f , x ) = − ( − ∆) α/ 2 f ( x )

Integro-Differential Equations, a powerful tool Theorem (Krylov-Safonov) Under (ASSUMPTIONS, listed below), there exists a universal γ , and C, so that any (appropriately defined) solution to ∂ t f − L ( f , x ) = g ( x ) in B 1 × ( − 1 , 0] , enjoys the estimate � � [ f ] C γ ( B 1 / 2 × ( − 1 / 2 , 0]) ≤ C � f � L ∞ ( R d × ( − 1 , 0]) + � g � L ∞ , where | f ( x , s ) − f ( y , t ) | [ f ] C γ ( B 1 / 2 × ( − 1 / 2 , 0]) = sup | x − y | + | s − t | 1 /α � γ � 0 < | x − y | < 1 / 2 , 0 < | t − s | < 1 / 2

Some Results for Kyrlov-Safonov all results assume a density: µ ( x , dh ) = k ( x , h ) dh symmetry: K ( x , − h ) = K ( x , h ) , LB: c 1 (2 − α ) | h | − d − α ≤ K ( x , h ) ≤ c 2 (2 − α ) | h | − d − α :UB

Some Results for Kyrlov-Safonov • Bass-Levin (2002): elliptic; b ≡ 0; symmetry, LB, UB (not robust) • Bass-Kassmann (2004): elliptic; b ≡ 0; variable α ; symmetry, LB, UB (not robust) • Silvestre (2006): elliptic; b ≡ 0; variable α ; slightly relaxed symmetry, LB, UB (not robust) • Caffarelli-Silvestre (2009): elliptic; b ≡ 0; symmetry, LB, UB (robust) • Chang Lara (2012): elliptic; nontrivial b ; non-symmetric, LB, UB (robust) • Change Lara - Davila (2016): parabolic; non-trivial b ; non-symmetric; LB, UB (robust) • Silvestre (2014): parabolic; non-trivial b ; non-symmetric; LB, UB (robust) • Schwab-Silvestre (2016): parabolic; non-trivial b ; non-symmetric, relaxed LB, UB only in integral sense (robust)

For Later Use – Result for Kyrlov-Safonov � � � ∂ t f − b ( x ) · ∇ + R d δ y f ( x ) K ( x , y ) dy = g ( x , t ) Theorem (Chang Lara - Davila 2016, also Chang Lara 2012 elliptic) Assume α ∈ [1 , 2) . Krylov-Safonov and C 1 ,γ for the class of equations where the pair ( b , K ) satisfies � � � � � r α − 1 sup � b + yK ( x , y ) dy � ≤ C � � � � r ∈ (0 , 1) B 1 \ B r and c 1 | y | − d − α ≤ K ( x , y ) ≤ c 2 | y | − d − α .

Krylov-Safonov, fully nonlinear Note, Krylov-Safonov holds for fully nonlinear equations, for f a viscosity solution of ∂ t f − F ( f , x ) = g ( x , t ) in B 1 × ( − 1 , 0] , where F is an operator that enjoys the structure, for some family of L ij as above, L ij ( f , x ) . F ( f , x ) = min max i j (There is much more to say, but not enough time)

C 1 ,γ , translation invariant If furthermore, F is translation invariant, i.e. F ( f ( · + z ) , x ) = F ( f , x + z ) (or, concretely, b ij and µ ij are independent of x ), Krylov-Safonov implies higher regularity Theorem ( C 1 ,γ , Assume F is translation invariant) There is a universal γ and C (depending upon the assumptions on F, above) so that if f is a viscosity solution of ∂ t f − F ( f , x ) = 0 in B 1 × ( − 1 , 0] , then ( ∗ )[ ∂ t f ( x , · )] C γ (( − 1 / 2 , 0]) + [ ∇ f ( · , t )] C γ ( B 1 / 2 ) ≤ C ( � u � L ∞ ( R d ) × ( − 1 , 0] + ( ∗∗ )) Sometimes ( ∗ ) is present and sometimes it is not, sometimes ( ∗∗ ) contains an extra term for the time behavior of u , and sometimes it doesn’t, depending upon the particular result. See: Chang Lara - Davila 2016 and Serra 2015 (also Kriventsov 2013)

Operators with the GCP Definition I : D ⊂ R X → R X is said to have the global comparison property (GCP) if f , g ∈ D and g touches f from above at x 0 ⇒ I ( f , x 0 ) ≤ I ( g , x 0 ) f ( x ) ≤ g ( x ) ∀ x ∈ X f ( x 0 ) = g ( x 0 )

Structure from GCP Theorem (Guillen-Schwab 2016 and 2019) (Generalizes to a complete, d-dimensional manifold) If I : C 2 ( R d ) → C 0 ( R d ) is Lipschitz, with the GCP, then ∀ u ∈ C 2 , x ∈ R d , I ( u , x ) = min max j { f ij ( x ) + L ij ( u , x ) } i where, for each pair of indices ij, we have • f ij ( x ) ∈ C 0 ( R d ) (uniformly) L ij ( u , x ) =Tr( A ij ( x ) D 2 u ) + B ij ( x ) · ∇ u + C ij ( x ) u � + R d ( u ( x + y ) − u ( x ) − y · ∇ u ( x ) ✶ B 1 ( y )) µ ij ( x , dy )

Structure from GCP Theorem (Guillen-Schwab 2016 and 2019) Furthermore if I : C 1 ,γ ( R d ) → C ( R d ) is Lipschitz and satisfies the GCP, then L ij ( u , x ) = C ij ( x ) u ( x ) + B ij ( x ) · ∇ u � + R d u ( x + y ) − u ( x ) − ∇ u ( x ) · y ✶ B 1 (0) µ ij ( x , dy ) and � min {| y | 1+ γ , 1 } µ ij ( x , dy ) < ∞ . sup sup ij x

What is the connection? Why are these three topics related? A heuristic answer is that the D-to-N on half space is the − ( − ∆) 1 / 2 operator. So you can think after flattening the domain, the D-to-N is like a − ( − ∆) 1 / 2 that depends in a nonlinear fashion on f .

Analysis of Hele-Shaw There is a more direct, but less obvious way to proceed. (Thanks to Hector!)

Level-set formulation First, let’s remind ourselves of the level-set interpretation of Hele-Shaw flow Let Ω( t ) ⊂ R N be a generic set which is said to have a boundary motion dictated by normal velocity( x ) = V ( x ) n ( x ) on ∂ Ω( t ) , where n ( x ) is the outward normal to Ω( t ) at x , and V ( x ) is a scalar. Assume Φ is some function so that Ω( t ) = { Φ( · , t ) > 0 } and ∂ Ω( t ) = { Φ( · , t ) = 0 } . so n ( x ) = −∇ Φ( x , t ) . |∇ Φ |

Level-set formulation Assume that γ : (0 , 1) → R N such that ∀ t , γ ( t ) ∈ ∂ Ω( t ). Hence 0 = ∂ t (Φ ◦ γ ) = ∂ t Φ + ∇ Φ · ˙ γ, and since we are assuming that (˙ γ ) n = V , 0 = ∂ t Φ + ( − n ( x ) |∇ Φ | ) · ˙ γ = ∂ t Φ − V |∇ Φ | , so ∂ t Φ = V |∇ Φ | .

Choices for Φ Now, let’s go back to Hele-Shaw. The first (most obvious) choice of Φ is Φ = U , which gives ∂ t U f = ( ∂ n U f ) |∇ U f | on ∂ { U f ( · , t ) > 0 } The drawbacks are that: • not obviously an equation for f , so no reduction of complexity, • the domain is a manifold and is not fixed in time!

Choices for Φ A different choice of Φ is Φ = x d +1 − f ( x , t ), so that � 1 + |∇ f ( x , t ) | 2 on R d × (0 , T ] . ∂ t f ( x , t ) = ( ∂ n U f ( x , f ( x , t ))) This is already an improvement, but still not perfect • CLOSER to an equation on only f • takes place on the nice, fixed domain, R d , so a reduction of variables!

Hele-Shaw as a parabolic integro-differential equation Again, why is this related to the integro-differential equations ?!?!? Focus on the following map (thanks, Hector!) f �→ ∂ n U f , I ( f , x ) := ∂ n U f ( x , f ( x )) .

Hele-Shaw as a parabolic integro-differential equation Lemma The operator I ( f , x ) = ∂ n U f ( x , f ( x ))) has the GCP.

Hele-Shaw as a parabolic integro-differential equation � 1 + |∇ f | 2 we need is the following. The other property of H ( f ) = I ( f ) Lemma (Chang Lara - Guillen - Schwab 2019) For each γ ∈ (0 , 1) , H is Lipschitz continuous as a map from C 1 ,γ ( R d ) b to C 0 b ( R d ) . Hence... using the structure of GCP... Theorem (Chang Lara - Guillen - Schwab 2019) There exists a family, a ij , c ij , b ij , µ i j, so that f is the unique viscosity solution of � a ij + c ij f ( x ) + b ij · ∇ f ( x ) + R d δ h f ( x ) µ ij ( dh ) . ∂ t f = min max i j