Blow-up by aggregation in chemotaxis Manuel del Pino University of - PowerPoint PPT Presentation

Blow-up by aggregation in chemotaxis Manuel del Pino University of Bath Singular problems associated to quasilinear equations, In honor of Marie Fran¸ coise Bidaut-V´ eron and Laurent V´ eron June 2nd, 2020

The Keller-Segel system in R 2 . in R 2 × (0 , ∞ ) ,  u t =∆ u − ∇ · ( u ∇ v )    v =( − ∆) − 1 u := 1 � 1  R 2 log | x − z | u ( z , t ) dz ( KS ) 2 π    in R 2 . u ( · , 0) = u 0 ≥ 0  is the classical diffusion model for chemotaxis , the motion of a population of bacteria driven by standard diffusion and a nonlocal drift given by the gradient of a chemoatractant, a chemical the bacteria produce. u ( x , t )= population density. v ( x , t )= the chemoatractant

Basic properties. For a regular solution u ( x , t ) defined up to a time T > 0, in R 2 × (0 , T ) � u t = ∇ · ( u ∇ (log u − v )) − ∆ v = u • Conservation of mass d � � R 2 u ( x , t ) dx = lim u ∇ (log u − v ) · ν d σ dt R →∞ ∂ B R � = lim ( ∇ u · ν ) − u ( ∇ v · ν ) d σ = 0 R →∞ ∂ B R � • The second moment identity . Let M = R 2 u ( x , t ) dx , then d 1 − M � � � R 2 | x | 2 u ( x , t ) dx = 4 M dt 8 π

d � � R 2 | x | 2 u ( x , t ) dx = R 2 | x | 2 (∆ u − ∇ · ( u ∇ v )) dx dt � � R 2 ∆( | x | 2 ) udx + = R 2 (2 x · ∇ v ) udx � =4 M + 2 R 2 u ( x · ∇ v ) dx . 2 π log 1 1 From v ( · , t ) = | · | ∗ u ( · , t ) we get � R 2 u ( x · ∇ v ) dx = 1 � � R 2 u ( x , t ) u ( y , t ) x · ( x − y ) − 2 | x − y | 2 dx dy π R 2 = 1 � � R 2 u ( x , t ) u ( y , t )( x − y ) · ( x − y ) dx dy | x − y | 2 2 π R 2 = M 2 2 π . Hence R 2 | x | 2 u ( x , t ) dx = 4 M − M 2 d � dt 2 π

Then, if the initial second moment is finite we have � � R 2 u ( x , 0) | x | 2 dx + 4 M (1 − M R 2 u ( x , t ) | x | 2 dx = 8 π ) t . As a consequence, • If M > 8 π the solution cannot remain smooth beyond some time. u ( x , t ) blows-up in finite time. • If M = 8 π The second moment of the solution is preserved in time. • If M < 8 π second moment grows linearly in time while mass is preserved (as in heat equation): the solution “diffuses”

 u t =∆ u − ∇ · ( u ∇ v )    v =( − ∆) − 1 u := 1 2 π log 1  | · | ∗ u    u ( · , 0) = u 0 ≥ 0 .  • If M ≤ 8 π the solution exists classically at all times t ∈ (0 , ∞ ). • If M < 8 π then u ( x , t ) goes to zero and spreads in self-similar way. (Blanchet-Dolbeault-Perthame (2006); J¨ ager-Luckhaus (1992).)

• If M > 8 π blow-up is expected to take place by aggregation which means that at a finite time u ( x , t ) concentrates and forms a set of Dirac masses with mass at least 8 π at a blow-up point. • Examples of blow-up with precise asymptotics, and mass slightly above 8 π were found by • Herrero-Vel´ azquez (1996), Vel´ azquez (2002, 2006) Raphael and Schweyer (2014). • Collot-Ghoul-Masmoudi-Nguyen (2019): New method, precise asymptotics and nonradial stability of the blow-up phenomenon.

u t = ∇ · ( u ∇ (log u − ( − ∆) − 1 u )) What special thing happens exactly at the critical mass 8 π ? � R 2 u (log u − ( − ∆) − 1 u ) dx E ( u ) := is a Lyapunov functional for (KS). Along a solution u ( x , t ), � R 2 u |∇ (log u − ( − ∆) − 1 u ) | 2 dx ≤ 0 . ∂ t E ( u ( · , t )) = − and this vanishes only at the steady states v = log u or − ∆ v = e v = u in R 2 the Liouville equation .

− ∆ v = e v = u in R 2 � All solutions with finite mass R 2 u < + ∞ are known: � x − ξ � 8 U λ,ξ ( x ) = λ − 2 U 0 , U 0 ( x ) = (1 + | x | 2 ) 2 . λ � R 2 U λ,ξ ( x ) dx = 8 π, E ( U λ,ξ ) = E ( U 0 ) for all λ, ξ , and U λ,ξ ⇀ 8 πδ ξ as λ → 0 + .

The functions U λ,ξ are the extremals for the log-HLS inequality R 2 u =8 π E ( u ) = E ( U 0 ) min � The functional E ( u ) loses the P.S. condition along this family, which makes possible the presence of bubbling phenomena along the flow. The problem is critical .

� The critical mass case R 2 u 0 = 8 π • Blanchet-Carlen-Carrillo (2012), Carlen-Figalli (2013): Asymptotic stability of the family of steady states under finite second moment perturbations. • Lopez Gomez-Nagai-Yamada oscillatory (2014) instabilities. • Blanchet-Carrillo-Masmoudi (2008) If in addition to critical mass we assume finite second moment � R 2 | x | 2 u 0 ( x ) dx < + ∞ then the solution u ( x , t ) aggregates in infinite time : for some λ ( t ) → 0 and some point q we have that (near q ) 1 � x − q � u ( x , t ) ≈ λ ( t ) 2 U 0 as t → + ∞ λ ( t ) no information about the rate.

• Chavanis-Sire (2006), Campos (2012) formal analysis to derive the rate λ ( t ). • Ghoul-Masmoudi (2019) Construction of a radial solution with this profile that confirms formal rate 1 λ ( t ) ∼ √ log t as t → + ∞ . Stability of the phenomenon inside the radial class is found. Full stability left as an open problem.

Theorem (D´ avila, del Pino, Dolbeault, Musso, Wei, Arxiv 2019) There exists a function u ∗ 0 ( x ) with � � R 2 u ∗ R 2 | x | 2 u ∗ 0 ( x ) dx = 8 π, 0 ( x ) dx < + ∞ such that for any initial condition in (KS) that is a small perturbation of u ∗ 0 and has mass 8 π , the solution has the form 1 � x − q � u ( x , t ) = λ ( t ) 2 U 0 + o (1) λ ( t ) 1 λ ( t ) = √ log t (1 + o (1)) , as t → + ∞ .

Let us explain the mechanism in Theorem 1. We look for a solution of S ( u ) := − u t + ∇ · ( u ∇ (log u − ( − ∆) − 1 u )) = 0 λ 2 U 0 ( y ), y = x 1 which is close to λ where 0 < λ ( t ) → 0 is a parameter function to be determined. Let U ( x , t ) = α y = x λ 2 U 0 ( y ) χ, λ. Here χ ( x , t ) = χ 0 ( | x | / √ t ), where χ 0 is smooth with χ 0 ( s ) = 1 � λ 2 � for s < 1 and = 0 for s > 2 and α ( t ) = 1 + O is such that t � R 2 U dx = 8 π . We look for a local correction of the form u = U + ϕ where ϕ ( x , t ) = 1 y = x λ 2 φ ( y , t ) , λ.

1 We compute for ϕ ( x , t ) = λ 2 φ ( y , t ), S ( U + ϕ ) = S ( U ) + L U [ ϕ ] − ϕ t + O ( � ϕ � 2 ) L U [ ϕ ] = ∆ ϕ − ∇ V · ∇ ϕ − ∇ U · ∇ ( − ∆) − 1 ϕ ≈ λ − 4 L 0 [ φ ] and for | x | ≪ √ t we have ( V 0 = log U 0 ) L 0 [ φ ] = ∆ y φ − ∇ V 0 · ∇ φ − ∇ U 0 · ∇ ( − ∆) − 1 φ We will have obtained an improvement of the approximation if we solve L 0 [ φ ] + λ 4 S ( U ) = 0 , φ ( y , t ) = O ( | y | − 4 − σ )

Let us consider the elliptic problem L 0 [ φ ] = E ( y ) = O ( | y | − 6 − σ ) in R 2 Which can be written as g = φ − ( − ∆) − 1 φ ∇ · ( U 0 ∇ g ) = E ( y ) , U 0 � Assume E radial E = E ( | y | ) and R 2 E = 0. We solve as � ∞ � ∞ d ρ E ( r ) rdr = O ( | y | − σ ) . g ( y ) = ρ U 0 ( ρ ) | y | ρ

Now we solve, setting ψ = ( − ∆) − 1 φ , in R 2 . ∆ ψ + U 0 ψ = − U 0 g It can be solved for ψ = O ( | y | − 2 − σ ) (Fredholm alternative) iff � R 2 gZ 0 = 0 , where Z 0 = ( y · ∇ U 0 + 2 U 0 ). Now, � � � R 2 | y | 2 E ( y ) dy = R 2 ∇ · ( U 0 ∇| y | 2 ) gdy = 2 R 2 Z 0 gdy

Hence we can solve as desired ( φ = O ( | y | − 4 − σ ) if � � R 2 | y | 2 E ( y ) dy = 0 = R 2 E ( y ) dy Now, the equation we need to solve is L 0 [ φ ] + λ 4 S ( U ) = 0 , φ ( y , t ) = O ( | y | − 4 − σ ) So we need � � R 2 S ( U ) | x | 2 dx = R 2 S ( U ) dx = 0 .

S ( U ) = − U t + ∇ · ( U ∇ (log U − ( − ∆) − 1 U ) = S 1 + S 2 � We clearly have R 2 S ( U ) = 0. � A direct computation (that uses R 2 U = 8 π ) gives � R 2 | x | 2 S 2 = 0 . Finally � � � R 2 S 1 | x | 2 = − R 2 U t | x | 2 = ∂ t ( R 2 U | x | 2 ) = 0 R 2 U | x | 2 = constant . � if and only if

We have � ∞ √ � R 2 U ( x , t ) | x | 2 dx = t ) U ( ρ/λ ) λ − 2 ρ 3 d ρ χ 0 ( s / 0 √ t √ � λ U ( r ) r 3 dr ∼ λ 2 log( ∼ t /λ ) 0 Thus the requirement is λ 2 log( √ t /λ ) = c 2 , and we get c � log(log t ) � λ ( t ) = √ log t + O log t

For the actual proof: We let λ ( t ) , α ( t ) be parameter functions with 1 α ( t ) = 1 + o ( t − 1 ) . λ ( t ) = √ log t (1 + o (1)) , � x The function U = αλ − 2 U 0 � χ is defined as before. We look for λ a solution of the form u ( x , t ) = U [ λ, α ] + ϕ ϕ ( x , t ) = η 1 λ 2 φ in � x � + φ out ( x , t ) λ, t � � | x | where η ( x , t ) = χ 0 . √ t The pair ( φ in , φ out ) is imposed to solve a coupled system, the inner-outer gluing system that leads to u ( x , t ) be a solution

The system involves the main part of the linear operator near the core and far away from it. L U [ ϕ ] = ∆ x ϕ − ∇ x V · ∇ ϕ − ∇ x U · ∇ ( − ∆) − 1 ϕ Near 0, L U [ ϕ ] ≈ λ − 4 L 0 [ φ ] for ϕ = λ − 2 φ ( y , t ) , y = x λ . Away: ∇ x U ∼ 4 λ 2 x −∇ x V ∼ 4 x | x | 2 , | x | 5 So, setting r = | x | , far away from the core the operator looks like L U [ ϕ ] ≈ ∆ x ϕ + 4 r ∂ r ϕ (for radial functions ϕ ( r ), L U [ ϕ ] ≈ ϕ ′′ + 5 r ϕ ′ , a 6 d -Laplacian).