A uniform null controllability result for the Keller …...A uniform null controllability result for the Keller-Segel system F.W. Chaves-Silva BCAM-Basque Center for Applied Mathematics

A uniform null controllability result for the Keller-Segelsystem

F.W. Chaves-Silva

BCAM-Basque Center for Applied MathematicsJoint work with S. Guerrero

MTM 2011-2096 Meeting

(BCAM) The Keller-Segel system June 12, Bilbao 1 / 23

Chemotaxis: description of the change of motion when a population formed ofindividuals (such as amoebae, bacteria, endothelial cells, etc.) reacts in response(taxis) to an external chemical stimulus spread in the environment where theyreside.

tumor angiogenesis

inflammatory response

wound healing

embryology

bacterial colony growth, pattern formation

...


The (one species) Keller-Segel system of chemotaxis:∣∣∣∣∣∣∣∣ut −∆u = −∇ · (u∇v) in Q := Ω× (0,T ),εvt −∆v = au − bv in Q,∂u∂ν = ∂v

∂ν = 0 on Σ := ∂Ω× (0,T ),u(x , 0) = u0; v(x , 0) = v0 in Ω,

(1)

where a, b > 0 and ε is a small positive parameter, which is intended to go to zero.

Here:

u = u(x , t) ≥ 0: population density;

v = v(x , t) ≥ 0: density of the chemical.

We assume:Ω ⊂ RN , N = 2, 3.


For all t > 0, we have the mass preservation:∫Ω

u(x , t)dx =

∫Ω

u0(x)dx . (2)


An important aspect of system (1) is the expected onset of chemotactic collapse.In other words, under suitable circumstances, the whole population concentrate ina single point (spora) in finite time.In mathematical terms, this means formation of a Dirac delta-type singularity infinite time, i.e.,

u(x , t) −→ Mδ(x0) as t −→ T , (3)

for some T <∞, where x0 is the point where the spora develops, and

M =

∫Ω

u(x , t)dx =

∫Ω

u0(x)dx .


Singularities (infinite mass at a point in space) occur in finite time for large data,while smooth solutions exist globally for small data.


The control problem

Given ε > 0, T > 0 and (u, v), solution of (1), find a control g ε such that theassociated solution (u, v) of∣∣∣∣∣∣∣∣

ut −∆u = −∇ · (u∇v) in Q,εvt −∆v = au − bv + g εχ in Q,∂u∂ν = ∂v

∂ν = 0 on Σu(x , 0) = u0; v(x , 0) = v0 in Ω,

(4)

satisfiesu(T ) = u(T ), v(T ) = v(T ) (5)

andg ε is bounded independently from ε. (6)

Here, χ : RN → R is a C∞ function such that supp χ ⊂⊂ ω, 0 ≤ χ ≤ 1 andχ ≡ 1 in ω′, for some ω′ ⊂⊂ ω ⊂ Ω.


The importance of (6) is due to the fact that in many applications1 system (1) isapproximated, for sufficiently small ε, by∣∣∣∣∣∣∣∣

ut −∆u = −∇ · (u∇v) in Q,−∆v = au − bv in Q,∂u∂ν = ∂v

∂ν = 0 on Σ,u(x , 0) = u0; v(x , 0) = v0 in Ω.

(7)

1See [1] and [7].(BCAM) The Keller-Segel system June 12, Bilbao 8 / 23

Main Result

Theorem (Chaves-Silva & Guerreo, 2014)

Let 0 < ε ≤ 1 and let (M1,M2) ∈ R2+ be such that aM1 − bM2 = 0. Then there

exists δ > 0 such that, for any (u0, v0) ∈ H1(Ω)× H2(Ω) with u0, v0 ≥ 0,satisfying

∫Ω

u0dx = M1, ∂v0

∂ν = 0 on ∂Ω and||(u0 −M1, v0 −M2)||H1(Ω)×H2(Ω) ≤ δ, there exists g ε ∈ L2(0,T ; H1(Ω)), with‖g ε‖L2(0,T ;H1(Ω)) bounded independently from ε, such that the associated solution

(u, v) of (1) satisfies:

(u(T ), v(T )) = (M1,M2) in Ω.


How do we do?

Linearization + (Uniform) Inverse mapping theorem


Linearization

We linearize system (4) around (M1,M2):∣∣∣∣∣∣∣∣ut −∆u = −M1∆v + h1 in Q,εvt −∆v = au − bv + g εχ+ h2 in Q,∂u∂ν = ∂v

∂ν = 0 on Σ,u(x , 0) = u0; v(x , 0) = v0 in Ω,

(8)

where h1 and h2 are given exterior forces belonging to appropriate Banach spacesand having exponential decay at t = T .

We prove the following result for (8):


Theorem (2)

Let 0 < ε ≤ 1 and (M1,M2) ∈ R2 be such that aM1 − bM2 = 0. Assume that:

(u0, v0) ∈ H1(Ω)× H2(Ω),

∫Ω

u0dx = 0,∂v0

∂ν= 0 on ∂Ω (9)

and

h1, h2 have “good” exponential decay at t = T . (10)

Then there exists a control g ε ∈ L2(0,T ; H1(Ω)), bounded independently from ε,such that, if (u, v) is the associated solution of (8), then

u(T ) = v(T ) = 0.

Theorem (2) is consequence of a Carleman inequality for the adjoint system of (8).


The adjoint system of (8) reads:∣∣∣∣∣∣∣∣∣∣−ϕt −∆ϕ = aξ + f1 in Q,−εξt −∆ξ = −bξ −M1∆ϕ+ f2 in Q,∂ϕ∂ν = ∂ξ

∂ν = 0 on Σ,ϕ(x ,T ) = ϕT ; ξ(x ,T ) = ξT in Ω,∫

ΩϕT (x)dx = 0.

(11)

The (uniform) Carleman inequality we prove for (11) is the following.

Theorem

Given 0 < ε ≤ 1, there exists C = C (Ω, ω) such that, for any (ϕT , ξT ) ∈ L2(Ω)2

and any f1, f2 ∈ L2(Q), the solution (ϕ, ξ) of system (11) satisfies:∫∫Q

e−C/tm

|ϕ−(ϕ)

Ω(t)|2dxdt +

∫∫Q

e−C/tm

|ξ|2dxdt

≤ C

( ∫∫ω×(0,T )

e−C/tm

|ξ|2dxdt +

∫∫Q

e−C/tm

(|f1|2 + |f2|2)dxdt

),

for some m > 1, where(ϕ)

Ω(t) = 1

|Ω|∫

Ωϕ(x , t)dx .




ΩϕT (x)dx = 0.

(11)


Theorem



e−C/tm

|ϕ−(ϕ)

Ω(t)|2dxdt +

∫∫Q

e−C/tm

|ξ|2dxdt

≤ C

( ∫∫ω×(0,T )

e−C/tm

|ξ|2dxdt +

∫∫Q

e−C/tm

(|f1|2 + |f2|2)dxdt

),


Ω(t) = 1

|Ω|∫

Ωϕ(x , t)dx .




ΩϕT (x)dx = 0.

(11)


Theorem



e−C/tm

|ϕ−(ϕ)

Ω(t)|2dxdt +

∫∫Q

e−C/tm

|ξ|2dxdt

≤ C

( ∫∫ω×(0,T )

e−C/tm

|ξ|2dxdt +

∫∫Q

e−C/tm

(|f1|2 + |f2|2)dxdt

),


Ω(t) = 1

|Ω|∫

Ωϕ(x , t)dx .


The proof of this Carleman inequality is divided into 3 steps.Step 1. Carleman Inequality for ∆ϕ.We write e−C/t

m

ϕ = η+ψ, where η solves a heat equation with f1 as a right-handside term and ψ solves a heat equation with a right-hand side in H1(0,T ; H2(Ω).We consider the equation satisfied by ψ and apply the usual Carleman inequalityfor the heat equation with Neumann boundary for ψ and combine it with theusual energy estimates for the equation satisfied by η. This gives a global estimateof ∆ϕ in terms of a local integral of ∆ψ and global integrals of ∆ξ and f1.

Step 2. Carleman inequality for ξ.In the second step we obtain a Carleman inequality for ξ, with a precisedependence of the degenerating parameter multiplying the time derivative andcombine it with the Carleman inequality obtained in step 1. This gives a globalestimate of ξ and ∆ϕ in terms of local integrals of ξ and ∆ψ and global integralsof f1 and f2.



m





m




Step 3. Estimate of the local integral of ∆ψ.In the last step we estimate a local integral of ∆ψ in terms of a local integral of ξand global integrals of f1 and f2. Combining steps 2 and 3 we finish the proof ofthe Carleman inequality.


Inverse mapping theorem

We change our controllability problem for the Keller-Segel system into a nullcontrollability problem by writing the solution (u, v) of the Keller-Segel system asu = M1 + z and v = M2 + w , where (z ,w) solves∣∣∣∣∣∣

L(z ,w) = (−∇ · (z∇w), gχ) in Q,∂z∂ν = ∂w

∂ν = 0 on Σ,z(x , 0) = u0 −M1; w(x , 0) = v0 −M2 in Ω

(12)

withL(u, v) = (ut −∆u + M1∆v , εvt −∆v + bv − au). (13)

Note that (u(T ), v(T )) = (M1,M2) if and only if (z(T ),w(T )) = 0.

We apply Liusternik’s (Graves) inverse mapping theorem:




L(z ,w) = (−∇ · (z∇w), gχ) in Q,∂z∂ν = ∂w


(12)







L(z ,w) = (−∇ · (z∇w), gχ) in Q,∂z∂ν = ∂w


(12)







L(z ,w) = (−∇ · (z∇w), gχ) in Q,∂z∂ν = ∂w


(12)





Theorem

Let E and G be two Banach spaces and let A : E → G be a continuous functionfrom E to G defined in Bη(0) for some η > 0 with A(0) = 0. Let Λ be acontinuous and linear operator from E onto Y and suppose there exists C0 > 0such that

||e||E ≤ C0||Λ(e)||G (14)

and that there exists δ < C−10 such that

||A(e1)−A(e2)− Λ(e1 − e2)|| ≤ δ||e1 − e2|| (15)

whenever e1, e2 ∈ Bη(0). Then the equation A(e) = h has a solution e ∈ Bη(0)whenever ||h||G ≤ cη, where c = C−1

0 − δ.


We define:G = X × Y ,

X = (h1, h2); eC/tmh1 ∈ L2(Q), eC/tmh2 ∈ L2(0,T ; H1(Ω))

and

∫Ω

h1(x , t)dx = 0 a. e. t ∈ (0,T ), (16)

Y = (z0,w0) ∈ H1(Ω)× H2(Ω);

∫Ω

z0dx = 0 and∂w0

∂ν= 0 on ∂Ω, (17)

E =

(u, v , g) : eC/tmgχ, eC/tm

(L(u, v)

)1∈ L2(Q),

eC/tm((

L(u, v))

2−gχ

)∈ L2(0,T ; H1(Ω)),∫

Ω

(L(u, v)

)1dx = 0 and

∂u

∂ν=∂v

∂ν= 0 on Σ

and, for each 0 < ε ≤ 1, the operator

A(z ,w , g) = (L(u, v) + ((∇ · (z∇w),−gχ)), z(., 0),w(., 0)) ∀(z ,w , g) ∈ E .


We have

A′(0, 0, 0) = (L(u, v) + (0,−gχ)), z(., 0),w(., 0)) ∀(z ,w , g) ∈ E .

It is not difficult to show that the operator A fits the regularity required to applyLiusternik’s theorem and obtain the uniform null controllability of the Keller-Segelsystem around (M1,M2).


Comments

We are able to avoid the blow up for regular data.

It would be interesting to analyze the controllability of the Keller-Segelsystem (1) around non constant trajectories, even for a fixed ε. Thelinearization around (u, v) reads:∣∣∣∣∣∣∣∣

ut −∆u = ∇ · (u∇v) +∇ · (u∇v) + h1 in Q,εvt −∆v = au − bv + g εχ+ h2 in Q,∂u∂ν = ∂v

∂ν = 0 on Σ,u(x , 0) = u0; v(x , 0) = v0 in Ω.

(18)


Other related systems

Keller-Segel + Stokes:∣∣∣∣∣∣∣∣ut + y · ∇u −∆u = ∇ · (u∇v) in Q,vt + y · ∇v −∆v = au − bv in Q,yt −∆y +∇Π = 0 in Q∇ · y = 0 in Q.

(19)

Two species Keller-Segel:∣∣∣∣∣∣u1t −∆u1 = −∇ · (u1∇v) in Q,

u2t −∆u2 = −∇ · (u2∇v) in Q,

vt −∆v = a1u1 + a2u2 − bv in Q.(20)


References

P. Biler, L. Brandolese, On the parabolic-elliptic limit of the doubly ParabolicKeller-Segel system modeling chemotaxis, Studia Mathematica, 193(3)(2009), 241–261.

F. W. Chaves-Silva, S. Guerrero, J.-P. Puel, Controllability of fast diffusioncoupled parabolic systems, to appear MCRF.

E. Feireisl, P. Laurencot, H. Petzeltova, On convergence to equilibria for theKeller-Segel chemotaxis model, J. Diff. Equations, 236 (2007), 551–569.

D. Horstmann, From 1970 until present: the Keller-Segel model inchemotaxis and its consequences, I. Jahresber. DMV, 105 (2003), 103–165.

T. Hillen, D. Painter, A user’s guide to PDE models of chemotaxis, J. Math.Biol., 58 (2009), 183–217.

E. F. Keller, L. A. Segel, Initiation of slime mold aggregation viewed as aninstability, J. Theor. Biol., 26 (1970), 399–415.

P.G. Lemaire-Rieusset, Small data in a optimal Banach space for theparabolic-parabolic and parabolic-elliptic Keller-Segel equations in the wholespace, preprint.


Thank you!!


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A uniform null controllability result for the Keller …...A uniform null controllability result for the Keller-Segel system F.W. Chaves-Silva BCAM-Basque Center for Applied Mathematics