Some new results of global existence for - - PowerPoint PPT Presentation

Some new results of global existence for reaction-diffusion-advection systems

Michel Pierre

Ecole Normale Sup´ erieure de Rennes and Institut de Recherche Math´ ematique de Rennes, France

Workshop ”New Trends in Modeling, Control and Inverse Problems”

Toulouse, June 16th-19th, 2014.

Introduction: a family of systems

◮

∂tui + Aiui = fi(u1, ..., um) on (0, ∞) × Ω, ui(0, ·) = u0

i ≥ 0,

Introduction: a family of systems

◮

∂tui + Aiui = fi(u1, ..., um) on (0, ∞) × Ω, ui(0, ·) = u0

i ≥ 0, ◮ Ai are various ”diffusion-advection” operators, possibly

Ai = Ai(t)

Introduction: a family of systems

◮

∂tui + Aiui = fi(u1, ..., um) on (0, ∞) × Ω, ui(0, ·) = u0

i ≥ 0, ◮ Ai are various ”diffusion-advection” operators, possibly

Ai = Ai(t)

◮ fi : [0, ∞)m → I

R are regular nonlinearities such that :

• (P): the positivity of the solutions is preserved for all time:

f = (f1, ..., fm) is quasi-positive

• (M): some mass dissipativity conditions holds like
• i fi(u) ≤ 0

Introduction: a family of systems

◮

∂tui + Aiui = fi(u1, ..., um) on (0, ∞) × Ω, ui(0, ·) = u0

i ≥ 0, ◮ Ai are various ”diffusion-advection” operators, possibly

Ai = Ai(t)

◮ fi : [0, ∞)m → I

R are regular nonlinearities such that :

• (P): the positivity of the solutions is preserved for all time:

f = (f1, ..., fm) is quasi-positive

• (M): some mass dissipativity conditions holds like
• i fi(u) ≤ 0

◮ or more general mass control property

• i fi(u) ≤ C[1 +

i ui].

A simple choice for the Ai

◮

   ∂tui−di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

where di ∈ (0, ∞). Local existence of nonnegative solutions

• n some maximal interval (0, T ∗) always holds for

u0

i ∈ L∞(Ω)+.

A simple choice for the Ai

◮

   ∂tui−di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

where di ∈ (0, ∞). Local existence of nonnegative solutions

• n some maximal interval (0, T ∗) always holds for

u0

i ∈ L∞(Ω)+. ◮ If the di = d are all equal and i fi(u) ≤ 0, then

∂t

• i

ui

• − d∆
• i

ui

• ≤ 0,

so that, ∀t ∈ (0, T ∗)

A simple choice for the Ai

◮

   ∂tui−di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

where di ∈ (0, ∞). Local existence of nonnegative solutions

• n some maximal interval (0, T ∗) always holds for

u0

i ∈ L∞(Ω)+. ◮ If the di = d are all equal and i fi(u) ≤ 0, then

∂t

• i

ui

• − d∆
• i

ui

• ≤ 0,

so that, ∀t ∈ (0, T ∗)

◮

• i

ui(t)L∞(Ω) ≤

• i

u0

i L∞(Ω)

which implies T ∗ = +∞ and global existence on [0, ∞).

The known results

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ In all cases, we keep L1(Ω)-estimates uniform in time, namely

∂t

• Ω
• i

ui − 0 =

• Ω
• i

fi(u) ≤ 0, ⇒

• i

ui(t)L1(Ω) ≤

• i

u0

i L1(Ω)

⇒ ∀ t ∈ [0, T ∗), max

i

ui(t)L1(Ω) ≤

• i

u0

i L1(Ω).

The known results

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ In all cases, we keep L1(Ω)-estimates uniform in time, namely

∂t

• Ω
• i

ui − 0 =

• Ω
• i

fi(u) ≤ 0, ⇒

• i

ui(t)L1(Ω) ≤

• i

u0

i L1(Ω)

⇒ ∀ t ∈ [0, T ∗), max

i

ui(t)L1(Ω) ≤

• i

u0

i L1(Ω). ◮ Negative result: if the di are not equal, then L∞(Ω)-blow up

may occur in finite time (in any dimension)+(for any

superquadratic growth and high dimension).

The known results

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ In all cases, we keep L1(Ω)-estimates uniform in time, namely

∂t

• Ω
• i

ui − 0 =

• Ω
• i

fi(u) ≤ 0, ⇒

• i

ui(t)L1(Ω) ≤

• i

u0

i L1(Ω)

⇒ ∀ t ∈ [0, T ∗), max

i

ui(t)L1(Ω) ≤

• i

u0

i L1(Ω). ◮ Negative result: if the di are not equal, then L∞(Ω)-blow up

may occur in finite time (in any dimension)+(for any

superquadratic growth and high dimension).

◮ Positive results of global existence: two main families.

The known results: 1) strong solutions

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ Theorem. Assume f = (f1, ..., fm) satisfies (P),(M) and has

a triangular structure which means: ∀u ∈ [0, ∞)m, Qf (u) ≤ 0 [or Qf (u) ≤ b(1+

• i

ui), b ∈ I Rm], for some (lower) triangular matrix Q , with nonnegative entries and invertible, and if the growth of the fi is at most polynomial, then the system (S) has a global classical solution.

The known results: 1) strong solutions

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ Theorem. Assume f = (f1, ..., fm) satisfies (P),(M) and has

a triangular structure which means: ∀u ∈ [0, ∞)m, Qf (u) ≤ 0 [or Qf (u) ≤ b(1+

• i

ui), b ∈ I Rm], for some (lower) triangular matrix Q , with nonnegative entries and invertible, and if the growth of the fi is at most polynomial, then the system (S) has a global classical solution.

◮ A typical example with m = 2 where α, β ≥ 1:

• f1(u) = −uα

1 uβ 2 ,

f1(u) ≤ 0 f2(u) = uα

1 uβ 2

f1(u) + f2(u) = 0. Q = 1 1 1

The known results: 1) strong solutions

       ∂tu1 − d1∆u1 = −uα

1 uβ 2 ,

∂tu2 − d2∆u2 = uα

1 uβ 2 ,

∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ We obviously have on the maximum interval (0, T ∗)

u1L∞(QT∗) ≤ u0

1L∞(Ω).

The known results: 1) strong solutions

       ∂tu1 − d1∆u1 = −uα

1 uβ 2 ,

∂tu2 − d2∆u2 = uα

1 uβ 2 ,

∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ We obviously have on the maximum interval (0, T ∗)

u1L∞(QT∗) ≤ u0

1L∞(Ω). ◮ Next, a main estimate is that

∂tu2 − d2∆u2 = − [∂tu1 − d1∆u1] implies the existence of C = C(p, T, Ω) such that: ∀p ∈ (1, ∞), u2Lp(QT ) ≤ C u1Lp(QT ) [QT = (0, T) × Ω]. Follows from the Lp-regularity theory for the heat operator.

The known results: 1) strong solutions

       ∂tu1 − d1∆u1 = −uα

1 uβ 2 ,

∂tu2 − d2∆u2 = uα

1 uβ 2 ,

∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ We obviously have on the maximum interval (0, T ∗)

u1L∞(QT∗) ≤ u0

1L∞(Ω). ◮ Next, a main estimate is that

∂tu2 − d2∆u2 = − [∂tu1 − d1∆u1] implies the existence of C = C(p, T, Ω) such that: ∀p ∈ (1, ∞), u2Lp(QT ) ≤ C u1Lp(QT ) [QT = (0, T) × Ω]. Follows from the Lp-regularity theory for the heat operator.

◮ This implies that u2 is bounded in Lp(QT ∗) for all

p < ∞...and also in L∞(QT ∗) thanks to the polynomial growth of uα

1 uβ 2 . Whence T ∗ = +∞.

The known results: 1)The Lp-estimate by duality

◮

∂tu2 − d2∆u2 ≤ − [∂tu1 − d1∆u1] , u2 ≥ 0, implies the existence of C = C(p, T, Ω) such that: ∀p ∈ (1, ∞), u2Lp(QT ) ≤ C u1Lp(QT ).

The known results: 1)The Lp-estimate by duality

◮

∂tu2 − d2∆u2 ≤ − [∂tu1 − d1∆u1] , u2 ≥ 0, implies the existence of C = C(p, T, Ω) such that: ∀p ∈ (1, ∞), u2Lp(QT ) ≤ C u1Lp(QT ).

◮ Solve the dual problem

− (∂tψ + d2∆ψ) = Θ ∈ C ∞

0 (QT), Θ ≥ 0,

ψ(T) = 0, ∂νψ = 0 on ΣT.

The known results: 1)The Lp-estimate by duality

◮

∂tu2 − d2∆u2 ≤ − [∂tu1 − d1∆u1] , u2 ≥ 0, implies the existence of C = C(p, T, Ω) such that: ∀p ∈ (1, ∞), u2Lp(QT ) ≤ C u1Lp(QT ).

◮ Solve the dual problem

− (∂tψ + d2∆ψ) = Θ ∈ C ∞

0 (QT), Θ ≥ 0,

ψ(T) = 0, ∂νψ = 0 on ΣT.

◮

• QT

u2Θ ≤

• Ω

(u0

1 + u0 2)ψ(0) + (d1 − d2)

• QT

u1∆ψ.

The known results: 1)The Lp-estimate by duality

◮

∂tu2 − d2∆u2 ≤ − [∂tu1 − d1∆u1] , u2 ≥ 0, implies the existence of C = C(p, T, Ω) such that: ∀p ∈ (1, ∞), u2Lp(QT ) ≤ C u1Lp(QT ).

◮ Solve the dual problem

− (∂tψ + d2∆ψ) = Θ ∈ C ∞

0 (QT), Θ ≥ 0,

ψ(T) = 0, ∂νψ = 0 on ΣT.

◮

• QT

u2Θ ≤

• Ω

(u0

1 + u0 2)ψ(0) + (d1 − d2)

• QT

u1∆ψ.

◮ By the Lp′-regularity theory

∆ψLp′(QT ) + ψ(0)Lp′(Ω) ≤ CΘLp′(QT ).

The known results: 1)The Lp-estimate by duality

◮

∂tu2 − d2∆u2 ≤ − [∂tu1 − d1∆u1] , u2 ≥ 0, implies the existence of C = C(p, T, Ω) such that: ∀p ∈ (1, ∞), u2Lp(QT ) ≤ C u1Lp(QT ).

◮ Solve the dual problem

− (∂tψ + d2∆ψ) = Θ ∈ C ∞

0 (QT), Θ ≥ 0,

ψ(T) = 0, ∂νψ = 0 on ΣT.

◮

• QT

u2Θ ≤

• Ω

(u0

1 + u0 2)ψ(0) + (d1 − d2)

• QT

u1∆ψ.

◮ By the Lp′-regularity theory

∆ψLp′(QT ) + ψ(0)Lp′(Ω) ≤ CΘLp′(QT ).

◮ ⇒

• QT u2Θ
• ≤ CΘLp′(QT ) ⇒ Lp(QT)-estimate on u2 by

duality.

The known results: 2) weak solutions

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ Theorem. Assume f = (f1, ..., fm) satisfies (P),(M) and

assume there is an L1-a priori estimate on the nonlinearities for the solutions of (S): sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗) Then, there exists a global weak solution to (S) for all u0

i ∈ L1(Ω)+.

The known results: 2) weak solutions

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

Main ingredients of the proof:

◮ Truncating the fi → f n

i

→ global approximate solutions un

i

The known results: 2) weak solutions

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

Main ingredients of the proof:

◮ Truncating the fi → f n

i

→ global approximate solutions un

i

◮ Compactness of the mapping

(g, w0) ∈ L1(QT) × L1(Ω) → w ∈ L1(QT) where ∂tw − d∆w = g on QT, w(0, ·) = w0, ∂νw = 0 on ∂Ω. so that un

i → ui in L1(QT) and a.e.

The known results: 2) weak solutions

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

Main ingredients of the proof:

◮ Truncating the fi → f n

i

→ global approximate solutions un

i

◮ Compactness of the mapping

(g, w0) ∈ L1(QT) × L1(Ω) → w ∈ L1(QT) where ∂tw − d∆w = g on QT, w(0, ·) = w0, ∂νw = 0 on ∂Ω. so that un

i → ui in L1(QT) and a.e.

◮ We first prove that the limit ui is a supersolution.

The known results: 2) weak solutions

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

Main ingredients of the proof:

◮ Truncating the fi → f n

i

→ global approximate solutions un

i

◮ Compactness of the mapping

(g, w0) ∈ L1(QT) × L1(Ω) → w ∈ L1(QT) where ∂tw − d∆w = g on QT, w(0, ·) = w0, ∂νw = 0 on ∂Ω. so that un

i → ui in L1(QT) and a.e.

◮ We first prove that the limit ui is a supersolution. ◮ For this, we use the equation satisfied by Tk

• un

i + η j=i un j

• where Tk(r) = min{r, k}, η > 0.

The known results: 2) weak solutions

(S)    ∂tun

i − di∆un i = f n i (un 1, ..., un m) on (0, ∞) × Ω,

∂νun

i = 0 on (0, ∞) × ∂Ω,

un

i (0, ·) = u0 i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

◮ If m = 1: ∂tTk(un 1) − d1∆Tk(un 1)≥T ′ k(un 1)f n 1 (un 1).

The known results: 2) weak solutions

(S)    ∂tun

i − di∆un i = f n i (un 1, ..., un m) on (0, ∞) × Ω,

∂νun

i = 0 on (0, ∞) × ∂Ω,

un

i (0, ·) = u0 i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

◮ If m = 1: ∂tTk(un 1) − d1∆Tk(un 1)≥T ′ k(un 1)f n 1 (un 1). ◮ ⇒ ∂tTk(u1) − d1∆Tk(u1) ≥ T ′ k(u1)f1(u1).

Then k → ∞ ⇒ u1 is a supersolution

The known results: 2) weak solutions

(S)    ∂tun

i − di∆un i = f n i (un 1, ..., un m) on (0, ∞) × Ω,

∂νun

i = 0 on (0, ∞) × ∂Ω,

un

i (0, ·) = u0 i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

◮ If m = 1: ∂tTk(un 1) − d1∆Tk(un 1)≥T ′ k(un 1)f n 1 (un 1). ◮ ⇒ ∂tTk(u1) − d1∆Tk(u1) ≥ T ′ k(u1)f1(u1).

Then k → ∞ ⇒ u1 is a supersolution

◮ Let wn i := Tk

• un

i + η j=i un j

• ,

∂twn

i − di∆wn i ≥ T ′ k(wn i )fi(un 1, ..., un m) + Rn i (η, k). ◮ The limit ui is a supersolution by letting successively:

n → ∞, η → 0, k → +∞.

The known results: 2) weak solutions

(S)    ∂tun

i − di∆un i = f n i (un 1, ..., un m) on (0, ∞) × Ω,

∂νun

i = 0 on (0, ∞) × ∂Ω,

un

i (0, ·) = u0 i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

◮ If m = 1: ∂tTk(un 1) − d1∆Tk(un 1)≥T ′ k(un 1)f n 1 (un 1). ◮ ⇒ ∂tTk(u1) − d1∆Tk(u1) ≥ T ′ k(u1)f1(u1).

Then k → ∞ ⇒ u1 is a supersolution

◮ Let wn i := Tk

• un

i + η j=i un j

• ,

∂twn

i − di∆wn i ≥ T ′ k(wn i )fi(un 1, ..., un m) + Rn i (η, k). ◮ The limit ui is a supersolution by letting successively:

n → ∞, η → 0, k → +∞.

◮ Main estimate for η → 0 :

• [un

i ≤k] |∇un

i |2 ≤ C k

The known results: 2) weak solutions

(S)    ∂tun

i − di∆un i = f n i (un 1, ..., un m) on (0, ∞) × Ω,

∂νun

i = 0 on (0, ∞) × ∂Ω,

un

i (0, ·) = u0 i ≥ 0,

sup

i

fi(u)L1(QT ) ≤ C(T) for all T > 0. (∗)

◮ If m = 1: ∂tTk(un 1) − d1∆Tk(un 1)≥T ′ k(un 1)f n 1 (un 1). ◮ ⇒ ∂tTk(u1) − d1∆Tk(u1) ≥ T ′ k(u1)f1(u1).

Then k → ∞ ⇒ u1 is a supersolution

◮ Let wn i := Tk

• un

i + η j=i un j

• ,

∂twn

i − di∆wn i ≥ T ′ k(wn i )fi(un 1, ..., un m) + Rn i (η, k). ◮ The limit ui is a supersolution by letting successively:

n → ∞, η → 0, k → +∞.

◮ Main estimate for η → 0 :

• [un

i ≤k] |∇un

i |2 ≤ C k ◮ Condition (M) easily implies that the ui are also subsolutions.

The known results: 2) weak solutions

◮ A typical example (where λ < 1):

(S)

• ∂tu1 − d1∆u1 = −uα

1 uβ 2 + λuγ 1uδ 2 on (0, ∞) × Ω,

∂tu2 − d2∆u2 = uα

1 uβ 2 − uγ 1uδ 2 on (0, ∞) × Ω,

Here : f1 + f2 ≤ 0 and f1 + λf2 ≤ 0. This implies the L1(QT)-a priori estimate on each uα

1 uβ 2 , uγ 1uδ 2

The known results: 2) weak solutions

◮ A typical example (where λ < 1):

(S)

• ∂tu1 − d1∆u1 = −uα

1 uβ 2 + λuγ 1uδ 2 on (0, ∞) × Ω,

∂tu2 − d2∆u2 = uα

1 uβ 2 − uγ 1uδ 2 on (0, ∞) × Ω,

Here : f1 + f2 ≤ 0 and f1 + λf2 ≤ 0. This implies the L1(QT)-a priori estimate on each uα

1 uβ 2 , uγ 1uδ 2 ◮ For general systems, the L1(QT)-a priori estimate on the fi(u)

holds if, for some invertible matrix Q with nonnegative entries ∀u ∈ [0, ∞)m, Q f (u) ≤ b[1 +

• i

ui].

The known results: 2) weak solutions

◮ A typical example (where λ < 1):

(S)

• ∂tu1 − d1∆u1 = −uα

1 uβ 2 + λuγ 1uδ 2 on (0, ∞) × Ω,

∂tu2 − d2∆u2 = uα

1 uβ 2 − uγ 1uδ 2 on (0, ∞) × Ω,

Here : f1 + f2 ≤ 0 and f1 + λf2 ≤ 0. This implies the L1(QT)-a priori estimate on each uα

1 uβ 2 , uγ 1uδ 2 ◮ For general systems, the L1(QT)-a priori estimate on the fi(u)

holds if, for some invertible matrix Q with nonnegative entries ∀u ∈ [0, ∞)m, Q f (u) ≤ b[1 +

• i

ui].

◮ The L1(QT)-a priori estimate on the fi(u) holds if the

nonlinearities fi are at most quadratic

The known results: 3) an L2(QT)-a priori estimate

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ Theorem. If f is quasi-positive =(P) and satisfy

• i fi(u) ≤ 0 =(M), then

uiL2(QT ) ≤ C = C

• T, di, u0

i L2(Ω)

• for the solutions of (S).

The known results: 3) an L2(QT)-a priori estimate

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ Theorem. If f is quasi-positive =(P) and satisfy

• i fi(u) ≤ 0 =(M), then

uiL2(QT ) ≤ C = C

• T, di, u0

i L2(Ω)

• for the solutions of (S).

◮ Whence fi(u) ∈ L1(QT) if the growth of fi is at most

quadratic ⇒ global existence of weak solutions.

The known results: 3) an L2(QT)-a priori estimate

(S)    ∂tui − di∆ui = fi(u1, ..., um) on (0, ∞) × Ω, ∂νui = 0 on (0, ∞) × ∂Ω, ui(0, ·) = u0

i ≥ 0,

◮ Theorem. If f is quasi-positive =(P) and satisfy

• i fi(u) ≤ 0 =(M), then

uiL2(QT ) ≤ C = C

• T, di, u0

i L2(Ω)

• for the solutions of (S).

◮ Whence fi(u) ∈ L1(QT) if the growth of fi is at most

quadratic ⇒ global existence of weak solutions.

◮ This L2-estimate comes from the remark

∂t

• i

ui

• − ∆
• A
• i

ui

• ≤ 0, A =
• i diui
• i ui

0 < min

i

di ≤ A ≤ max

i

di < +∞

An even sharper estimate

(by J.A. Canizo, L. Desvillettes, K. Fellner):

There exists ǫ(N) > 0 such that

• i

uiL2+ǫ(QT) ≤ C

• i

ui(0)L2+ǫ(Ω).

Uses the Riesz-Thorin interpolation theorem.

The known results: 4) two typical examples

◮ ui = ui(t, x) = concentration of Ui

U1 + U2 k+ ⇋ k− U3 f1(u) = f2(u) = −f3(u) = −k+u1u2 + k−u3 Triangular structure : f1(u) = f2(u) ≤ k−u3, f1(u)+f3(u) = 0 Global existence of classical solutions.

The known results: 4) two typical examples

◮ ui = ui(t, x) = concentration of Ui

U1 + U2 k+ ⇋ k− U3 f1(u) = f2(u) = −f3(u) = −k+u1u2 + k−u3 Triangular structure : f1(u) = f2(u) ≤ k−u3, f1(u)+f3(u) = 0 Global existence of classical solutions.

◮

U1 + U2 k+ ⇋ k− U3 + U4 f1(u) = f2(u) = −f3(u) = −f4(u) = −k+u1u2 + k−u3u4. Global existence of weak solutions. Classical if N = 1, 2 [Pr¨ uss, Goudon, Vasseur]. Open for classical when N ≥ 3 (bound on the size of points with possible blow up [Goudon-Vasseur]).

A first extension for global bounded solutions

(S)        ∂tui − div (Di(t, x)∇ui + Vi(t, x)ui) = fi(t, x, u), (Di(t, x)∇ui + Vi(t, x)ui) · ν = 0 on ∂Ω, ui(0, ·) = u0

i ≥ 0,

Di =

• dlk

i

• 1≤k,l≤N symmetric elliptic, Vi ∈ I

RN.

◮ Theorem. [D. Bothe, A. Fischer, M.P., G. Rolland] Assume that

f = (f1, ..., fm) satisfies (P), (M), the triangular structure and with growth at most polynomial. Assume also that, Vi, ∇dlk

i

∈ L∞ (0, T; Lr(Ω)) for some r > max{2, N}, dlk

i

∈ C(QT), ∀ T > 0. Then, there are global bounded solutions for (S).

A first extension for global bounded solutions

(S)        ∂tui − div (Di(t, x)∇ui + Vi(t, x)ui) = fi(t, x, u), (Di(t, x)∇ui + Vi(t, x)ui) · ν = 0 on ∂Ω, ui(0, ·) = u0

i ≥ 0,

Di =

• dlk

i

• 1≤k,l≤N symmetric elliptic, Vi ∈ I

RN.

◮ Theorem. [D. Bothe, A. Fischer, M.P., G. Rolland] Assume that

f = (f1, ..., fm) satisfies (P), (M), the triangular structure and with growth at most polynomial. Assume also that, Vi, ∇dlk

i

∈ L∞ (0, T; Lr(Ω)) for some r > max{2, N}, dlk

i

∈ C(QT), ∀ T > 0. Then, there are global bounded solutions for (S).

◮ The assumptions are so that Lp′-regularity theory holds for

each dual problem [H. Amann, R. Denk-M. Hieber-J. Pr¨

uss] − [∂tΨ + div (Di(t, x)∇Ψ)] + Vi(t, x) · ∇Ψ = Θ ∈ C ∞

0 ((τ, τ + δ),

Di(τ, x)∇Ψ · ν = θ ∈ C ∞ ((τ, τ + δ) × ∂Ω) , where δ is small.

Extensions to Wentzell boundary conditions

(S)        ∂tui − di∆ui = fi(u1, ..., um), ∂tui − δi∆∂Ωui + di∂νui = gi(u) on ∂Ω, ui(0, ·) = u0

i ≥ 0,

∆∂Ω = Laplace − Beltrami operator on ∂Ω, δi ∈ {0, 1}.

◮ [G. Goldstein, J. Goldstein, M. Meyries, M.P.] Assume f , g satisfy

(P), (M), the triangular structure and have at most polynomial growth. Then, System (S) has a global classical solution.

Extensions to Wentzell boundary conditions

(S)        ∂tui − di∆ui = fi(u1, ..., um), ∂tui − δi∆∂Ωui + di∂νui = gi(u) on ∂Ω, ui(0, ·) = u0

i ≥ 0,

∆∂Ω = Laplace − Beltrami operator on ∂Ω, δi ∈ {0, 1}.

◮ [G. Goldstein, J. Goldstein, M. Meyries, M.P.] Assume f , g satisfy

(P), (M), the triangular structure and have at most polynomial growth. Then, System (S) has a global classical solution.

◮ The main tool is that Lp′(QT)-regularity theory holds for the

dual linear problems    − (∂tψ + di∆ψ) = Θ ∈ C ∞

0 (QT),

− (∂tψ + δi∆∂Ωψ) + di∂νψ = θ ∈ C ∞

0 ((0, T) × ∂Ω) ,

ψ(T) = 0.

[R. Denk-J. Pr¨ uss-R. Zacher, D. Mugnolo-S. Romanelli, M. Meyries]

Extensions to nonlinear diffusions [D. Bothe, G. Rolland]

(S)        ∂tui − div (di(t, x, u)∇ui) = fi(t, x, u), ∂νui = 0 on ∂Ω, ui(0, ·) = u0

i ≥ 0,

0 < dm ≤ di ≤ dM < +∞.

◮ m = 3, f1(u) = f2(u) = −f3(u) = −k+u1u2 + k−u3, di

regular, non degenerate, N ≤ 5. Then, global existence of classical solutions. With Sobolev embeddings, Lq-estimates and bootstrap techniques.

Extensions to nonlinear diffusions [D. Bothe, G. Rolland]

(S)        ∂tui − div (di(t, x, u)∇ui) = fi(t, x, u), ∂νui = 0 on ∂Ω, ui(0, ·) = u0

i ≥ 0,

0 < dm ≤ di ≤ dM < +∞.

◮ m = 3, f1(u) = f2(u) = −f3(u) = −k+u1u2 + k−u3, di

regular, non degenerate, N ≤ 5. Then, global existence of classical solutions. With Sobolev embeddings, Lq-estimates and bootstrap techniques.

◮ m chemical species, R reactions

Uj1 + Uj2 k+

j

⇋ k−

j

Uj3, j = 1, ..., R, ji ∈ {1, ..., m} with atomic conservation law. Global existence of classical solutions when N ≤ 3.

Extensions to nonlinear diffusions [D. Bothe, G. Rolland]

(S)        ∂tui − div (di(ui)∇ui) = fi(t, x, u), ∂νui = 0 on ∂Ω, ui(0, ·) = u0

i ≥ 0,

0 < dm ≤ di ≤ dM < +∞.

◮ Then, global existence of classical solutions if N ≤ 9 for

m = 3, f1(u) = f2(u) = −f3(u) = −k+u1u2 + k−u3, and for N ≤ 5 for the more general system. Here, a priori L2(QT)-estimates hold.

Extensions to nonlinear diffusions [D. Bothe, G. Rolland]

(S)        ∂tui − div (di(ui)∇ui) = fi(t, x, u), ∂νui = 0 on ∂Ω, ui(0, ·) = u0

i ≥ 0,

0 < dm ≤ di ≤ dM < +∞.

◮ Then, global existence of classical solutions if N ≤ 9 for

m = 3, f1(u) = f2(u) = −f3(u) = −k+u1u2 + k−u3, and for N ≤ 5 for the more general system. Here, a priori L2(QT)-estimates hold.

◮ di = di(ui), f at most quadratic. Then, global existence of

weak solutions, even with u0

i ∈ L1(Ω) only.

Based on L2(QT) estimates, localized in time uiL2((τ,T)×Ω) ≤ C/τ N/4.

Global existence in electro-diffusion-reaction systems

◮

∂tui − di div (∇ui + ziui∇Φ) = fi(u), −∆Φ = m

i=1 ziui,

Global existence in electro-diffusion-reaction systems

◮

∂tui − di div (∇ui + ziui∇Φ) = fi(u), −∆Φ = m

i=1 ziui, ◮ zi = charge number ∈ I

R, ui = ui(t, x) concentrations of chemical species which are ionized (if zi = 0) and which are in a dilute enough solution (electrolyte). Φ = Φ(t, x) = electrical potential

Global existence in electro-diffusion-reaction systems

◮

∂tui − di div (∇ui + ziui∇Φ) = fi(u), −∆Φ = m

i=1 ziui, ◮ zi = charge number ∈ I

R, ui = ui(t, x) concentrations of chemical species which are ionized (if zi = 0) and which are in a dilute enough solution (electrolyte). Φ = Φ(t, x) = electrical potential

◮ Nernst-Planck-Poisson model with a simple Fick diffusion

(di ∈ (0, ∞)), without fluid motion, and where the mass fluxes are Ji = −di∇ui − ziui∇Φ. The fi are the chemical reaction terms and are as before + (in general) the charge conservation:

i zi fi(u) = 0

Global existence in electro-diffusion-reaction systems

◮

∂tui − di div (∇ui + ziui∇Φ) = fi(u), −∆Φ = m

i=1 ziui, ◮ zi = charge number ∈ I

R, ui = ui(t, x) concentrations of chemical species which are ionized (if zi = 0) and which are in a dilute enough solution (electrolyte). Φ = Φ(t, x) = electrical potential

◮ Nernst-Planck-Poisson model with a simple Fick diffusion

(di ∈ (0, ∞)), without fluid motion, and where the mass fluxes are Ji = −di∇ui − ziui∇Φ. The fi are the chemical reaction terms and are as before + (in general) the charge conservation:

i zi fi(u) = 0 ◮

   (∇ui + ziui∇Φ) · ν = 0 on Σ, ∇Φ · ν + τ Φ = ξ on Σ, ui(0, ·) = u0

i .

Global existence in electro-diffusion-reaction systems

◮

∂tui − di div (∇ui + ziui∇Φ) = fi(u), −∆Φ = m

i=1 ziui, ◮ zi = charge number ∈ I

R, ui = ui(t, x) concentrations of chemical species which are ionized (if zi = 0) and which are in a dilute enough solution (electrolyte). Φ = Φ(t, x) = electrical potential

◮ Nernst-Planck-Poisson model with a simple Fick diffusion

(di ∈ (0, ∞)), without fluid motion, and where the mass fluxes are Ji = −di∇ui − ziui∇Φ. The fi are the chemical reaction terms and are as before + (in general) the charge conservation:

i zi fi(u) = 0 ◮

   (∇ui + ziui∇Φ) · ν = 0 on Σ, ∇Φ · ν + τ Φ = ξ on Σ, ui(0, ·) = u0

i .

◮ for Φ: the surface ∂Ω is charged and considered locally as

plate capacitor.

Known results:

∂tui − di div (∇ui + ziui∇Φ) = fi(u), −∆Φ = m

i=1 ziui,

◮ Local existence (and uniqueness) of strong solutions. Based

• n Lp-regularity theory.

Note: if ∇Φ is a ”good function”, then the main operator is good: ui → ∂tui − di div (∇ui + ziui∇Φ) .

Known results:

∂tui − di div (∇ui + ziui∇Φ) = fi(u), −∆Φ = m

i=1 ziui,

◮ Local existence (and uniqueness) of strong solutions. Based

• n Lp-regularity theory.

Note: if ∇Φ is a ”good function”, then the main operator is good: ui → ∂tui − di div (∇ui + ziui∇Φ) .

◮ In dimension 1 et 2, global existence for reversible chemical

reactions with at most quadratic growth (by Lp-estimates and Sobolev embeddings on (0, T ∗)).

Known results:

∂tui − di div (∇ui + ziui∇Φ) = fi(u), −∆Φ = m

i=1 ziui,

◮ Local existence (and uniqueness) of strong solutions. Based

• n Lp-regularity theory.

Note: if ∇Φ is a ”good function”, then the main operator is good: ui → ∂tui − di div (∇ui + ziui∇Φ) .

◮ In dimension 1 et 2, global existence for reversible chemical

reactions with at most quadratic growth (by Lp-estimates and Sobolev embeddings on (0, T ∗)).

◮ A few results in higher dimensions m = 2, z1 = −z2 = 1 (see

also Debye-H¨ uckel models, semi-conductors,..) [also if all the zi have the same sign].

Some references

◮ H. Amann-M. Renardy (with neutrality assumption

• i ziui = 0)

◮ Several contributions by H. Gajewski, A. Glitzsky, K. Gr¨

• ger,
• R. H¨

unlich, J. Rehberg, : global existence and asymptotic behavior in dimension 1,2; nonlinear models with multi-dimensional results, non isotropic diffusions, ...

◮ Y.S. Choi-R. Lui: d = 2 ◮ P. Biler, J. Dolbeault, W. Hebisch, T. Nadzieja: mainly m = 2

+ asymptotic behavior

◮ D. Bothe, A. Fischer, J. Saal: d = 2 coupled with

Navier-Stokes

◮ M. Schmuck, J.W. Jerome-R. Sacco, m = 2, d ≤ 3, with

Navier-Stokes, and as well in R. Ryham, C. Deng-J. Zhao-S. Cui

Global existence in any dimension with fi ≡ 0

           ∂tui − di div (∇ui + ziui∇Φ) = 0, (∇ui + ziui∇Φ) · ν = 0 on ∂Ω, −∆Φ = m

i=1 ziui

∇Φ · ν + τ Φ = ξ on ∂Ω, ui(0, ·) = u0

i ≥ 0.

• Theorem. [D. Bothe, A. Fischer, G. Rolland, M.P.] Existence of a

global weak solution with the following properties for all T > 0

◮ ui ∈ C([0, ∞); L1(Ω)) ∩ L1

0, T; W 1,1

loc (Ω)

• ,

Φ ∈ L∞(0, ∞; H1(Ω)) ∩ L2(0, T; H2

loc(Ω)) ∩ C([0, ∞); L2(Ω)),

Global existence in any dimension with fi ≡ 0

           ∂tui − di div (∇ui + ziui∇Φ) = 0, (∇ui + ziui∇Φ) · ν = 0 on ∂Ω, −∆Φ = m

i=1 ziui

∇Φ · ν + τ Φ = ξ on ∂Ω, ui(0, ·) = u0

i ≥ 0.

• Theorem. [D. Bothe, A. Fischer, G. Rolland, M.P.] Existence of a

global weak solution with the following properties for all T > 0

◮ ui ∈ C([0, ∞); L1(Ω)) ∩ L1

0, T; W 1,1

loc (Ω)

• ,

Φ ∈ L∞(0, ∞; H1(Ω)) ∩ L2(0, T; H2

loc(Ω)) ∩ C([0, ∞); L2(Ω)), ◮ ∇ui + ziui∇Φ ∈ L1(QT),

Global existence in any dimension with fi ≡ 0

           ∂tui − di div (∇ui + ziui∇Φ) = 0, (∇ui + ziui∇Φ) · ν = 0 on ∂Ω, −∆Φ = m

i=1 ziui

∇Φ · ν + τ Φ = ξ on ∂Ω, ui(0, ·) = u0

i ≥ 0.

• Theorem. [D. Bothe, A. Fischer, G. Rolland, M.P.] Existence of a

global weak solution with the following properties for all T > 0

◮ ui ∈ C([0, ∞); L1(Ω)) ∩ L1

0, T; W 1,1

loc (Ω)

• ,

Φ ∈ L∞(0, ∞; H1(Ω)) ∩ L2(0, T; H2

loc(Ω)) ∩ C([0, ∞); L2(Ω)), ◮ ∇ui + ziui∇Φ ∈ L1(QT), ◮ QT −∂tψ ui + di∇ψ (∇ui + ziui∇Φ) =

• Ω u0

i ψ(0)

• Ω ∇ψ∇Φ +
• ∂Ω ψ (τΦ − ξ) =
• Ω ψ m

i=1 ziui,

for all test-functions ψ ∈ C ∞(QT) with ψ(T) = 0.

Global existence in any dimension with fi ≡ 0

           ∂tui − di div (∇ui + ziui∇Φ) = 0, (∇ui + ziui∇Φ) · ν = 0 on ∂Ω, −∆Φ = m

i=1 ziui

∇Φ · ν + τ Φ = ξ on ∂Ω, ui(0, ·) = u0

i ≥ 0.

• Theorem. [D. Bothe, A. Fischer, G. Rolland, M.P.] Existence of a

global weak solution with the following properties for all T > 0

◮ ui ∈ C([0, ∞); L1(Ω)) ∩ L1

0, T; W 1,1

loc (Ω)

• ,

Φ ∈ L∞(0, ∞; H1(Ω)) ∩ L2(0, T; H2

loc(Ω)) ∩ C([0, ∞); L2(Ω)), ◮ ∇ui + ziui∇Φ ∈ L1(QT), ◮ QT −∂tψ ui + di∇ψ (∇ui + ziui∇Φ) =

• Ω u0

i ψ(0)

• Ω ∇ψ∇Φ +
• ∂Ω ψ (τΦ − ξ) =
• Ω ψ m

i=1 ziui,

for all test-functions ψ ∈ C ∞(QT) with ψ(T) = 0.

◮ If d ≤ 3: ∇ui, ui∇Φ ∈ L1(Q), Φ ∈ L2(0, T; H2(Ω)).

Main facts in the proof

◮ The free energy is defined as:

V (t) :=

• i
• Ω

ui log ui + 1 2

• Ω

|∇Φ|2 + τ 2

• ∂Ω

Φ2, V ′(t) = −

m

• i=1
• Ω

di |∇ui + ziui∇Φ|2 ui .

Main facts in the proof

◮ The free energy is defined as:

V (t) :=

• i
• Ω

ui log ui + 1 2

• Ω

|∇Φ|2 + τ 2

• ∂Ω

Φ2, V ′(t) = −

m

• i=1
• Ω

di |∇ui + ziui∇Φ|2 ui .

◮ It follows: supt V (t) ≤ V (0) < ∞ which implies

uiL∞(0,∞;L1(Ω)) + ΦL∞(0,∞;H1(Ω)) < +∞. And we also exploit the negative part of V ′(t)...

Main facts in the proof

◮ The free energy is defined as:

V (t) :=

• i
• Ω

ui log ui + 1 2

• Ω

|∇Φ|2 + τ 2

• ∂Ω

Φ2, V ′(t) = −

m

• i=1
• Ω

di |∇ui + ziui∇Φ|2 ui .

◮ It follows: supt V (t) ≤ V (0) < ∞ which implies

uiL∞(0,∞;L1(Ω)) + ΦL∞(0,∞;H1(Ω)) < +∞. And we also exploit the negative part of V ′(t)...

◮ Approximation by a nonlinear diffusion hi(ui) = ui + ηup i

∂tui − didiv (∇h(ui) + ziui∇Φ)) , p large with a free energy structure and a global strong solution for η > 0. Then, η → 0.

Extension to the electro-reaction-diffusion system

◮

           ∂tui − di div (∇ui + ziui∇Φ) = fi(u) in Q, (∇ui + ziui∇Φ) · ν = 0 sur Σ, −∆Φ = m

i=1 ziui in Q,

∇Φ · ν + τ Φ = ξ on Σ, u(0, ·) = u0.

Extension to the electro-reaction-diffusion system

◮

           ∂tui − di div (∇ui + ziui∇Φ) = fi(u) in Q, (∇ui + ziui∇Φ) · ν = 0 sur Σ, −∆Φ = m

i=1 ziui in Q,

∇Φ · ν + τ Φ = ξ on Σ, u(0, ·) = u0.

◮ Theorem:

We assume    (P) and (M), conservation of charges :

i zifi = 0,

entropy condition :

i ˆ

mi(log ui)fi(u) ≤ M.

Extension to the electro-reaction-diffusion system

◮

           ∂tui − di div (∇ui + ziui∇Φ) = fi(u) in Q, (∇ui + ziui∇Φ) · ν = 0 sur Σ, −∆Φ = m

i=1 ziui in Q,

∇Φ · ν + τ Φ = ξ on Σ, u(0, ·) = u0.

◮ Theorem:

We assume    (P) and (M), conservation of charges :

i zifi = 0,

entropy condition :

i ˆ

mi(log ui)fi(u) ≤ M.

◮ If there is an L1-a priori estimate

• i
• Q

|fi(u)| ≤ M < +∞, then, global existence of weak solutions holds.

Open problems

◮ For the electro-reaction-diffusion system, what about existence

• f global weak solution for quadratic reaction terms like

fi(c) = (−1)i(c1c3 − c2c4), i = 1, 2, 3, 4. ?

Open problems

◮ For the electro-reaction-diffusion system, what about existence

• f global weak solution for quadratic reaction terms like

fi(c) = (−1)i(c1c3 − c2c4), i = 1, 2, 3, 4. ?

◮ Even for classical reaction-diffusion , what about global

classical solution for these fi when N ≥ 3?

Open problems

◮ For the electro-reaction-diffusion system, what about existence

• f global weak solution for quadratic reaction terms like

fi(c) = (−1)i(c1c3 − c2c4), i = 1, 2, 3, 4. ?

◮ Even for classical reaction-diffusion , what about global

classical solution for these fi when N ≥ 3?

◮ What can be said for reversible reactions without L1(Q)- a

priori estimates like ∂tui − di∆ui = (−1)i(c2

1c2 3 − c2 2c2 4), i = 1, 2, 3, 4 ?

Open problems

◮ For the electro-reaction-diffusion system, what about existence

• f global weak solution for quadratic reaction terms like

fi(c) = (−1)i(c1c3 − c2c4), i = 1, 2, 3, 4. ?

◮ Even for classical reaction-diffusion , what about global

classical solution for these fi when N ≥ 3?

◮ What can be said for reversible reactions without L1(Q)- a

priori estimates like ∂tui − di∆ui = (−1)i(c2

1c2 3 − c2 2c2 4), i = 1, 2, 3, 4 ? ◮ Use of renormalized solutions?