Mathematical models in reaction-diffusion: from Turing instabilities - - PowerPoint PPT Presentation

Mathematical models in reaction-diffusion: from Turing instabilities to present questions

Michel Pierre

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

Congr` es Franco-Marocain de Math´ ematiques Appliqu´ ees Marrakech, 16-20 avril 2018

Goal of the talk

◮ Survey on global existence in time of solutions to

reaction-diffusion (RD) systems for which:

• positivity is preserved
• conservation or at least dissipation of the total mass

Goal of the talk

◮ Survey on global existence in time of solutions to

reaction-diffusion (RD) systems for which:

• positivity is preserved
• conservation or at least dissipation of the total mass

◮ This provides an a priori bound in L1, uniform in time.

QUESTION: how does this help for global existence ???

Goal of the talk

◮ Survey on global existence in time of solutions to

reaction-diffusion (RD) systems for which:

• positivity is preserved
• conservation or at least dissipation of the total mass

◮ This provides an a priori bound in L1, uniform in time.

QUESTION: how does this help for global existence ???

◮ OLD AND RECENT RESULTS- OPEN PROBLEMS

Goal of the talk

◮ Survey on global existence in time of solutions to

reaction-diffusion (RD) systems for which:

• positivity is preserved
• conservation or at least dissipation of the total mass

◮ This provides an a priori bound in L1, uniform in time.

QUESTION: how does this help for global existence ???

◮ OLD AND RECENT RESULTS- OPEN PROBLEMS ◮ The reactive part (= evolution without space variable) is a

good ODE for which global existence holds. Thus, a highly connected problem is: QUESTION: What happens to a ”good” system of ordinary differential equations (ODE) when a spatial variable is added and diffusion occurs (PDE)?

THE CHEMICAL BASIS OF MOKPHOGENESIS

BY A. M. TURING, F.R.S. University qf Manchester (Received 9 November 195 1-Revised 15 March 1952)

It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern

• r structure due to an instability of the homogeneous equilibrium, which is triggered off by

random disturbances. Such reaction-diffusion systems are considered in some detail in the case

• f an isolated ring of cells, a mathematically convenient, though biolo:~irall, unusual system.

The investigation is chiefly concerned with the onset of instability. It is faund that there are six essentially different forms which this may take. In the most interesting form stationary waves appear on the ring. It is suggested that this might account, for instance, for the tentacle patterns

• n Hydra and for whorled leaves. A system of reactions and diffusion on a sphere is also con-
• sidered. Such a system appears to account for gastrulation. Another reaction system in two

dimensions gives rise to patterns reminiscent of dappling. It is also suggested that stationary waves in two dimensions could account for the phenomena of phyllotaxis. The purpose of this paper is to discuss a possible mechanism by which the genes of a zygote may determine the anatomical structure of the resulting organism. The theory does not make any new hypotheses; it merely suggests that certain well-known physical laws are sufficient to account for many of the facts. The full understanding of the paper requires a good knowledge of mathe- matics, some biology, and some elementary chemistry. Since readers cannot be expected to be experts in all of these subjects, a number of elementary facts are explained, which can be found in text-books, but whose omission would make the paper difficult reading.

In this section a mathematical model of the growing embryo will be described. This model will be a simplification and an idealization, and consequently a falsification. It is to be hoped that the features retained for discussion are those of greatest importance in the present state of knowledge. The model takes two slightly different forms. In one of them the cell theory is recognized but the cells are idealized into geometrical points. In the other the matter of the organism is imagined as continuously distributed. The cells are not, however, completely ignored, for various physical and physico-chemical characteristics of the matter as a whole are assumed to have values appropriate to the cellular matter. With either of the models one proceeds as with a physical theory and defines an entity called 'the state of the system'. One then describes how that state is to be determined from the state at a moment very shortly before. With either model the description of the state consists of two parts, the mechanical and the chemical. The mechanical part of the state describes the positions, masses, velocities and elastic properties of the cells, and the forces between them. In the continuous form of the theory essentially the same information is given in the form of the stress, velocity, density and elasticity of the matter. The chemical part of the state is given (in the cell form of theory) as the chemical composition of each separate cell; the diffusibility of each substance between each two adjacent cells rnust also

VOL.237. B. 641. (Price 8s.) 5 14August I 952 [P~~btished

Turing instabilities

◮ The Chemical Basis of Morphogenesis

By A. M. Turing, University of Manchester In Phil. Trans. Royal Soc. London, 1952

Turing instabilities

◮ The Chemical Basis of Morphogenesis

By A. M. Turing, University of Manchester In Phil. Trans. Royal Soc. London, 1952

◮ Abstract: ”It is suggested that a system of chemical

substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances...”

Turing instabilities

◮ One example:

”Chemical morphogenetic process”=”Brusselator”, Prigogine

• u′

1 = −u1 u2 2 + b u2,

u′

2 =

u1u2

2 − (b + 1) u2 + a.

Data: a, b > 0 Unknown functions: u1, u2 : [0, ∞[→ [0, ∞[

Turing instabilities

◮ One example:

”Chemical morphogenetic process”=”Brusselator”, Prigogine

• u′

1 = −u1 u2 2 + b u2,

u′

2 =

u1u2

2 − (b + 1) u2 + a.

Data: a, b > 0 Unknown functions: u1, u2 : [0, ∞[→ [0, ∞[

◮ (u∗ 1, u∗ 2) = (b/a, a) is the unique stationary solution. The

matrix of the linearized system around (u∗

1, u∗ 2) is

A =

• −a2

−b a2 b − 1

• ; trace(A) = b − (1 + a2); det(A) = a2

Turing instabilities

◮ One example:

”Chemical morphogenetic process”=”Brusselator”, Prigogine

• u′

1 = −u1 u2 2 + b u2,

u′

2 =

u1u2

2 − (b + 1) u2 + a.

Data: a, b > 0 Unknown functions: u1, u2 : [0, ∞[→ [0, ∞[

◮ (u∗ 1, u∗ 2) = (b/a, a) is the unique stationary solution. The

matrix of the linearized system around (u∗

1, u∗ 2) is

A =

• −a2

−b a2 b − 1

• ; trace(A) = b − (1 + a2); det(A) = a2

◮ (u∗ 1, u∗ 2) is asymptotically stable if b < 1 + a2.

The full reaction-diffusion system

◮ Notation: ui = ui(t, x), ui : [0, ∞) × Ω → [0, ∞), i = 1, 2

Ω ⊂ I RN regular open subset, QT = (0, T) × Ω, ΣT = (0, T) × ∂Ω

◮ d1, d2 ∈ (0, ∞)

   ∂tu1−d1∆xu1 = −u1u2

2 + b u2 in Q∞

∂tu2−d2∆xu2 = u1u2

2 − (b + 1) u2 + a in Q∞

∂νui = 0 on Σ∞, i = 1, 2.

The full reaction-diffusion system

◮ Notation: ui = ui(t, x), ui : [0, ∞) × Ω → [0, ∞), i = 1, 2

Ω ⊂ I RN regular open subset, QT = (0, T) × Ω, ΣT = (0, T) × ∂Ω

◮ d1, d2 ∈ (0, ∞)

   ∂tu1−d1∆xu1 = −u1u2

2 + b u2 in Q∞

∂tu2−d2∆xu2 = u1u2

2 − (b + 1) u2 + a in Q∞

∂νui = 0 on Σ∞, i = 1, 2.

◮ (u∗ 1, u∗ 2) = (b/a, a) is still a stationary solution.

The full reaction-diffusion system

◮ Notation: ui = ui(t, x), ui : [0, ∞) × Ω → [0, ∞), i = 1, 2

Ω ⊂ I RN regular open subset, QT = (0, T) × Ω, ΣT = (0, T) × ∂Ω

◮ d1, d2 ∈ (0, ∞)

   ∂tu1−d1∆xu1 = −u1u2

2 + b u2 in Q∞

∂tu2−d2∆xu2 = u1u2

2 − (b + 1) u2 + a in Q∞

∂νui = 0 on Σ∞, i = 1, 2.

◮ (u∗ 1, u∗ 2) = (b/a, a) is still a stationary solution. ◮ But it may loose its stability

for small x-dependent perturbations ⇒ ”Turing instabilities”

Turing instabilities

◮ Simple idea: the new linearized system is

∂t u1 u2

• =

d1 d2 ∆u1 ∆u2

• +

−a2 −b a2 b − 1 u1 u2

Turing instabilities

◮ Simple idea: the new linearized system is

∂t u1 u2

• =

d1 d2 ∆u1 ∆u2

• +

−a2 −b a2 b − 1 u1 u2

• ◮ Work in the spectral basis of the Laplace operator −∆.

Turing instabilities

◮ Simple idea: the new linearized system is

∂t u1 u2

• =

d1 d2 ∆u1 ∆u2

• +

−a2 −b a2 b − 1 u1 u2

• ◮ Work in the spectral basis of the Laplace operator −∆.

◮ In each direction, one has to look at the eigenvalues of the matrices

−λk d1 d2

• +

−a2 −b a2 b − 1

• where λk > 0, k ≥ 1 are the eigenvalues of −∆.

Recall that the real parts of the eigenvalues of the second matrix are negative if b < 1 + a2.

Turing instabilities

◮ Simple idea: the new linearized system is

∂t u1 u2

• =

d1 d2 ∆u1 ∆u2

• +

−a2 −b a2 b − 1 u1 u2

• ◮ Work in the spectral basis of the Laplace operator −∆.

◮ In each direction, one has to look at the eigenvalues of the matrices

−λk d1 d2

• +

−a2 −b a2 b − 1

• where λk > 0, k ≥ 1 are the eigenvalues of −∆.

Recall that the real parts of the eigenvalues of the second matrix are negative if b < 1 + a2.

◮ ...One the eigenvalues may become positive if d2 < d1 and

• 1 + a
• d2

d1 2 < b < 1 + a2.

Turing instabilities

◮ Simple idea: the new linearized system is

∂t u1 u2

• =

d1 d2 ∆u1 ∆u2

• +

−a2 −b a2 b − 1 u1 u2

• ◮ Work in the spectral basis of the Laplace operator −∆.

◮ In each direction, one has to look at the eigenvalues of the matrices

−λk d1 d2

• +

−a2 −b a2 b − 1

• where λk > 0, k ≥ 1 are the eigenvalues of −∆.

Recall that the real parts of the eigenvalues of the second matrix are negative if b < 1 + a2.

◮ ...One the eigenvalues may become positive if d2 < d1 and

• 1 + a
• d2

d1 2 < b < 1 + a2.

◮ Instability is interesting !

It may lead to non homogeneous equilibria (depending on x) and provide a great variety of asymptotic spatial patterns.

60

• A. M. TURING ON THE

was used and h was about 0.7. In the figure the set of points where f(x,y) is positive is shown

• black. The outlines of the black patches are somewhat less irregular than they should be

due to an inadequacy in the computation procedure.

L

I

FIGURE

• 2. An example of a 'dappled' pattern as resulting from a type (a) morphogen system.

A marker of unit length is shown. See text, $9, 11.

• 10. A NUMERICAL EXAMPLE

The numerous approximations and assumptions that have been made in the foregoing analysis may be rather confusing to many readers. In the present section it is proposed to consider in detail a single example of the case of most interest, (d). This will be made as specific as possible. It is unfortunately not possible to specify actual chemical reactions with the required properties, but it is thought that the reaction rates associated with the imagined reactions are not unreasonable. The detail to be specified includes (i) The number and dimensions of the cells of the ring. (ii) The diffusibilities of the morphogens. (iii) The reactions concerned. (iv) The rates at which the reactions occur. (v) Information about random disturbances. (vi) Information about the distribution, in space and time, of those morphogens which are of the nature of evocators. These will be taken in order. (i) It will be assumed that there are twenty cells in the ring, and that they have a diameter

• f 0.1 mm each. These cells are certainly on the large rather than the small side, but by

no means impossibly so. The number of cells in the ring has been chosen rather small in

• rder that it should not be necessary to make the approximation of continuous tissue.

(ii) Two morphogens are considered. They will be called X and Y, and the same letters will be used for their concentrations. This will not lead to any real confusion. The diffusion constant for X will be assumed to be 5 x 10- cm2 s-l and that for Y to be 2.5 x crn2s-l. With cells of diameter 0.01 cni this means that X flows between neighbouring cells at the

[Rui ¡Dilao] ¡Patterns ¡in ¡butterflies ¡compared ¡with ¡patterns ¡generated ¡from ¡equation ¡(23). ¡ ¡

Alan Turing 1912-1954

Ilia Prigogine [1917-2003], Chemistry Nobel prize in 1977

Is global existence destroyed by adding diffusion?

◮ Global existence holds for the Brusselator ODE

   u′

1 = −u1 u2 2 + b u2

u′

2 =

u1u2

2 − (b + 1) u2 + a

ui(0) = uo

i , uo i ∈ [0, +∞), i = 1, 2.

◮ Local existence holds on a maximal interval [0, T ∗) by

Cauchy-Lipschitz Theorem. Moreover solutions are nonnegative.

Is global existence destroyed by adding diffusion?

◮ Global existence holds for the Brusselator ODE

   u′

1 = −u1 u2 2 + b u2

u′

2 =

u1u2

2 − (b + 1) u2 + a

ui(0) = uo

i , uo i ∈ [0, +∞), i = 1, 2.

◮ Local existence holds on a maximal interval [0, T ∗) by

Cauchy-Lipschitz Theorem. Moreover solutions are nonnegative.

◮ And we have an a priori estimate:

(u1 + u2)′ = −u2 + a ≤ a ⇒ (u1 + u2)(t) ≤ a t + (uo

1 + uo 2) ⇒ T ∗ = +∞

Is global existence destroyed by adding diffusion?

◮ Global existence holds for the Brusselator ODE

   u′

1 = −u1 u2 2 + b u2

u′

2 =

u1u2

2 − (b + 1) u2 + a

ui(0) = uo

i , uo i ∈ [0, +∞), i = 1, 2.

◮ Local existence holds on a maximal interval [0, T ∗) by

Cauchy-Lipschitz Theorem. Moreover solutions are nonnegative.

◮ And we have an a priori estimate:

(u1 + u2)′ = −u2 + a ≤ a ⇒ (u1 + u2)(t) ≤ a t + (uo

1 + uo 2) ⇒ T ∗ = +∞

◮ Does one have global existence for the full system?

       ∂tu1−d1∆u1 = −u1u2

2 + b u2

∂tu2−d2∆u2 = u1u2

2 − (b + 1) u2

∂νui = 0 on ∂Ω, i = 1, 2 ui(0, ·) = uo

i (·) ≥ 0, uo = (uo 1, uo 2) ∈ L∞(Ω)2.

Does global existence hold?

       ∂tu1−d1∆u1 = −u1u2

2 + b u2

∂tu2−d2∆u2 = u1u2

2 − (b + 1) u2

∂νui = 0 on ∂Ω, i = 1, 2 ui(0, ·) = uo

i (·) ≥ 0, uo = (uo 1, uo 2) ∈ L∞(Ω)2.

Does global existence hold?

       ∂tu1−d1∆u1 = −u1u2

2 + b u2

∂tu2−d2∆u2 = u1u2

2 − (b + 1) u2

∂νui = 0 on ∂Ω, i = 1, 2 ui(0, ·) = uo

i (·) ≥ 0, uo = (uo 1, uo 2) ∈ L∞(Ω)2.

◮ First, we can forget the linear part for the question of global

existence and consider only    ∂tu1 − d1∆u1 = −u1u2

2

∂tu2 − d2∆u2 = u1u2

2

∂νui = 0 on ∂Ω, i = 1, 2

Does global existence hold?

       ∂tu1−d1∆u1 = −u1u2

2 + b u2

∂tu2−d2∆u2 = u1u2

2 − (b + 1) u2

∂νui = 0 on ∂Ω, i = 1, 2 ui(0, ·) = uo

i (·) ≥ 0, uo = (uo 1, uo 2) ∈ L∞(Ω)2.

◮ First, we can forget the linear part for the question of global

existence and consider only    ∂tu1 − d1∆u1 = −u1u2

2

∂tu2 − d2∆u2 = u1u2

2

∂νui = 0 on ∂Ω, i = 1, 2

◮ Using

• Ω ∆ui =
• ∂Ω ∂νui = 0, we have

d dt

• Ω

(u1 + u2)(t) =

• Ω

∂t(u1 + u2) = 0 ⇒

• Ω

u1(t) + u2(t) =

• Ω

uo

1 + uo 2

Does global existence hold?

       ∂tu1−d1∆u1 = −u1u2

2 + b u2

∂tu2−d2∆u2 = u1u2

2 − (b + 1) u2

∂νui = 0 on ∂Ω, i = 1, 2 ui(0, ·) = uo

i (·) ≥ 0, uo = (uo 1, uo 2) ∈ L∞(Ω)2.

◮ First, we can forget the linear part for the question of global

existence and consider only    ∂tu1 − d1∆u1 = −u1u2

2

∂tu2 − d2∆u2 = u1u2

2

∂νui = 0 on ∂Ω, i = 1, 2

◮ Using

• Ω ∆ui =
• ∂Ω ∂νui = 0, we have

d dt

• Ω

(u1 + u2)(t) =

• Ω

∂t(u1 + u2) = 0 ⇒

• Ω

u1(t) + u2(t) =

• Ω

uo

1 + uo 2

◮ ⇒

L1(Ω)-bound, uniform in time !

Does global existence hold?

       ∂tu1−d1∆u1 = −u1u2

2 + b u2

∂tu2−d2∆u2 = u1u2

2 − (b + 1) u2

∂νui = 0 on ∂Ω, i = 1, 2 ui(0, ·) = uo

i (·) ≥ 0, uo = (uo 1, uo 2) ∈ L∞(Ω)2.

◮ First, we can forget the linear part for the question of global

existence and consider only    ∂tu1 − d1∆u1 = −u1u2

2

∂tu2 − d2∆u2 = u1u2

2

∂νui = 0 on ∂Ω, i = 1, 2

◮ Using

• Ω ∆ui =
• ∂Ω ∂νui = 0, we have

d dt

• Ω

(u1 + u2)(t) =

• Ω

∂t(u1 + u2) = 0 ⇒

• Ω

u1(t) + u2(t) =

• Ω

uo

1 + uo 2

◮ ⇒

L1(Ω)-bound, uniform in time !

◮ How does this help for global existence?

Same question for the general family of systems:

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) ∂νui = 0 ui(0, ·) = u0

i (·) ≥ 0

di > 0, fi : [0, ∞)m → I R locally Lipschitz continuous where

◮ (P): Positivity (nonnegativity) is preserved

Same question for the general family of systems:

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) ∂νui = 0 ui(0, ·) = u0

i (·) ≥ 0

di > 0, fi : [0, ∞)m → I R locally Lipschitz continuous where

◮ (P): Positivity (nonnegativity) is preserved ◮ (M): 1≤i≤m fi ≤ 0

Same question for the general family of systems:

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) ∂νui = 0 ui(0, ·) = u0

i (·) ≥ 0

di > 0, fi : [0, ∞)m → I R locally Lipschitz continuous where

◮ (P): Positivity (nonnegativity) is preserved ◮ (M): 1≤i≤m fi ≤ 0 ◮ or more generally

(M’) ∀r ∈ [0, ∞[m,

1≤i≤m aifi(r) ≤ C[1 + 1≤i≤m ri]

for some ai > 0

◮

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

◮ (P) Preservation of Positivity: ∀i = 1, ..., m

∀r = (r1, ..., rm) ∈ [0, ∞[m, fi(r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0, = ” quasi-positivity ” of f = (fi)1≤i≤m.

✻

r2

✲r1 ✁ ✁ ✁ ✕

f1(0, r2)≥ 0 f2(0, r2)

❆ ❆ ❆ ❑

f1(r1, 0) f2(r1, 0)≥ 0

◮

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

◮ (P) Preservation of Positivity: ∀i = 1, ..., m

∀r ∈ [0, +∞[m, fi(r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.

◮ (M): 1≤i≤m fi(r1, ..., rm) ≤ 0 ⇒ ’Control of the Total

Mass’: ∀t ≥ 0,

• Ω
• 1≤i≤r

ui(t, x)dx ≤

• Ω
• 1≤i≤r

u0

i (x)dx

Add up, integrate on Ω, use

• Ω ∆ui =
• ∂Ω ∂νui = 0:
• Ω

∂t[

• ui(t)]dx =
• Ω
• i

fi(u)dx ≤ 0.

◮

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

◮ (P) Preservation of Positivity: ∀i = 1, ..., m

∀r ∈ [0, +∞[m, fi(r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.

◮ (M): 1≤i≤m fi(r1, ..., rm) ≤ 0 ⇒ ’Control of the Total

Mass’: ∀t ≥ 0,

• Ω
• 1≤i≤r

ui(t, x)dx ≤

• Ω
• 1≤i≤r

u0

i (x)dx

Add up, integrate on Ω, use

• Ω ∆ui =
• ∂Ω ∂νui = 0:
• Ω

∂t[

• ui(t)]dx =
• Ω
• i

fi(u)dx ≤ 0.

◮ ⇒ L1(Ω)- a priori estimates, uniform in time.

◮

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

◮ (P) Preservation of Positivity: ∀i = 1, ..., m

∀r ∈ [0, +∞[m, fi(r1, ..., ri−1, 0, ri+1, ..., rm) ≥ 0.

◮ (M): 1≤i≤m fi(r1, ..., rm) ≤ 0 ⇒ ’Control of the Total

Mass’: ∀t ≥ 0,

• Ω
• 1≤i≤r

ui(t, x)dx ≤

• Ω
• 1≤i≤r

u0

i (x)dx

Add up, integrate on Ω, use

• Ω ∆ui =
• ∂Ω ∂νui = 0:
• Ω

∂t[

• ui(t)]dx =
• Ω
• i

fi(u)dx ≤ 0.

◮ ⇒ L1(Ω)- a priori estimates, uniform in time. ◮ Remark: L1-bound for all time with (M’)

QUESTION: What about Global Existence of solutions under assumption (P)+(M)??

• r more generally (P)+ (M’) ??

Several approaches and techniques

◮ L∞-approach: local existence ◮ An Lp-approach ◮ Blow up may occur ◮ An L1-approach ◮ A surprising L2-estimate ◮ and even L2+ǫ... ◮ Renormalized solutions ◮ L Log L is also involved ◮ ◮ + OPEN PROBLEMS

Local existence in L∞ for systems

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

◮ Theorem (`

a la Cauchy-Lipschitz dans L∞). Let u0 ∈ L∞(Ω)+m. Then, there exist a maximum time T ∗ > 0 and u = (u1, ..., um) unique classical nonnegative solution of (S) on [0, T ∗). Moreover, sup

t∈[0,T ∗)

• u(t)L∞(Ω) + v(t)L∞(Ω)
• < +∞ ⇒ [T ∗ + ∞] .

Local existence in L∞ for systems

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

◮ Theorem (`

a la Cauchy-Lipschitz dans L∞). Let u0 ∈ L∞(Ω)+m. Then, there exist a maximum time T ∗ > 0 and u = (u1, ..., um) unique classical nonnegative solution of (S) on [0, T ∗). Moreover, sup

t∈[0,T ∗)

• u(t)L∞(Ω) + v(t)L∞(Ω)
• < +∞ ⇒ [T ∗ + ∞] .

◮ Corollary. If di = d for all i = 1, ..., m, then T ∗ = +∞.

Local existence in L∞ for systems

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

◮ Theorem (`

a la Cauchy-Lipschitz dans L∞). Let u0 ∈ L∞(Ω)+m. Then, there exist a maximum time T ∗ > 0 and u = (u1, ..., um) unique classical nonnegative solution of (S) on [0, T ∗). Moreover, sup

t∈[0,T ∗)

• u(t)L∞(Ω) + v(t)L∞(Ω)
• < +∞ ⇒ [T ∗ + ∞] .

◮ Corollary. If di = d for all i = 1, ..., m, then T ∗ = +∞. ◮ Proof: ∂t( i ui) − d ∆( i ui) ≤ 0.

⇒

• i

ui(t)L∞(Ω) ≤

• i

ui0L∞(Ω).

The Lp-approach

◮ Recall the Brusselator (β ∈ [1, +∞))

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2.

The Lp-approach

◮ Recall the Brusselator (β ∈ [1, +∞))

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2.

◮ By maximum principle u1(t)L∞(Ω) ≤ u0 1L∞(Ω).

The Lp-approach

◮ Recall the Brusselator (β ∈ [1, +∞))

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2.

◮ By maximum principle u1(t)L∞(Ω) ≤ u0 1L∞(Ω). ◮ We have : ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1).

The Lp-approach

◮ Recall the Brusselator (β ∈ [1, +∞))

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2.

◮ By maximum principle u1(t)L∞(Ω) ≤ u0 1L∞(Ω). ◮ We have : ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1). ◮ Or : ”u2 = (∂t − d2∆)−1 (− ∂t + d1∆) u1”

The Lp-approach

◮ Recall the Brusselator (β ∈ [1, +∞))

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2.

◮ By maximum principle u1(t)L∞(Ω) ≤ u0 1L∞(Ω). ◮ We have : ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1). ◮ Or : ”u2 = (∂t − d2∆)−1 (− ∂t + d1∆) u1” ◮ Lp-Fundamental Lemma:

∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ I R, u2 ≥ 0,

⇒ u2Lp(QT) ≤ Cu1Lp(QT)

for all 1 < p < +∞, all T > 0,

C = C(a, b, p, T, u1(0)p, u2(0)p).

The Lp-approach

◮ Recall the Brusselator (β ∈ [1, +∞))

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2.

◮ By maximum principle u1(t)L∞(Ω) ≤ u0 1L∞(Ω). ◮ We have : ∂tu2 − d2∆u2 = −(∂tu1 − d1∆u1). ◮ Or : ”u2 = (∂t − d2∆)−1 (− ∂t + d1∆) u1” ◮ Lp-Fundamental Lemma:

∂tu2 − d2∆u2 ≤ a∂tu1 + b∆u1, a, b ∈ I R, u2 ≥ 0,

⇒ u2Lp(QT) ≤ Cu1Lp(QT)

for all 1 < p < +∞, all T > 0,

C = C(a, b, p, T, u1(0)p, u2(0)p).

◮ Lemma=Dual statement of the Lp maximal regularity theory

for parabolic operators. Not valid for p = +∞.

The Lp-approach applied to the Brusselator

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2

◮ We know u1(t)L∞(Ω) ≤ u1(0)L∞(Ω) for all t ∈ [0, T ∗)

The Lp-approach applied to the Brusselator

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2

◮ We know u1(t)L∞(Ω) ≤ u1(0)L∞(Ω) for all t ∈ [0, T ∗) ◮ From the fundamental Lemma, for all p < +∞

u2Lp(QT∗) ≤ Cu1Lp(QT∗) ≤ Cu1L∞(QT∗) < +∞

The Lp-approach applied to the Brusselator

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2

◮ We know u1(t)L∞(Ω) ≤ u1(0)L∞(Ω) for all t ∈ [0, T ∗) ◮ From the fundamental Lemma, for all p < +∞

u2Lp(QT∗) ≤ Cu1Lp(QT∗) ≤ Cu1L∞(QT∗) < +∞

◮ u1uβ 2 Lq(QT∗) < +∞ for all q < +∞

The Lp-approach applied to the Brusselator

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2

◮ We know u1(t)L∞(Ω) ≤ u1(0)L∞(Ω) for all t ∈ [0, T ∗) ◮ From the fundamental Lemma, for all p < +∞

u2Lp(QT∗) ≤ Cu1Lp(QT∗) ≤ Cu1L∞(QT∗) < +∞

◮ u1uβ 2 Lq(QT∗) < +∞ for all q < +∞ ◮ Choosing q large enough (q > (N + 1)/2), this implies

u2L∞(QT∗) < +∞

The Lp-approach applied to the Brusselator

(S)    ∂tu1 − d1∆u1 = −u1uβ

2 (≤ 0)

∂tu2 − d2∆u2 = u1uβ

2

∂νui = 0 on ∂Ω, i = 1, 2

◮ We know u1(t)L∞(Ω) ≤ u1(0)L∞(Ω) for all t ∈ [0, T ∗) ◮ From the fundamental Lemma, for all p < +∞

u2Lp(QT∗) ≤ Cu1Lp(QT∗) ≤ Cu1L∞(QT∗) < +∞

◮ u1uβ 2 Lq(QT∗) < +∞ for all q < +∞ ◮ Choosing q large enough (q > (N + 1)/2), this implies

u2L∞(QT∗) < +∞

◮ ⇒

T ∗ = +∞

Extensions of the Lp-approach

◮ This approach extends to m × m systems

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

where fi have polynomial growth and        f1 ≤ 0 f1 + f2 ≤ 0 ... f1 + f2 + ... + fm ≤ 0

Extensions of the Lp-approach

◮ This approach extends to m × m systems

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

where fi have polynomial growth and        f1 ≤ 0 f1 + f2 ≤ 0 ... f1 + f2 + ... + fm ≤ 0

◮ or more generally if for an invertible triangular matrix Q with

nonnegative entries ∀r ∈ [0, ∞)m, Q f (r) ≤ [1 +

• 1≤i≤m

ri]b, for some b ∈ I Rm, f = (f1, ..., fm)t.

Application to the case of close di’s

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

◮ We may write with d := minj dj

∂t(

i ui) − d∆( i ui)

=

i(di − d)∆ui + i fi

≤ ∆ (

i(di − d)ui) .

Application to the case of close di’s

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

◮ We may write with d := minj dj

∂t(

i ui) − d∆( i ui)

=

i(di − d)∆ui + i fi

≤ ∆ (

i(di − d)ui) . ◮ We deduce from the Lp-Lemma that i uiLp(QT )

≤ C

i(di − d)uiLp(QT )

≤ C maxi{di − d}

i uiLp(QT ).

Application to the case of close di’s

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

◮ We may write with d := minj dj

∂t(

i ui) − d∆( i ui)

=

i(di − d)∆ui + i fi

≤ ∆ (

i(di − d)ui) . ◮ We deduce from the Lp-Lemma that i uiLp(QT )

≤ C

i(di − d)uiLp(QT )

≤ C maxi{di − d}

i uiLp(QT ). ◮ If M := C maxi{di − d} < 1, we deduce

• i

uiLp(QT ) ≤ (1 − M)−1 < +∞.

Application to the case of close di’s

(S)        ∀i = 1, ..., m ∂t ui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

:

◮ We may write with d := minj dj

∂t(

i ui) − d∆( i ui)

=

i(di − d)∆ui + i fi

≤ ∆ (

i(di − d)ui) . ◮ We deduce from the Lp-Lemma that i uiLp(QT )

≤ C

i(di − d)uiLp(QT )

≤ C maxi{di − d}

i uiLp(QT ). ◮ If M := C maxi{di − d} < 1, we deduce

• i

uiLp(QT ) ≤ (1 − M)−1 < +∞.

◮ Whence global existence if fi at most polynomial.

Extensions and limits of the Lp-approach

◮ Lp-approach does not apply to

• ∂tu1 − d1∆u1 = −u1eu2

2

∂tu2 − d2∆u2 = u1eu2

2

Extensions and limits of the Lp-approach

◮ Lp-approach does not apply to

• ∂tu1 − d1∆u1 = −u1eu2

2

∂tu2 − d2∆u2 = u1eu2

2

◮ neither to the system

∂tu1 − d1∆u1 = u3

1u2 2 − u2 1 u3 2

∂tu2 − d2∆u2 = u2

1 u3 2 − u3 1u2 2

Extensions and limits of the Lp-approach

◮ Lp-approach does not apply to

• ∂tu1 − d1∆u1 = −u1eu2

2

∂tu2 − d2∆u2 = u1eu2

2

◮ neither to the system

∂tu1 − d1∆u1 = u3

1u2 2 − u2 1 u3 2

∂tu2 − d2∆u2 = u2

1 u3 2 − u3 1u2 2

◮ and even not to the ”better” system with λ ∈ (0, 1)

• ∂tu1 − d1∆u1 = λu3

1u2 2 − u2 1 u3 2 [=: f1(u)]

∂tu2 − d2∆u2 = u2

1 u3 2 − u3 1u2 2 [=: f2(u)]

where : f1(u) + f2(u) ≤ 0, and also : f1(u) + λf2(u) ≤ 0

Finite time L∞-blow up may appear!

• ∂tu1 − d1∆u1 = f1(u1, u2)

∂tu2 − d2∆u2 = f2(u1, u2)

◮ Theorem: (D. Schmitt, MP) One can find polynomial

nonlinearities f , g satisfying (P) and (M) f + g ≤ 0, and also : ∃λ ∈ [0, 1[, f + λg ≤ 0, for which T ∗ < +∞ with lim

t→T ∗ u(t)L∞(Ω) = lim t→T ∗ v(t)L∞(Ω) = +∞.

Finite time L∞-blow up may appear!

• ∂tu1 − d1∆u1 = f1(u1, u2)

∂tu2 − d2∆u2 = f2(u1, u2)

◮ Theorem: (D. Schmitt, MP) One can find polynomial

nonlinearities f , g satisfying (P) and (M) f + g ≤ 0, and also : ∃λ ∈ [0, 1[, f + λg ≤ 0, for which T ∗ < +∞ with lim

t→T ∗ u(t)L∞(Ω) = lim t→T ∗ v(t)L∞(Ω) = +∞. ◮ Blow up may appear even in space dimension N = 1 (with

high degree polynomial nonlinearities)

Finite time L∞-blow up may appear!

• ∂tu1 − d1∆u1 = f1(u1, u2)

∂tu2 − d2∆u2 = f2(u1, u2)

◮ Theorem: (D. Schmitt, MP) One can find polynomial

nonlinearities f , g satisfying (P) and (M) f + g ≤ 0, and also : ∃λ ∈ [0, 1[, f + λg ≤ 0, for which T ∗ < +∞ with lim

t→T ∗ u(t)L∞(Ω) = lim t→T ∗ v(t)L∞(Ω) = +∞. ◮ Blow up may appear even in space dimension N = 1 (with

high degree polynomial nonlinearities)

◮ Blow up may appear with any superquadratic growth 2 + ǫ for

the fi (with high dimension N)

To proceed: Look for weak solutions which are allowed to go out of L∞(Ω) from time to time or even often (”Incomplete blow up”). We ask the nonlinearities to be at least in L1(QT): fi(u) ∈ L1(QT)

and the solution is understood in the sense of distributions or of Duhamel integral formula.

An L1-approach

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) ∂νui = 0 ui(0, ·) = u0

i (·) ≥ 0. ◮ L1-Theorem. Assume the two conditions (P)+ (M’) hold.

Assume moreover that the following a priori estimate holds: ∀i = 1, ..., m,

• QT

|fi(u)| ≤ C(T) < +∞, ∀T ∈ (0, +∞). Then, there exists a global weak solution for System (S), even for all u0 ∈ L1(Ω)+m !

An L1-approach

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) ∂νui = 0 ui(0, ·) = u0

i (·) ≥ 0. ◮ L1-Theorem. Assume the two conditions (P)+ (M’) hold.

Assume moreover that the following a priori estimate holds: ∀i = 1, ..., m,

• QT

|fi(u)| ≤ C(T) < +∞, ∀T ∈ (0, +∞). Then, there exists a global weak solution for System (S), even for all u0 ∈ L1(Ω)+m !

◮ Proof involves truncations Tk(r) := inf{r, k}, r ∈ [0, ∞) and

L1-type estimates like

• [0≤ui≤k]

di|∇ui|2 ≤ k

• |fi(u)| +
• Ω

u0

i

• .

L1-Theorem applies to many situations

• ∂tu1 − d1∆u1 = −u1eu2

2

∂tu2 − d2∆u2 = u1eu2

2

◮ Easy L1(QT)-estimate of the nonlinearity :

• Ω

u1(T) +

• QT

u1eu2

2 =

• Ω

u0

1.

L1-Theorem applies to many situations

• ∂tu1 − d1∆u1 = −u1eu2

2

∂tu2 − d2∆u2 = u1eu2

2

◮ Easy L1(QT)-estimate of the nonlinearity :

• Ω

u1(T) +

• QT

u1eu2

2 =

• Ω

u0

1. ◮ ⇒ Global existence of weak solutions

L1-Theorem applies to many situations

• ∂tu1 − d1∆u1 = −u1eu2

2

∂tu2 − d2∆u2 = u1eu2

2

◮ Easy L1(QT)-estimate of the nonlinearity :

• Ω

u1(T) +

• QT

u1eu2

2 =

• Ω

u0

1. ◮ ⇒ Global existence of weak solutions ◮ Existence holds for any u0 1, u0 2 ∈ L1(Ω)+ !!

L1-Theorem applies to many situations

• ∂tu1 − d1∆u1 = −u1eu2

2

∂tu2 − d2∆u2 = u1eu2

2

◮ Easy L1(QT)-estimate of the nonlinearity :

• Ω

u1(T) +

• QT

u1eu2

2 =

• Ω

u0

1. ◮ ⇒ Global existence of weak solutions ◮ Existence holds for any u0 1, u0 2 ∈ L1(Ω)+ !! ◮ Recall that the equation

∂tu2 − ∆u2 = eu2

2, u2(0) = u0

2,

does not have even local solutions in general when u0

2 ∈ L1(Ω)+ only.

L1-Theorem applies to many situations

• ∂tu1 − d1∆u1 = −u1eu2

2

∂tu2 − d2∆u2 = u1eu2

2

◮ Easy L1(QT)-estimate of the nonlinearity :

• Ω

u1(T) +

• QT

u1eu2

2 =

• Ω

u0

1. ◮ ⇒ Global existence of weak solutions ◮ Existence holds for any u0 1, u0 2 ∈ L1(Ω)+ !! ◮ Recall that the equation

∂tu2 − ∆u2 = eu2

2, u2(0) = u0

2,

does not have even local solutions in general when u0

2 ∈ L1(Ω)+ only. ◮ OPEN PROBLEM: are the solutions classical ?

L1-Theorem applies to many situations

◮ Let λ ∈ (−∞, 1) and

∂tu1 − d1∆u1 = λu3

1u2 2 − u2 1 u3 2 in QT,

∂tu2 − d2∆u2 = u2

1 u3 2 − u3 1u2 2 in QT

L1-Theorem applies to many situations

◮ Let λ ∈ (−∞, 1) and

∂tu1 − d1∆u1 = λu3

1u2 2 − u2 1 u3 2 in QT,

∂tu2 − d2∆u2 = u2

1 u3 2 − u3 1u2 2 in QT ◮ λ ∈ (−∞, 0] ⇒ OK ! (triangular structure).

L1-Theorem applies to many situations

◮ Let λ ∈ (−∞, 1) and

∂tu1 − d1∆u1 = λu3

1u2 2 − u2 1 u3 2 in QT,

∂tu2 − d2∆u2 = u2

1 u3 2 − u3 1u2 2 in QT ◮ λ ∈ (−∞, 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0, 1) ?

• Ω

(u1 + u2)(T) = −(1 − λ)

• QT

u3

1u2 2 +

• Ω

u0

1 + u0 2,

• Ω

(u1 + λu2) = −(1 − λ)

• QT

u2

1u3 2 +

• Ω

u0

1 + λu0 2.

L1-Theorem applies to many situations

◮ Let λ ∈ (−∞, 1) and

∂tu1 − d1∆u1 = λu3

1u2 2 − u2 1 u3 2 in QT,

∂tu2 − d2∆u2 = u2

1 u3 2 − u3 1u2 2 in QT ◮ λ ∈ (−∞, 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0, 1) ?

• Ω

(u1 + u2)(T) = −(1 − λ)

• QT

u3

1u2 2 +

• Ω

u0

1 + u0 2,

• Ω

(u1 + λu2) = −(1 − λ)

• QT

u2

1u3 2 +

• Ω

u0

1 + λu0 2. ◮ ⇒

• QT u3

1u2 2,

• QT u2

1u3 2 ≤ C ⇒ Global weak solutions

L1-Theorem applies to many situations

More generally, the same method applies if there exists an invertible matrix Q with nonnegative entries such that ∀r ∈ [0, ∞)m, Q f (r) ≤ [1 +

• 1≤i≤m

ri]b, for some b ∈ I Rm, f = (f1, ..., fm)t. In other words: there are m independent inequalities between the fi’s (not necessarily triangular)

L1-Theorem does not solve everything

◮ Let λ ∈ (−∞, 1) and

∂tu − d1∆u = λu3v2 − u2 v3 in QT, ∂tv − d2∆v = u2 v3 − u3v2 in QT

◮ λ ∈ (−∞, 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0, 1) ⇒ Global weak solutions

L1-Theorem does not solve everything

◮ Let λ ∈ (−∞, 1) and

∂tu − d1∆u = λu3v2 − u2 v3 in QT, ∂tv − d2∆v = u2 v3 − u3v2 in QT

◮ λ ∈ (−∞, 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0, 1) ⇒ Global weak solutions ◮ OPEN PROBLEMS

◮ For λ ∈ (0, 1): is the solution classical ???(i.e. bounded)

L1-Theorem does not solve everything

◮ Let λ ∈ (−∞, 1) and

∂tu − d1∆u = λu3v2 − u2 v3 in QT, ∂tv − d2∆v = u2 v3 − u3v2 in QT

◮ λ ∈ (−∞, 0] ⇒ OK ! (triangular structure). ◮ λ ∈ (0, 1) ⇒ Global weak solutions ◮ OPEN PROBLEMS

◮ For λ ∈ (0, 1): is the solution classical ???(i.e. bounded) ◮ What happens for λ = 1???

A surprising L2-estimate for these systems

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

◮ L2-Theorem. Assume (P)+(M’). Then, the following a

priori estimate holds for the solutions of (S): ∀T > 0,

• QT

m

• i=1

u2

i ≤ C

• Ω

m

• i=1

u0

i

2 , C = C(T, (di)).

A surprising L2-estimate for these systems

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0.

◮ L2-Theorem. Assume (P)+(M’). Then, the following a

priori estimate holds for the solutions of (S): ∀T > 0,

• QT

m

• i=1

u2

i ≤ C

• Ω

m

• i=1

u0

i

2 , C = C(T, (di)).

◮ The proof uses only the sum of the equations

∂t(

• i

ui) − ∆(

• i

diui) ≤ 0.

Idea of the proof of the L2-estimate

◮

∂t

• i

ui

• − ∆
• i

diui

• ≤ 0.

Idea of the proof of the L2-estimate

◮

∂t

• i

ui

• − ∆
• i

diui

• ≤ 0.

◮

∂tW − ∆ (a W ) ≤ 0, W =

• i

ui a =

• i diui
• i ui

Idea of the proof of the L2-estimate

◮

∂t

• i

ui

• − ∆
• i

diui

• ≤ 0.

◮

∂tW − ∆ (a W ) ≤ 0, W =

• i

ui a =

• i diui
• i ui

◮

0 ≤ min

i

di ≤ a =

• i diui
• i ui

≤ max

i

di < +∞

Idea of the proof of the L2-estimate

◮

∂t

• i

ui

• − ∆
• i

diui

• ≤ 0.

◮

∂tW − ∆ (a W ) ≤ 0, W =

• i

ui a =

• i diui
• i ui

◮

0 ≤ min

i

di ≤ a =

• i diui
• i ui

≤ max

i

di < +∞

◮ The operator W → ∂tW − ∆(aW ) is not of divergence form

and a is not continuous, but bounded from above and from below so that the operator is parabolic and this implies W L2(QT ) ≤ CW0L2(Ω).

Idea of the proof of the L2-estimate

◮

∂t

• i

ui

• − ∆
• i

diui

• ≤ 0.

◮

∂tW − ∆ (a W ) ≤ 0, W =

• i

ui a =

• i diui
• i ui

◮

0 ≤ min

i

di ≤ a =

• i diui
• i ui

≤ max

i

di < +∞

◮ The operator W → ∂tW − ∆(aW ) is not of divergence form

and a is not continuous, but bounded from above and from below so that the operator is parabolic and this implies W L2(QT ) ≤ CW0L2(Ω).

◮ Seen on the dual operator ψ → − (∂tψ + a∆ψ) which is

regularizing and satisfies L2-maximal regularity.

Extensions of the L2-estimate for such systems

◮ Variable coefficients di = di(t, x),

nonlinear diffusions −∆di(ui)

Extensions of the L2-estimate for such systems

◮ Variable coefficients di = di(t, x),

nonlinear diffusions −∆di(ui)

◮ W0 ∈ L1(Ω) only !: The L2-estimate may be localized for

∂tW − ∆(aW ) ≤ 0, d ≤ a ≤ d. W L2((τ,T)×Ω) ≤ C(d, d, T) τ N/4 W0L1(Ω).

Extensions of the L2-estimate for such systems

◮ Variable coefficients di = di(t, x),

nonlinear diffusions −∆di(ui)

◮ W0 ∈ L1(Ω) only !: The L2-estimate may be localized for

∂tW − ∆(aW ) ≤ 0, d ≤ a ≤ d. W L2((τ,T)×Ω) ≤ C(d, d, T) τ N/4 W0L1(Ω).

◮ [J.A. Ca˜

nizo, L. Desvillettes, K. Fellner]: there exists ǫ(N) > 0 such that W L2+ǫ(QT ) ≤ CW0L2+ǫ(Ω).

Applications to quadratic systems

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0, u0 ∈ L1(Ω)m.

◮ Corollary of the L1 and L2 Theorems. Assume (P)+(M’) and

the fi are at most quadratic, i.e. ∀1 ≤ i ≤ m, ∀r ∈ [0, ∞)2, |fi(r)| ≤ C[1 +

• r 2

j ].

Then, (S) has a global weak solution.

Applications to quadratic systems

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0, u0 ∈ L1(Ω)m.

◮ Corollary of the L1 and L2 Theorems. Assume (P)+(M’) and

the fi are at most quadratic, i.e. ∀1 ≤ i ≤ m, ∀r ∈ [0, ∞)2, |fi(r)| ≤ C[1 +

• r 2

j ].

Then, (S) has a global weak solution.

◮ Proof if u0

i ∈ L2(Ω). By L2-estimate + at most quadratic growth

• f fi, we have the a priori estimate for the solutions of (S)

∀1 ≤ i ≤ m,

• QT

|fi(u)| ≤ C(T), Then, we apply the L1-theorem.

Applications to quadratic systems

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in QT ∂νui = 0

• n ΣT

ui(0, ·) = u0

i (·) ≥ 0, u0 ∈ L1(Ω)m.

◮ Corollary of the L1 and L2 Theorems. Assume (P)+(M’) and

the fi are at most quadratic, i.e. ∀1 ≤ i ≤ m, ∀r ∈ [0, ∞)2, |fi(r)| ≤ C[1 +

• r 2

j ].

Then, (S) has a global weak solution.

◮ Proof if u0

i ∈ L2(Ω). By L2-estimate + at most quadratic growth

• f fi, we have the a priori estimate for the solutions of (S)

∀1 ≤ i ≤ m,

• QT

|fi(u)| ≤ C(T), Then, we apply the L1-theorem.

◮ Proof when u0

i ∈ L1(Ω): use the local L2-estimates and work a

little more near t = 0.

Application to the Lotka-Volterra system

◮ Applies to the (quadratic) Lotka-Volterra system: for all

u0 ∈ L1(Ω)+m, there exists a global weak solution to the system: For all i = 1, ..., m, ∂tui − di∆ui = eiui +  

1≤j≤m

pijuj   ui =: fi(u), where ei ∈ I R, pij ∈ I R + ”Dissipation”.

Application to the Lotka-Volterra system

◮ Applies to the (quadratic) Lotka-Volterra system: for all

u0 ∈ L1(Ω)+m, there exists a global weak solution to the system: For all i = 1, ..., m, ∂tui − di∆ui = eiui +  

1≤j≤m

pijuj   ui =: fi(u), where ei ∈ I R, pij ∈ I R + ”Dissipation”.

◮ ”Dissipation”: we assume that, for some (ai) ∈ (0, +∞)m

∀ξ ∈ [0, ∞)m,

m

• i,j=1

aipijξiξj ≤ 0 ⇒

• i

aifi(u) ≤

• i

aieiui (whence (M’))

Applications to chemical quadratic systems

◮ ui = ui(t, x) = concentration of Ui

U1 + U3 k+ ⇋ k− U2 + U4 for i = 1, 2, 3, 4, ∂tui − di∆ui = (−1)i(k+u1u3 − k−u2u4).

Applications to chemical quadratic systems

◮ ui = ui(t, x) = concentration of Ui

U1 + U3 k+ ⇋ k− U2 + U4 for i = 1, 2, 3, 4, ∂tui − di∆ui = (−1)i(k+u1u3 − k−u2u4).

◮ L1 + L2-theorem ⇒ Global weak solutions for u0 ∈ L1(Ω)+4.

Applications to chemical quadratic systems

◮ ui = ui(t, x) = concentration of Ui

U1 + U3 k+ ⇋ k− U2 + U4 for i = 1, 2, 3, 4, ∂tui − di∆ui = (−1)i(k+u1u3 − k−u2u4).

◮ L1 + L2-theorem ⇒ Global weak solutions for u0 ∈ L1(Ω)+4. ◮ See also [Desvillettes-Fellner-P-Vovelle] using entropy decay

• i

fi(u) log ui ≤ 0 (+u0

i log u0 i ∈ L1(Ω))

Applications to chemical quadratic systems

◮ ui = ui(t, x) = concentration of Ui

U1 + U3 k+ ⇋ k− U2 + U4 for i = 1, 2, 3, 4, ∂tui − di∆ui = (−1)i(k+u1u3 − k−u2u4).

◮ L1 + L2-theorem ⇒ Global weak solutions for u0 ∈ L1(Ω)+4. ◮ See also [Desvillettes-Fellner-P-Vovelle] using entropy decay

• i

fi(u) log ui ≤ 0 (+u0

i log u0 i ∈ L1(Ω))

◮ What about global classical solutions when u0 ∈ L∞(Ω)m?

Applications to chemical quadratic systems

◮ ui = ui(t, x) = concentration of Ui

U1 + U3 k+ ⇋ k− U2 + U4 for i = 1, 2, 3, 4, ∂tui − di∆ui = (−1)i(k+u1u3 − k−u2u4).

◮ L1 + L2-theorem ⇒ Global weak solutions for u0 ∈ L1(Ω)+4. ◮ See also [Desvillettes-Fellner-P-Vovelle] using entropy decay

• i

fi(u) log ui ≤ 0 (+u0

i log u0 i ∈ L1(Ω))

◮ What about global classical solutions when u0 ∈ L∞(Ω)m?

◮ OK if N = 1, 2 [Pruess;Goudon-Vasseur;

Ca˜ nizo-Desvillettes-Fellner]

Applications to chemical quadratic systems

◮ ui = ui(t, x) = concentration of Ui

U1 + U3 k+ ⇋ k− U2 + U4 for i = 1, 2, 3, 4, ∂tui − di∆ui = (−1)i(k+u1u3 − k−u2u4).

◮ L1 + L2-theorem ⇒ Global weak solutions for u0 ∈ L1(Ω)+4. ◮ See also [Desvillettes-Fellner-P-Vovelle] using entropy decay

• i

fi(u) log ui ≤ 0 (+u0

i log u0 i ∈ L1(Ω))

◮ What about global classical solutions when u0 ∈ L∞(Ω)m?

◮ OK if N = 1, 2 [Pruess;Goudon-Vasseur;

Ca˜ nizo-Desvillettes-Fellner]

◮ With Ω = I

RN: OK for any N [Kanel; Caputo-Goudon-Vasseur]

Applications to chemical quadratic systems

◮ ui = ui(t, x) = concentration of Ui

U1 + U3 k+ ⇋ k− U2 + U4 for i = 1, 2, 3, 4, ∂tui − di∆ui = (−1)i(k+u1u3 − k−u2u4).

◮ L1 + L2-theorem ⇒ Global weak solutions for u0 ∈ L1(Ω)+4. ◮ See also [Desvillettes-Fellner-P-Vovelle] using entropy decay

• i

fi(u) log ui ≤ 0 (+u0

i log u0 i ∈ L1(Ω))

◮ What about global classical solutions when u0 ∈ L∞(Ω)m?

◮ OK if N = 1, 2 [Pruess;Goudon-Vasseur;

Ca˜ nizo-Desvillettes-Fellner]

◮ With Ω = I

RN: OK for any N [Kanel; Caputo-Goudon-Vasseur]

◮ With Ω bounded :OK for any N [Ph. Souplet] (very recent)

Global classical solutions for L2 growth [Ph. Souplet]

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in (0, +∞) × Ω ∂νui = 0 on ∂Ω, ui(0, ·) = u0

i (·) ≥ 0 ∈ L∞(Ω)+m.

Assume that for all r ∈ [0, +∞)m

◮ Quadratic growth

|fi(r)| ≤ C[1 +

• i

(ri)2],

Global classical solutions for L2 growth [Ph. Souplet]

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in (0, +∞) × Ω ∂νui = 0 on ∂Ω, ui(0, ·) = u0

i (·) ≥ 0 ∈ L∞(Ω)+m.

Assume that for all r ∈ [0, +∞)m

◮ Quadratic growth

|fi(r)| ≤ C[1 +

• i

(ri)2],

◮ Mass dissipation

• i

fi(u) ≤ 0,

Global classical solutions for L2 growth [Ph. Souplet]

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in (0, +∞) × Ω ∂νui = 0 on ∂Ω, ui(0, ·) = u0

i (·) ≥ 0 ∈ L∞(Ω)+m.

Assume that for all r ∈ [0, +∞)m

◮ Quadratic growth

|fi(r)| ≤ C[1 +

• i

(ri)2],

◮ Mass dissipation

• i

fi(u) ≤ 0,

◮ Entropy dissipation

• i

fi(u) log ui ≤ 0.

Global classical solutions for L2 growth [Ph. Souplet]

(S)        ∀i = 1, ..., m ∂tui − di∆ui = fi(u1, u2, ..., um) in (0, +∞) × Ω ∂νui = 0 on ∂Ω, ui(0, ·) = u0

i (·) ≥ 0 ∈ L∞(Ω)+m.

Assume that for all r ∈ [0, +∞)m

◮ Quadratic growth

|fi(r)| ≤ C[1 +

• i

(ri)2],

◮ Mass dissipation

• i

fi(u) ≤ 0,

◮ Entropy dissipation

• i

fi(u) log ui ≤ 0.

◮ Then, System (S) has a global classical solution for all N.

What about general (=highly superquadratic) chemical reactions?

p1U1 + p2U2 + ... + pmUm k+ ⇋ k− q1U1 + q2U2 + ... + qmUm,

◮ The general associated reaction-diffusion system is

∀1 ≤ i ≤ m, ∂tui − di∆ui = (pi − qi)[k−

j

uqj

j − k+ j

upj

j ]

What about general (=highly superquadratic) chemical reactions?

p1U1 + p2U2 + ... + pmUm k+ ⇋ k− q1U1 + q2U2 + ... + qmUm,

◮ The general associated reaction-diffusion system is

∀1 ≤ i ≤ m, ∂tui − di∆ui = (pi − qi)[k−

j

uqj

j − k+ j

upj

j ] ◮ Existence of global classical and even weak solutions is open

in general.

What about general (=highly superquadratic) chemical reactions?

p1U1 + p2U2 + ... + pmUm k+ ⇋ k− q1U1 + q2U2 + ... + qmUm,

◮ The general associated reaction-diffusion system is

∀1 ≤ i ≤ m, ∂tui − di∆ui = (pi − qi)[k−

j

uqj

j − k+ j

upj

j ] ◮ Existence of global classical and even weak solutions is open

in general.

◮ Existence of global so-called ”renormalized solutions” ”`

a la DiPerna-Lions”, is proved by [J. Fischer] for homogeneous Neumann boundary conditions and u0 log u0 ∈ L1(Ω)

Existence of global renormalized solutions

   ∀1 ≤ i ≤ m, ∂tui − di∆ui = (pi − qi)[

j u qj j − j u pj j ] := fi(u),

∂νui = 0 on Σ∞(precisely!) ui(0) = u0

i ≥ 0 with u0 i log u0 i ∈ L1(Ω).

◮ Let uǫ be solution of ∂tuǫ − di∆uǫ = fi(uǫ)/[1 + ǫ

j |fj(uǫ)|].

Existence of global renormalized solutions

   ∀1 ≤ i ≤ m, ∂tui − di∆ui = (pi − qi)[

j u qj j − j u pj j ] := fi(u),

∂νui = 0 on Σ∞(precisely!) ui(0) = u0

i ≥ 0 with u0 i log u0 i ∈ L1(Ω).

◮ Let uǫ be solution of ∂tuǫ − di∆uǫ = fi(uǫ)/[1 + ǫ

j |fj(uǫ)|].

◮ THEOREM [J. Fischer, ’14] The approximate solution uǫ

converges (up to a subsequence) a.e. on Q∞ to some u with

ui ∈ L∞(0, T; L1(Ω)), √ui ∈ L2(0, T; H1(Ω)), such that for all ξ : [0, ∞)m → I R with Dξ compactly supported ∂tξ(u) =

• i

∂iξ(u)∂tui =

• i

∂iξ(u)[di∆ui + fi(u)],

in the sense of distributions.

Existence of global renormalized solutions

   ∀1 ≤ i ≤ m, ∂tui − di∆ui = (pi − qi)[

j u qj j − j u pj j ] := fi(u),

∂νui = 0 on Σ∞(precisely!) ui(0) = u0

i ≥ 0 with u0 i log u0 i ∈ L1(Ω).

◮ Let uǫ be the solution of ∂tuǫ − di∆uǫ = fi(uǫ)/[1 + ǫ

j |fj(uǫ)|].

◮ THEOREM [J. Fischer, ’14] The approximate solution uǫ

converges (up to a subsequence) a.e. on Q∞ to some u with

ui ∈ L∞(0, T; L1(Ω)), √ui ∈ L2(0, T; H1(Ω)), such that for all ξ : [0, ∞)m → I R with Dξ compactly supported ∂tξ(u) =

• i

∂iξ(u)∂tui =

• i

∂iξ(u)[di∆ui + fi(u)],

◮ What about weak solutions?? ⇔ fi(u) ∈ L1(QT) ??? ◮ A fortiori: what about classical solutions ???

Or counterexamples ???

◮ Case of Dirichlet boundary conditions ?

Extensions to nonlinear (degenerate) diffusions

◮ What about systems with nonlinear diffusions?

(Sβ)        ∀i = 1, ..., m ∂tui − ∆(uβi

i ) = fi(u1, u2, ..., um),

βi ∈ [1, +∞) ui = 0 ui(0, ·) = u0

i (·) ≥ 0

Extensions to nonlinear (degenerate) diffusions

◮ What about systems with nonlinear diffusions?

(Sβ)        ∀i = 1, ..., m ∂tui − ∆(uβi

i ) = fi(u1, u2, ..., um),

βi ∈ [1, +∞) ui = 0 ui(0, ·) = u0

i (·) ≥ 0 ◮ An extension of the L1-theorem: [E. Laamri, M.P. ’15] if f

satisfies (P)+(M) and

• QT |fi(u)| ≤ C, and if βi ∈ [1, 2),

then global existence of weak solutions holds.

Extensions to nonlinear (degenerate) diffusions

◮ What about systems with nonlinear diffusions?

(Sβ)        ∀i = 1, ..., m ∂tui − ∆(uβi

i ) = fi(u1, u2, ..., um),

βi ∈ [1, +∞) ui = 0 ui(0, ·) = u0

i (·) ≥ 0 ◮ An extension of the L1-theorem: [E. Laamri, M.P. ’15] if f

satisfies (P)+(M) and

• QT |fi(u)| ≤ C, and if βi ∈ [1, 2),

then global existence of weak solutions holds.

◮ Proof similar to the linear case, with L1-compactness for the

porous media operator, use of truncations, ...But one has only

• [ui≤k] |∇ui|2 ≤ C k2−βi.

Extensions to nonlinear (degenerate) diffusions

◮ What about systems with nonlinear diffusions?

(Sβ)        ∀i = 1, ..., m ∂tui − ∆(uβi

i ) = fi(u1, u2, ..., um),

βi ∈ [1, +∞) ui = 0 ui(0, ·) = u0

i (·) ≥ 0 ◮ An extension of the L1-theorem: [E. Laamri, M.P. ’15] if f

satisfies (P)+(M) and

• QT |fi(u)| ≤ C, and if βi ∈ [1, 2),

then global existence of weak solutions holds.

◮ Proof similar to the linear case, with L1-compactness for the

porous media operator, use of truncations, ...But one has only

• [ui≤k] |∇ui|2 ≤ C k2−βi.

◮ OPEN problem: What happens when βi ≥ 2 ?

Extensions to nonlinear (degenerate) diffusions

◮ What about systems with nonlinear diffusions?

(Sβ)        ∀i = 1, ..., m ∂tui − ∆(uβi

i ) = fi(u1, u2, ..., um),

βi ∈ [1, +∞) ui = 0 ui(0, ·) = u0

i (·) ≥ 0 ◮ An extension of the L1-theorem: [E. Laamri, M.P. ’15] if f

satisfies (P)+(M) and

• QT |fi(u)| ≤ C, and if βi ∈ [1, 2),

then global existence of weak solutions holds.

◮ Proof similar to the linear case, with L1-compactness for the

porous media operator, use of truncations, ...But one has only

• [ui≤k] |∇ui|2 ≤ C k2−βi.

◮ OPEN problem: What happens when βi ≥ 2 ? ◮ ...Some partial (significant) results...

Extensions to nonlinear (degenerate) diffusions

◮ Consider the (favorite) case of reversible chemical reactions

U1 + U3 k+ ⇋ k− U2 + U4 modelized by ’Mass Action Law’ for the reactive terms and Nonlinear porous media type Diffusion: (S)        ∀i = 1, 2, 3, 4 ∂tui − ∆(uβi

i ) = (−1)i[u1u3 − u2u4]

ui = 0 ui(0, ·) = u0

i (·) ≥ 0, u0 i ∈ L∞(Ω).

Extensions to nonlinear (degenerate) diffusions

◮ Consider the (favorite) case of reversible chemical reactions

U1 + U3 k+ ⇋ k− U2 + U4 modelized by ’Mass Action Law’ for the reactive terms and Nonlinear porous media type Diffusion: (S)        ∀i = 1, 2, 3, 4 ∂tui − ∆(uβi

i ) = (−1)i[u1u3 − u2u4]

ui = 0 ui(0, ·) = u0

i (·) ≥ 0, u0 i ∈ L∞(Ω).

◮ Theorem. [E. Laamri, M.P. ’15] System (S) has a global weak

solution for all βi ∈ [1, ∞).

Extensions to nonlinear (degenerate) diffusions

◮ Consider the (favorite) case of reversible chemical reactions

U1 + U3 k+ ⇋ k− U2 + U4 modelized by ’Mass Action Law’ for the reactive terms and Nonlinear porous media type Diffusion: (S)        ∀i = 1, 2, 3, 4 ∂tui − ∆(uβi

i ) = (−1)i[u1u3 − u2u4]

ui = 0 ui(0, ·) = u0

i (·) ≥ 0, u0 i ∈ L∞(Ω).

◮ Theorem. [E. Laamri, M.P. ’15] System (S) has a global weak

solution for all βi ∈ [1, ∞).

◮ One of the ideas is that the L2(QT)-estimate is replaced by an

Lβi+1(QT)-estimate for ui. The proof is the same as for the L2-estimate.

Extensions to nonlinear (degenerate) diffusions

◮ More generally, for the model of reversible reaction

p1U1 + p2U2 + ... + pmUm k+ ⇋ k− q1U1 + q2U2 + ... + qmUm, ∀1 ≤ i ≤ m, ∂tui − ∆(uβi

i ) = (pi − qi)[

• j

uqj

j −

• j

upj

j ]

Global existence of weak solutions holds if [E. Laamri,M.P. ’15]

• i

pi βi + 1 ≤ 1,

• i

qi βi + 1 ≤ 1.

Extensions to nonlinear (degenerate) diffusions

◮ More generally, for the model of reversible reaction

p1U1 + p2U2 + ... + pmUm k+ ⇋ k− q1U1 + q2U2 + ... + qmUm, ∀1 ≤ i ≤ m, ∂tui − ∆(uβi

i ) = (pi − qi)[

• j

uqj

j −

• j

upj

j ]

Global existence of weak solutions holds if [E. Laamri,M.P. ’15]

• i

pi βi + 1 ≤ 1,

• i

qi βi + 1 ≤ 1.

◮ What about existence of weak solutions in general when

L1(QT)-bounds hold ?

Extensions to nonlinear (degenerate) diffusions

◮ More generally, for the model of reversible reaction

p1U1 + p2U2 + ... + pmUm k+ ⇋ k− q1U1 + q2U2 + ... + qmUm, ∀1 ≤ i ≤ m, ∂tui − ∆(uβi

i ) = (pi − qi)[

• j

uqj

j −

• j

upj

j ]

Global existence of weak solutions holds if [E. Laamri,M.P. ’15]

• i

pi βi + 1 ≤ 1,

• i

qi βi + 1 ≤ 1.

◮ What about existence of weak solutions in general when

L1(QT)-bounds hold ?

◮ What about existence of at least renormalized solutions in

general ?

Approximating by stationary reaction-diffusion systems

◮ Recall that the implicit time-discretisation of

∂tui − di∆uβi

i

= fi(u), i = 1, ..., m, is given, for h := tn+1 − tn by ui(tn+1) − h di∆ui(tn+1)βi = h fi(u(tn+1)) + ui(tn).

Approximating by stationary reaction-diffusion systems

◮ Recall that the implicit time-discretisation of

∂tui − di∆uβi

i

= fi(u), i = 1, ..., m, is given, for h := tn+1 − tn by ui(tn+1) − h di∆ui(tn+1)βi = h fi(u(tn+1)) + ui(tn).

◮ ⇒ Question of existence for stationary reaction-diffusion

systems: for all i = 1, ..., m ui − δi∆uβi

i

= Fi(u) + gi, gi ∈ L1(Ω)+, with nonlinearities Fi as before and various boundary conditions.

Approximating by stationary reaction-diffusion systems

◮ Recall that the implicit time-discretisation of

∂tui − di∆uβi

i

= fi(u), i = 1, ..., m, is given, for h := tn+1 − tn by ui(tn+1) − h di∆ui(tn+1)βi = h fi(u(tn+1)) + ui(tn).

◮ ⇒ Question of existence for stationary reaction-diffusion

systems: for all i = 1, ..., m ui − δi∆uβi

i

= Fi(u) + gi, gi ∈ L1(Ω)+, with nonlinearities Fi as before and various boundary conditions.

◮ Or more generally

ui + Aiui = Fi(u) + gi, gi ∈ L1(Ω)+, where Ai are ”good” nonlinear diffusion operators in L1(Ω) (=m-accretive).

Some results on stationary reaction-diffusion systems [E.

Laamri-M.P.]

◮ (1) Existence of weak solutions (Fi(u) ∈ L1) for

ui + Aiui = Fi(u) + gi, gi ∈ L1(Ω)+, with ”good” m-accretive operators Ai in L1 and (Fi) with m independent inequalities. Includes Aiui = −∆uβi

i , βi ∈ [1, +∞) with classical boundary

conditions.

Some results on stationary reaction-diffusion systems [E.

Laamri-M.P.]

◮ (1) Existence of weak solutions (Fi(u) ∈ L1) for

ui + Aiui = Fi(u) + gi, gi ∈ L1(Ω)+, with ”good” m-accretive operators Ai in L1 and (Fi) with m independent inequalities. Includes Aiui = −∆uβi

i , βi ∈ [1, +∞) with classical boundary

conditions.

◮ (2) For linear diffusions and ”chemical” nonlinearities

Fi(u) = λi[

• j

uqj

j −

• j

upj

j ], λi(pi − qi) > 0.

Existence of weak solutions for gi log gi ∈ L1(Ω) and m ≤ 5.

Some results on stationary reaction-diffusion systems [E.

Laamri-M.P.]

◮ (1) Existence of weak solutions (Fi(u) ∈ L1) for

ui + Aiui = Fi(u) + gi, gi ∈ L1(Ω)+, with ”good” m-accretive operators Ai in L1 and (Fi) with m independent inequalities. Includes Aiui = −∆uβi

i , βi ∈ [1, +∞) with classical boundary

conditions.

◮ (2) For linear diffusions and ”chemical” nonlinearities

Fi(u) = λi[

• j

uqj

j −

• j

upj

j ], λi(pi − qi) > 0.

Existence of weak solutions for gi log gi ∈ L1(Ω) and m ≤ 5.

◮ OPEN PROBLEMS:

• extension of (2) to any m, gi ∈ L1, nonlinear diffusions,
• exploit the L1-estimates on the stationary case to go back to

global existence for the evolution problem.

...and Chemical-Electro-Reaction-Diffusion...

◮ The Nernst-Planck system

   ∂tui − didiv (∇ui + ziui∇Φ) = fi(u), −∆φ =

i ziui,

+boundary conditions where zi ∈ I R = charge of the ionized species Ui and Φ is the electrical potential and same structure for the fi’s

...and Chemical-Electro-Reaction-Diffusion...

◮ The Nernst-Planck system

   ∂tui − didiv (∇ui + ziui∇Φ) = fi(u), −∆φ =

i ziui,

+boundary conditions where zi ∈ I R = charge of the ionized species Ui and Φ is the electrical potential and same structure for the fi’s

◮ Coupling of the previous kind of RD-systems with an electrical

environment ! Applications (among others) to the electrodeposition of nickel-iron alloy.

...and Chemical-Electro-Reaction-Diffusion...

◮ The Nernst-Planck system

   ∂tui − didiv (∇ui + ziui∇Φ) = fi(u), −∆φ =

i ziui,

+boundary conditions where zi ∈ I R = charge of the ionized species Ui and Φ is the electrical potential and same structure for the fi’s

◮ Coupling of the previous kind of RD-systems with an electrical

environment ! Applications (among others) to the electrodeposition of nickel-iron alloy.

◮ Much work on mathematical analysis and numerical

simulations in Marrakech. See e.g. the recent thesis defended in March by Fatima AQEL under the supervising of Noureddine Alaa. ...and many more interesting contributions...