SEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTIONweb.iku.edu.tr/ias/documents/ias_presentation.pdf · Evolution Equations Semigroup theory Kato's approach Examples

Evolution EquationsSemigroup theoryKato's approach

ExamplesReferences

SEMIGROUP APPROACH FOR PARTIAL

DIFFERENTIAL EQUATIONS OF EVOLUTION

Nilay Duruk Mutluba³

Istanbul Kemerburgaz University

Istanbul Analysis Seminars

24 October 2014

Sabanc� University Karaköy Communication Center

Nilay Duruk Mutluba³ Semigroup Approach


ExamplesReferences

1 Evolution Equations

2 Semigroup theory

3 Kato's approach

4 Examples

5 References



ExamplesReferences

Evolution Equations

u(x , t)

(i) Does it exist?

(ii) Is it unique?

(iii) Does it depend continuously on the initial data?

⇒ Well-posed?



ExamplesReferences

Evolution Equations

u(x , t)

(i) Does it exist?

(ii) Is it unique?


⇒ Well-posed?



ExamplesReferences

Evolution Equations

u(x , t)

(i) Does it exist?

(ii) Is it unique?


⇒ Well-posed?



ExamplesReferences

Evolution Equations

u(x , t)

(i) Does it exist?

(ii) Is it unique?


⇒ Well-posed?



ExamplesReferences

Evolution Equations

u(x , t)

(i) Does it exist?

(ii) Is it unique?


⇒ Well-posed?



ExamplesReferences

Solutions

Classical solution: A solution of a "PDE of order k" which is at

least k times continuously di�erentiable.

Weak solution!!

Sobolev space : Hilbert space which involves weak solutions

Hk(R).



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Solutions



Weak solution!!


Hk(R).



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Solutions



Weak solution!!


Hk(R).



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Simple example

du

dt= f (u), u(0) = u0 (1)

Two Banach spaces : X and Y , with Y ⊂ X .

f : continuous from Y to X .

Assume that for all u0, there exist a real number T > 0 and a

unique solution u ∈ C ([0,T ],Y ) satisfying equation (1).

⇒ dudt∈ C ([0,T ],X )).

Assume that u0 7→ u is continuous (Y 7→ C ([0,T ],Y ).⇒ locally well-posedness in Y .

Very strong!!



ExamplesReferences

Simple example

du

dt= f (u), u(0) = u0 (1)





⇒ dudt∈ C ([0,T ],X )).


Very strong!!



ExamplesReferences

Simple example

du

dt= f (u), u(0) = u0 (1)





⇒ dudt∈ C ([0,T ],X )).


Very strong!!



ExamplesReferences

Simple example

du

dt= f (u), u(0) = u0 (1)





⇒ dudt∈ C ([0,T ],X )).


Very strong!!



ExamplesReferences

Simple example

du

dt= f (u), u(0) = u0 (1)





⇒ dudt∈ C ([0,T ],X )).


Very strong!!



ExamplesReferences

Linear Evolution Equations

Homogeneous equation:

u′(t) + Au(t) = 0, 0 ≤ t ≤ T ,

u(0) = u0



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∀ u0, solution u exists and is unique:

u(t) = etAu0

Here, etA =∑∞

n=0(tA)n

n! .



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Nonhomogeneous equation:

u′(t) + Au(t) = f (t), 0 ≤ t ≤ T ,

u(0) = u0



ExamplesReferences


u(t) = etAu0 +

∫ t

0

e(t−s)Af (s)ds

Lions, Yosida, Pazy...

KATO! Semigroup theory!!



ExamplesReferences

De�nitions

De�nition

Let X be a Banach space. If the mapping T (t) : R+ → L(X )satis�es the following properties, then it is a C0 semigroup:

(i) T (0) = I

(ii) ∀ t, s ≥ 0, T (t + s) = T (t)T (s)

(iii) limt→0 T (t)x = x , x ∈ X .



ExamplesReferences

De�nition

De�nition

Let {T (t)}t≥0 be a C0 semigroup.

If the limit Ax := limt→0T (t)x−x

tis de�ned, then the in�nitesimal

generator of T (t) is A. Domain of A is denoted by D(A):

D(A) := {x ∈ X : limt→0

T (t)x − x

tlimit is in X .}



ExamplesReferences

Cauchy Problem

u′(t) = Au(t), u(0) = u0,

(i) If a continuously di�erentiable function u satis�es u(t) ∈ D(A)for all t ≥ 0 and satis�es the equation, then it is a classical

solution,

(ii) If for a continuous function u,∫ t

0u(s)ds ∈ D(A) and

A∫ t

0u(s)ds = u(t)− x , then it is a mild solution.



ExamplesReferences

Cauchy Problem

u′(t) = Au(t), u(0) = u0,


solution,



A∫ t




ExamplesReferences

Cauchy Problem

u′(t) = Au(t), u(0) = u0,


solution,



A∫ t




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Idea

u′(t) + A(u, t)u = 0, u(0) = u0

→ u(t) = etAu0 → u(t) = T (t)u0

u′(t) + A(ω, t)u = 0, u(0) = u0

→ u(t) = Tω(t)u0ω 7→ Φ(ω) = Tω(t)u0 = u

Φ(ω) contraction mapping ⇒ unique solution u (Banach �xed

point theorem)



ExamplesReferences

Idea

u′(t) + A(u, t)u = 0, u(0) = u0

→ u(t) = etAu0 → u(t) = T (t)u0

u′(t) + A(ω, t)u = 0, u(0) = u0

→ u(t) = Tω(t)u0

ω 7→ Φ(ω) = Tω(t)u0 = u


point theorem)



ExamplesReferences

Idea

u′(t) + A(u, t)u = 0, u(0) = u0

→ u(t) = etAu0 → u(t) = T (t)u0

u′(t) + A(ω, t)u = 0, u(0) = u0

→ u(t) = Tω(t)u0ω 7→ Φ(ω) = Tω(t)u0 = u


point theorem)



ExamplesReferences

Notation

B(X ;Y ): Set of all bounded linear operators on X to Y

G (X ): All negative generators of C0 semigroups on X

G (X , µ, β): −A generates a C0 semigroup {e−tA} with||e−tA|| ≤ µeβt , 0 ≤ t <∞

A ∈ G (X , 1, β) ⇒ quasi-m-accretive



ExamplesReferences

Notation

B(X ;Y ): Set of all bounded linear operators on X to Y

G (X ): All negative generators of C0 semigroups on X

G (X , µ, β): −A generates a C0 semigroup {e−tA} with||e−tA|| ≤ µeβt , 0 ≤ t <∞

A ∈ G (X , 1, β) ⇒ quasi-m-accretive



ExamplesReferences

Heat equation

ut − κuxx = 0, u(x , 0) = u0(x), x ∈ R

Solution by classical method:

u(x , t) =1√4πκt

∫ ∞−∞

e−(x−y)2

4κt u0(y)dy

By new notation,

ut + Au = 0, u(0) = u0

where A = −κD2x and T (t)u0 = 1√

4πκt(e−

x2

4κt ∗ u0)(x).

For all t, ||u(t)|| ≤ u0 ⇒ −κD2x ∈ G (L2, 1, 0).



ExamplesReferences

Heat equation

ut − κuxx = 0, u(x , 0) = u0(x), x ∈ R


u(x , t) =1√4πκt

∫ ∞−∞

e−(x−y)2

4κt u0(y)dy

By new notation,

ut + Au = 0, u(0) = u0


4πκt(e−

x2

4κt ∗ u0)(x).




ExamplesReferences

Heat equation

ut − κuxx = 0, u(x , 0) = u0(x), x ∈ R


u(x , t) =1√4πκt

∫ ∞−∞

e−(x−y)2

4κt u0(y)dy

By new notation,

ut + Au = 0, u(0) = u0


4πκt(e−

x2

4κt ∗ u0)(x).




ExamplesReferences

Heat equation

ut − κuxx = 0, u(x , 0) = u0(x), x ∈ R


u(x , t) =1√4πκt

∫ ∞−∞

e−(x−y)2

4κt u0(y)dy

By new notation,

ut + Au = 0, u(0) = u0


4πκt(e−

x2

4κt ∗ u0)(x).




ExamplesReferences

General approach

Evolution equation depending on time variable t and spatial

variable x

→ Ordinary di�erential equation depending on time variable t in an

in�nite dimensional Banach space

Choose an appropriate Banach space.

Choose the corresponding operator A.

Spatial derivatives and other restrictions are captured by either

the Banach space or the operator.



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General approach


variable x









ExamplesReferences

General approach


variable x









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Quasi-linear equations

ut + A(u)u = f (u), t ≥ 0, u(0) = u0. (2)

X and Y Banach spaces

Y ⊂ X continuous and densely injected

S : Y 7→ X topological isomorphism



ExamplesReferences

Assumptions

(A1) For given C > 0 and for all y ∈ Y satisfying ||y ||Y ≤ C , A(y)is quasi-m-accretive. In other words, −A(y) generates a

quasi-contractive C0 semigroup {T (t)}t≥0 in X satisfying

||T (t)|| ≤ ewt .

(A2) For all y ∈ Y , A(y) is a bounded linear operator from Y to X

and

||(A(y)− A(z))w ||X ≤ c1||y − z ||X ||w ||Y , y , z ,w ∈ Y .



ExamplesReferences

Assumptions

(A1) For given C > 0 and for all y ∈ Y satisfying ||y ||Y ≤ C , A(y)is quasi-m-accretive. In other words, −A(y) generates a

quasi-contractive C0 semigroup {T (t)}t≥0 in X satisfying

||T (t)|| ≤ ewt .

(A2) For all y ∈ Y , A(y) is a bounded linear operator from Y to X

and

||(A(y)− A(z))w ||X ≤ c1||y − z ||X ||w ||Y , y , z ,w ∈ Y .



ExamplesReferences

Assumptions

(A3) SA(y)S−1 = A(y) + B(y), where B(y) ∈ L(X ) is uniformly

bounded on {y ∈ Y : ||y ||Y ≤ M} and||(B(y)− B(z))w ||X ≤ c2||y − z ||Y ||w ||X , w ∈ X .

(A4) The function f is bounded on bounded subsets

{y ∈ Y : ||y ||Y ≤ M} of Y for each M > 0, and is Lipschitz

in X and Y :

||f (y)− f (z)||X ≤ c3||y − z ||X , ∀y , z ∈ X

and

||f (y)− f (z)||Y ≤ c4||y − z ||Y , ∀y , z ∈ Y .

Here, c1, c2, c3 and c4 are constants depending only on

max{||y ||Y , ||z ||Y }.



ExamplesReferences

Assumptions

(A3) SA(y)S−1 = A(y) + B(y), where B(y) ∈ L(X ) is uniformly

bounded on {y ∈ Y : ||y ||Y ≤ M} and||(B(y)− B(z))w ||X ≤ c2||y − z ||Y ||w ||X , w ∈ X .

(A4) The function f is bounded on bounded subsets

{y ∈ Y : ||y ||Y ≤ M} of Y for each M > 0, and is Lipschitz

in X and Y :

||f (y)− f (z)||X ≤ c3||y − z ||X , ∀y , z ∈ X

and

||f (y)− f (z)||Y ≤ c4||y − z ||Y , ∀y , z ∈ Y .

Here, c1, c2, c3 and c4 are constants depending only on

max{||y ||Y , ||z ||Y }.Nilay Duruk Mutluba³ Semigroup Approach


ExamplesReferences

Conclusion

1

Theorem

Assume (A1), (A2), (A3), (A4) hold. Given u0 ∈ Y , there is a

maximal T > 0, depending on u0 and a unique solution u to (2)

such that

u = (u0, .) ∈ C ([0,T ),Y ) ∩ C 1([0,T ),X ).

Moreover, the map u0 7→ u(u0, .) is continuous from Y to

C ([0,T ),Y ) ∩ C 1([0,T ),X ).

1T. Kato, Quasi-linear equations of evolution, with applications to partial

di�erential equations, in: Spectral theory and di�erential equations, Lecture

Notes in Math. 448, Springer-Verlag, Berlin, 1975, pp. 25− 70.Nilay Duruk Mutluba³ Semigroup Approach


ExamplesReferences

Quasilinear wave equation

ut + a(u)ux = b(u)u

X = L2(R), Y = H2(R), S = 1− D2x

A(u, t) = A(u) = a(u)∂x

f (u, t) = b(u)u

B(u) = −[a′(u)uxx + a′′(u)u2x ]∂S−1 − 2a′(u)ux∂2S−1

Alternatively,

X = L2(R), Y = Hs(R), s ≥ 2,

S = (1− D2x )s/2 or S = (1− Dx)s

Gain regularity !!



ExamplesReferences

Quasilinear wave equation

ut + a(u)ux = b(u)u

X = L2(R), Y = H2(R), S = 1− D2x

A(u, t) = A(u) = a(u)∂x

f (u, t) = b(u)u

B(u) = −[a′(u)uxx + a′′(u)u2x ]∂S−1 − 2a′(u)ux∂2S−1

Alternatively,

X = L2(R), Y = Hs(R), s ≥ 2,

S = (1− D2x )s/2 or S = (1− Dx)s

Gain regularity !!



ExamplesReferences

Model equation

ut + ux +3

2εuux −

3

8ε2u2ux +

3

16ε3u3ux +

µ

12(uxxx − uxxt)

+7ε

24µ(uuxxx + 2uxuxx) = 0, x ∈ R, t > 0. (3)

Here, u(x , t) denotes free surface elevation, ε and µ represent the

amplitude and shallowness parameters, respectively.



ExamplesReferences

Quasilinear equation

Rewrite equation (3) in the following way and de�ne the periodic

Cauchy problem:

ut + A(u)u = f (u) x ∈ R, t > 0, (4)

u(x , 0) = u0(x) x ∈ R, (5)

u(x , t) = u(x + 1, t) x ∈ R, t > 0, (6)

where

A(u) = −(1 +7

2εu)∂x , (7)

f (u) = −(1− µ

12∂2x )−1∂x [2u +

5

2εu2 − 1

8ε2u3 +

3

64ε3u4 − 7

48εµu2x ].

(8)



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Setting

(i) X = L2[0, 1],

(ii) Y = Hsper , s > 3/2,

(iii) S = Λs , Λ = (1− ∂2x )1/2.



ExamplesReferences

Assumption 1

If u ∈ Hsper , s > 3/2, then the operator A(u) = −(1 + u)∂x

with the following domain,

D(A) = {ω ∈ L2[0, 1] : −(1 + u)∂xω ∈ L2[0, 1]} ⊂ L2[0, 1]

is quasi-m-accretive.



ExamplesReferences

Assumption 1

If

(a) For all w ∈ D(A), there exists a real number β such that

(Aw ,w)X ≥ −β||w ||2X ,(b) The range of A + λI is all of X for some (or all) λ > β,

then we are done.



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Assumption 2

For every ω ∈ Hsper , s > 3/2, A(u) is bounded linear operator

from Hsper to L2[0, 1] and

||(A(u)− A(v))ω||L2[0,1] ≤ c1||u − v ||L2[0,1]||ω||Hsper.



ExamplesReferences

Assumption 3

The operator

B(u) = SA(u)S−1 − A(u)

= Λs(u∂x + ∂x)Λ−s − (u∂x + ∂x)

= [Λs , u∂x + ∂x ]Λ−s

is bounded in L2[0, 1] for u ∈ Hsper with s > 3/2.



ExamplesReferences

Assumption 4

Let f (u) be the function (8). Then:

(i) f is bounded on bounded subsets {u ∈ Hsper : ||u||Hs

per≤ M}

of Hsper for each M > 0.

(ii) ||f (u)− f (v)||L2[0,1] ≤ c3||u − v ||L2[0,1].(iii) ||f (u)− f (v)||Hs

per≤ c4||u − v ||Hs

per, s > 3/2.



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Local well-posedness

Given u0 ∈ Hsper , s >

32, there exists a unique solution u to (3)-(6)

depending on u0 as follows:

u = u(u0, .) ∈ C ([0,T ),Hsper ) ∩ C 1([0,T ), L2[0, 1]).

Moreover, the mapping u0 ∈ Hsper 7→ u(u0, .) is continuous from

Hsper to

C ([0,T ),Hsper ) ∩ C 1([0,T ), L2[0, 1]).



ExamplesReferences

References

N. Duruk Mutluba³, Local well-posedness and wave breaking

results for periodic solutions of a shallow water equation for

waves of moderate amplitude, Nonlinear Anal. 97, 2014,

145-154.

T. Kato, Quasi-linear equations of evolution, with applications

to partial di�erential equations, in:Spectral theory and

di�erential equations, Lecture Notes in Math. 448,

Springer-Verlag, Berlin, 1975, pp. 25�70.

A. Pazy, Semigroups of Linear Operators and Applications to

Partial Di�erential Equations, Applied Mathematical Sciences

44, Springer-Verlag, New York, U.S.A., 1983.


Documents

SEMIGROUP APPROACH FOR PARTIAL DIFFERENTIAL EQUATIONS OF EVOLUTIONweb.iku.edu.tr/ias/documents/ias_presentation.pdf · Evolution Equations Semigroup theory Kato's approach Examples