semigroup approach for partial differential equations of...

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).

ExamplesReferences

Solutions

Weak solution!!

Hk(R).

ExamplesReferences

Solutions

Weak solution!!

Hk(R).

ExamplesReferences

Simple example

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

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

⇒ dudt∈ C ([0,T ],X )).

Very strong!!

ExamplesReferences

Simple example

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

⇒ dudt∈ C ([0,T ],X )).

Very strong!!

ExamplesReferences

Simple example

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

⇒ dudt∈ C ([0,T ],X )).

Very strong!!

ExamplesReferences

Simple example

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

ExamplesReferences

∀ u0, solution u exists and is unique:

u(t) = etAu0

Here, etA =∑∞

n=0(tA)n

ExamplesReferences

Nonhomogeneous equation:

u′(t) + Au(t) = f (t), 0 ≤ t ≤ T ,

u(0) = u0

ExamplesReferences

u(t) = etAu0 +

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

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

ExamplesReferences

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

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

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−

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−

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−

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−

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.

ExamplesReferences

General approach

variable x

ExamplesReferences

General approach

variable x

ExamplesReferences

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

||(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

||(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

||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

||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

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 −

8ε2u2ux +

16ε3u3ux +

12(uxxx − uxxt)

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)

A(u) = −(1 +7

2εu)∂x , (7)

f (u) = −(1− µ

12∂2x )−1∂x [2u +

2εu2 − 1

8ε2u3 +

64ε3u4 − 7

48εµu2x ].

ExamplesReferences

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

(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.

ExamplesReferences

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.

ExamplesReferences

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.

semigroup approach for partial differential equations of...

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