semigroup approach for partial differential equations of...
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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
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
1 Evolution Equations
2 Semigroup theory
3 Kato's approach
4 Examples
5 References
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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?
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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?
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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?
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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?
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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?
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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).
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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).
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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).
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
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!!
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
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!!
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
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!!
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
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!!
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
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!!
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
Linear Evolution Equations
Homogeneous equation:
u′(t) + Au(t) = 0, 0 ≤ t ≤ T ,
u(0) = u0
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
Linear Evolution Equations
∀ u0, solution u exists and is unique:
u(t) = etAu0
Here, etA =∑∞
n=0(tA)n
n! .
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
Linear Evolution Equations
Nonhomogeneous equation:
u′(t) + Au(t) = f (t), 0 ≤ t ≤ T ,
u(0) = u0
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
Linear Evolution Equations
u(t) = etAu0 +
∫ t
0
e(t−s)Af (s)ds
Lions, Yosida, Pazy...
KATO! Semigroup theory!!
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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 .
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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 .}
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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
Φ(ω) contraction mapping ⇒ unique solution u (Banach �xed
point theorem)
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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
Φ(ω) contraction mapping ⇒ unique solution u (Banach �xed
point theorem)
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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
Φ(ω) contraction mapping ⇒ unique solution u (Banach �xed
point theorem)
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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).
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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).
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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).
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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).
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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 .
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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 .
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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
Evolution EquationsSemigroup theoryKato's approach
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
Evolution EquationsSemigroup theoryKato's 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
Evolution EquationsSemigroup theoryKato's 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 !!
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's 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 !!
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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)
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
ExamplesReferences
Setting
(i) X = L2[0, 1],
(ii) Y = Hsper , s > 3/2,
(iii) S = Λs , Λ = (1− ∂2x )1/2.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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]).
Nilay Duruk Mutluba³ Semigroup Approach
Evolution EquationsSemigroup theoryKato's approach
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.
Nilay Duruk Mutluba³ Semigroup Approach