The heat equation in a non-cylindrical domain governed by a - - PowerPoint PPT Presentation
The heat equation in a non-cylindrical domain governed by a subdifferential inclusion Jos e Alberto Murillo Hern andez Universidad Polit ecnica de Cartagena Spain J.A. Murillo (UPCT) The heat equation in non-cylindrical domains,
Jos´ e Alberto Murillo Hern´ andez Universidad Polit´ ecnica de Cartagena Spain
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20
1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe
3 The heat equation in a time-varying domain described
4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20
1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe
3 The heat equation in a time-varying domain described
4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20
1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe
3 The heat equation in a time-varying domain described
4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20
1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe
3 The heat equation in a time-varying domain described
4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20
1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe
3 The heat equation in a time-varying domain described
4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 1 / 20
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω (CP) where
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω (CP) where Q = Ω × ]0, T[ ⊂ I RN+1
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω (CP) where Q = Ω × ]0, T[ ⊂ I RN+1 Ω ⊂ I RN spatial domain
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω (CP) where Q = Ω × ]0, T[ ⊂ I RN+1 Ω ⊂ I RN spatial domain Σ = {(x, t) : 0 ≤ t < T, x ∈ Γ}
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω0 (NCP) where
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω0 (NCP) where Q = {(x, t) : 0 < t < T, x ∈ Ωt} ⊂ I RN+1
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω0 (NCP) where Q = {(x, t) : 0 < t < T, x ∈ Ωt} ⊂ I RN+1 Ωt ⊂ I RN, spatial domain changing along time
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω0 (NCP) where Q = {(x, t) : 0 < t < T, x ∈ Ωt} ⊂ I RN+1 Ωt ⊂ I RN, spatial domain changing along time Σ = {(x, t) : 0 ≤ t < T, x ∈ Γt}
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 2 / 20
Q ⊂ I RN+1 spatio-temporal domain
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 3 / 20
Q ⊂ I RN+1 spatio-temporal domain Cylindrical case
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 3 / 20
Q ⊂ I RN+1 spatio-temporal domain Cylindrical case Non-cylindrical case
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 3 / 20
Usually Ωt is assumed to be generated by the flow of a nonautonomous vector field V : [0, T] × I RN − → I RN
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 4 / 20
Usually Ωt is assumed to be generated by the flow of a nonautonomous vector field V : [0, T] × I RN − → I RN That is Ωt = Tt(Ω0), where ∂Tt(x) ∂t = V(t, Tt(x)) T0(x) = x
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 4 / 20
Usually Ωt is assumed to be generated by the flow of a nonautonomous vector field V : [0, T] × I RN − → I RN That is Ωt = Tt(Ω0), where ∂Tt(x) ∂t = V(t, Tt(x)) T0(x) = x
Cannarsa, Da Prato & Zol´ esio (1989, 1990) Zol´ esio (2004) Burdzy, Chen & Sylvester (2004) etc...
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 4 / 20
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 5 / 20
The classical procedure to solve (NCP) is: Show that Tt is a diffeomorphism for any t, Lipschitz w.r. to t. Use Tt to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω0 × ]0, T[. Establish the existence of solution ω(x, t) of the parabolic problem. The map u(x, t) = ω(T −1
t
(x), t) provides the solution of the original non-cylindrical problem.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20
The classical procedure to solve (NCP) is: Show that Tt is a diffeomorphism for any t, Lipschitz w.r. to t. Use Tt to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω0 × ]0, T[. Establish the existence of solution ω(x, t) of the parabolic problem. The map u(x, t) = ω(T −1
t
(x), t) provides the solution of the original non-cylindrical problem.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20
The classical procedure to solve (NCP) is: Show that Tt is a diffeomorphism for any t, Lipschitz w.r. to t. Use Tt to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω0 × ]0, T[. Establish the existence of solution ω(x, t) of the parabolic problem. The map u(x, t) = ω(T −1
t
(x), t) provides the solution of the original non-cylindrical problem.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20
The classical procedure to solve (NCP) is: Show that Tt is a diffeomorphism for any t, Lipschitz w.r. to t. Use Tt to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω0 × ]0, T[. Establish the existence of solution ω(x, t) of the parabolic problem. The map u(x, t) = ω(T −1
t
(x), t) provides the solution of the original non-cylindrical problem.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20
The classical procedure to solve (NCP) is: Show that Tt is a diffeomorphism for any t, Lipschitz w.r. to t. Use Tt to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω0 × ]0, T[. Establish the existence of solution ω(x, t) of the parabolic problem. The map u(x, t) = ω(T −1
t
(x), t) provides the solution of the original non-cylindrical problem.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20
The classical procedure to solve (NCP) is: Show that Tt is a diffeomorphism for any t, Lipschitz w.r. to t. Use Tt to transform the heat equation into a parabolic one with variable coefficients defined in the reference cylinder Ω0 × ]0, T[. Establish the existence of solution ω(x, t) of the parabolic problem. The map u(x, t) = ω(T −1
t
(x), t) provides the solution of the original non-cylindrical problem.
L´ ımaco, Medeiros & Zuazua (2002)
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 6 / 20
Focussing on the evolution of the spatial domain: Can we consider non-cylindrical problems for domains evolving in a more general way? Is it possible to solve problems where the velocity (in some sense) of the domain depends on its global shape? Perturbations involving subdifferential inclusions could be allowed?
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 7 / 20
Focussing on the evolution of the spatial domain: Can we consider non-cylindrical problems for domains evolving in a more general way? Is it possible to solve problems where the velocity (in some sense) of the domain depends on its global shape? Perturbations involving subdifferential inclusions could be allowed?
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 7 / 20
Focussing on the evolution of the spatial domain: Can we consider non-cylindrical problems for domains evolving in a more general way? Is it possible to solve problems where the velocity (in some sense) of the domain depends on its global shape? Perturbations involving subdifferential inclusions could be allowed?
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 7 / 20
Focussing on the evolution of the spatial domain: Can we consider non-cylindrical problems for domains evolving in a more general way? Is it possible to solve problems where the velocity (in some sense) of the domain depends on its global shape? Perturbations involving subdifferential inclusions could be allowed?
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 7 / 20
Focussing on the evolution of the spatial domain: Can we consider non-cylindrical problems for domains evolving in a more general way? Is it possible to solve problems where the velocity (in some sense) of the domain depends on its global shape? Perturbations involving subdifferential inclusions could be allowed?
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 7 / 20
1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe
3 The heat equation in a time-varying domain described
4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 8 / 20
Let I RN be endowed with the usual (Euclidean) norm | · | and let K(I RN) be the family of all its nonempty compact subsets. Equipped with the Hausdorff distance, d lH(K, M) := max
x∈K
dM(x), sup
z∈M
dK(z)
K(I RN) is a complete separable metric space, also satisfying that closed balls are compact. However, K(I RN) has not a linear (vector) structure at all!.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 9 / 20
Let I RN be endowed with the usual (Euclidean) norm | · | and let K(I RN) be the family of all its nonempty compact subsets. Equipped with the Hausdorff distance, d lH(K, M) := max
x∈K
dM(x), sup
z∈M
dK(z)
K(I RN) is a complete separable metric space, also satisfying that closed balls are compact. However, K(I RN) has not a linear (vector) structure at all!.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 9 / 20
Let I RN be endowed with the usual (Euclidean) norm | · | and let K(I RN) be the family of all its nonempty compact subsets. Equipped with the Hausdorff distance, d lH(K, M) := max
x∈K
dM(x), sup
z∈M
dK(z)
K(I RN) is a complete separable metric space, also satisfying that closed balls are compact. However, K(I RN) has not a linear (vector) structure at all!.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 9 / 20
Let I RN be endowed with the usual (Euclidean) norm | · | and let K(I RN) be the family of all its nonempty compact subsets. Equipped with the Hausdorff distance, d lH(K, M) := max
x∈K
dM(x), sup
z∈M
dK(z)
K(I RN) is a complete separable metric space, also satisfying that closed balls are compact. However, K(I RN) has not a linear (vector) structure at all!.
Aubin (1999) Delfour & Zol´ esio (2001) Lorenz (2010)
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 9 / 20
Shape mutations: a tool to define the evolving velocity of sets
Let C0,1(I RN; I RN) be the family of all Lipschitz vector fields. For V ∈ C0,1(I RN; I RN), ϑV(t, K) = {Tt(x) : x ∈ K} is the reachable set at time t associated to the solutions ∂Tt(x) ∂t = V(Tt(x)), T0(x) = x starting from K ⊂ I RN. The map h ❀ ϑV(h, K) provides a curve (a shape transition) on K(I RN). These transitions will play the role of “directions” in K(I RN). Since d lH (ϑV(t + h, K), ϑV(h, ϑV(t, K))) h → 0, as h → 0, we can see the field V as the “velocity at time t” of the tube ϑV(·, K).
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 10 / 20
Shape mutations: a tool to define the evolving velocity of sets
Let C0,1(I RN; I RN) be the family of all Lipschitz vector fields. For V ∈ C0,1(I RN; I RN), ϑV(t, K) = {Tt(x) : x ∈ K} is the reachable set at time t associated to the solutions ∂Tt(x) ∂t = V(Tt(x)), T0(x) = x starting from K ⊂ I RN. The map h ❀ ϑV(h, K) provides a curve (a shape transition) on K(I RN). These transitions will play the role of “directions” in K(I RN). Since d lH (ϑV(t + h, K), ϑV(h, ϑV(t, K))) h → 0, as h → 0, we can see the field V as the “velocity at time t” of the tube ϑV(·, K).
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 10 / 20
Shape mutations: a tool to define the evolving velocity of sets
Let C0,1(I RN; I RN) be the family of all Lipschitz vector fields. For V ∈ C0,1(I RN; I RN), ϑV(t, K) = {Tt(x) : x ∈ K} is the reachable set at time t associated to the solutions ∂Tt(x) ∂t = V(Tt(x)), T0(x) = x starting from K ⊂ I RN. The map h ❀ ϑV(h, K) provides a curve (a shape transition) on K(I RN). These transitions will play the role of “directions” in K(I RN). Since d lH (ϑV(t + h, K), ϑV(h, ϑV(t, K))) h → 0, as h → 0, we can see the field V as the “velocity at time t” of the tube ϑV(·, K).
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 10 / 20
Shape mutations: a tool to define the evolving velocity of sets
Let C0,1(I RN; I RN) be the family of all Lipschitz vector fields. For V ∈ C0,1(I RN; I RN), ϑV(t, K) = {Tt(x) : x ∈ K} is the reachable set at time t associated to the solutions ∂Tt(x) ∂t = V(Tt(x)), T0(x) = x starting from K ⊂ I RN. The map h ❀ ϑV(h, K) provides a curve (a shape transition) on K(I RN). These transitions will play the role of “directions” in K(I RN). Since d lH (ϑV(t + h, K), ϑV(h, ϑV(t, K))) h → 0, as h → 0, we can see the field V as the “velocity at time t” of the tube ϑV(·, K).
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 10 / 20
Shape mutations: a tool to define the evolving velocity of sets
Given a tube K : I ⊂ I R − → K(I RN), it is said that V ∈ C0,1(I RN; I RN) belongs to the shape mutation of K(·) at t in the forward direction if lim
h→0+
d lH (ϑV(h, K(t)), K(t + h)) h = 0 Then we will write V ∈
The set
RN; I RN) can be regarded as the “velocity” of K(·) at time t. A map V : I ⊂ I R − → C0,1(I RN; I RN) will be a “shape primitive” of K(·) if for any t the field V(t) belongs to its shape mutation at t. We will write
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 11 / 20
Shape mutations: a tool to define the evolving velocity of sets
Given a tube K : I ⊂ I R − → K(I RN), it is said that V ∈ C0,1(I RN; I RN) belongs to the shape mutation of K(·) at t in the forward direction if lim
h→0+
d lH (ϑV(h, K(t)), K(t + h)) h = 0 Then we will write V ∈
The set
RN; I RN) can be regarded as the “velocity” of K(·) at time t. A map V : I ⊂ I R − → C0,1(I RN; I RN) will be a “shape primitive” of K(·) if for any t the field V(t) belongs to its shape mutation at t. We will write
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 11 / 20
Shape mutations: a tool to define the evolving velocity of sets
Given a tube K : I ⊂ I R − → K(I RN), it is said that V ∈ C0,1(I RN; I RN) belongs to the shape mutation of K(·) at t in the forward direction if lim
h→0+
d lH (ϑV(h, K(t)), K(t + h)) h = 0 Then we will write V ∈
The set
RN; I RN) can be regarded as the “velocity” of K(·) at time t. A map V : I ⊂ I R − → C0,1(I RN; I RN) will be a “shape primitive” of K(·) if for any t the field V(t) belongs to its shape mutation at t. We will write
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 11 / 20
Shape mutations: a tool to define the evolving velocity of sets
Remark (Doyen, 1995) . The shape mutation of a tube is usually set-valued. For instance if K(t) = B is constant and equal to the closed unit ball in I R2, it is clear that for every t, 0, V(x, y) = (−y, x) ∈
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 12 / 20
Shape mutations: a tool to define the evolving velocity of sets
Remark (Doyen, 1995) . The shape mutation of a tube is usually set-valued. For instance if K(t) = B is constant and equal to the closed unit ball in I R2, it is clear that for every t, 0, V(x, y) = (−y, x) ∈
R − → C0,1(I RN; I RN) is a shape primitive of the reachable tube t ❀ ϑV(t, K), K ∈ K(I RN).
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 12 / 20
Morphological shape equations
We are ready to define morphological shape equations as a generalization of
For a map V : I R+ ×K(I RN) − → C0,1(I RN, I RN), a solution of the morphological shape equation
(♠)
R+ will be a compact-valued Lipschitz tube, K(·) such that, for all t ∈ I, V(t, K(t)) belongs to
lim
h→0+
d lH ` ϑV(t,K(t))(h, K(t)), K(t + h) ´ h = 0 Solutions of (♠) satisfy the recurrence law K(t) = {x(t) : ˙ x(s) = V(s, K(s))(x(s)), x(0) ∈ K(0)}
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 13 / 20
Morphological shape equations
We are ready to define morphological shape equations as a generalization of
For a map V : I R+ ×K(I RN) − → C0,1(I RN, I RN), a solution of the morphological shape equation
(♠)
R+ will be a compact-valued Lipschitz tube, K(·) such that, for all t ∈ I, V(t, K(t)) belongs to
lim
h→0+
d lH ` ϑV(t,K(t))(h, K(t)), K(t + h) ´ h = 0 Solutions of (♠) satisfy the recurrence law K(t) = {x(t) : ˙ x(s) = V(s, K(s))(x(s)), x(0) ∈ K(0)}
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 13 / 20
Morphological shape equations
We are ready to define morphological shape equations as a generalization of
For a map V : I R+ ×K(I RN) − → C0,1(I RN, I RN), a solution of the morphological shape equation
(♠)
R+ will be a compact-valued Lipschitz tube, K(·) such that, for all t ∈ I, V(t, K(t)) belongs to
lim
h→0+
d lH ` ϑV(t,K(t))(h, K(t)), K(t + h) ´ h = 0 Solutions of (♠) satisfy the recurrence law K(t) = {x(t) : ˙ x(s) = V(s, K(s))(x(s)), x(0) ∈ K(0)}
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 13 / 20
Morphological shape equations
∂Tt(x) ∂t = V(t, Tt(x)) T0(x) = x 9 = ; is the solution of the morphological shape equation
solutions of morphological shape equations contains all the tubes described by flows of vector fields.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 14 / 20
Morphological shape equations
∂Tt(x) ∂t = V(t, Tt(x)) T0(x) = x 9 = ; is the solution of the morphological shape equation
solutions of morphological shape equations contains all the tubes described by flows of vector fields.
global shape are allowed. For instance taking V(t, K) = W(t) + ϕ(K)F with W(t), F ∈ C0,1(I RN, I RN) and ϕ(K) = ( 0, if d lH(K, M) ≥ δ δ − d lH(K, M),
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 14 / 20
Existence theorem for morphological shape equations
Theorem 1 (Cauchy-Lipschitz for shape equations, Aubin, 1999)
Let V : I R+ ×K(I RN) − → C0,1(I RN; I RN) be continuous with respect to the first variable t, λ-Lipschitz with respect to the second one K, that is, V (t, K) − V (t, B)∞ ≤ λd lH(K, B) and satisfying α := sup
0<t<T,K∈K(I RN)
sup
x=y
„|V(t, K)(x) − V(t, K)(y)| |x − y| « < ∞ then for any K ∈ K(I RN) there exists a unique solution of
satisfying K(0) = K.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 15 / 20
Existence theorem for morphological shape equations
Theorem 1 (Cauchy-Lipschitz for shape equations, Aubin, 1999)
Let V : I R+ ×K(I RN) − → C0,1(I RN; I RN) be continuous with respect to the first variable t, λ-Lipschitz with respect to the second one K, that is, V (t, K) − V (t, B)∞ ≤ λd lH(K, B) and satisfying α := sup
0<t<T,K∈K(I RN)
sup
x=y
„|V(t, K)(x) − V(t, K)(y)| |x − y| « < ∞ then for any K ∈ K(I RN) there exists a unique solution of
satisfying K(0) = K. Moreover, if K(·) and B(·) are solutions starting from K and B respectively, then ∀ t, d lH (K(t), B(t)) ≤ e(α+λ)td lH (K, B)
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 15 / 20
Existence theorem for morphological shape equations
Let φ : I RN − → I R be convex and l.s.c.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 16 / 20
Existence theorem for morphological shape equations
Let φ : I RN − → I R be convex and l.s.c. By means of its subdifferential ∂φ we can perturb the equation
(♠)
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 16 / 20
Existence theorem for morphological shape equations
Let φ : I RN − → I R be convex and l.s.c. By means of its subdifferential ∂φ we can perturb the equation to get the morphological shape inclusion
(♠)
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 16 / 20
Existence theorem for morphological shape equations
Let φ : I RN − → I R be convex and l.s.c. By means of its subdifferential ∂φ we can perturb the equation to get the morphological shape inclusion
(♣)
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 16 / 20
Existence theorem for morphological shape equations
Let φ : I RN − → I R be convex and l.s.c. By means of its subdifferential ∂φ we can perturb the equation to get the morphological shape inclusion
(♣)
Theorem 2
Assuming hypotheses of Theorem 1, for any K ∈ K(I RN), there exists a unique solution
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 16 / 20
Existence theorem for morphological shape equations
Let φ : I RN − → I R be convex and l.s.c. By means of its subdifferential ∂φ we can perturb the equation to get the morphological shape inclusion
(♣)
Theorem 2
Assuming hypotheses of Theorem 1, for any K ∈ K(I RN), there exists a unique solution
K(t) = {x(t) : ˙ x(s) ∈ −∂φ(x(s)) + V(s, K(s))(x(s)), x(0) ∈ K}
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 16 / 20
Existence theorem for morphological shape equations
Let φ : I RN − → I R be convex and l.s.c. By means of its subdifferential ∂φ we can perturb the equation to get the morphological shape inclusion
(♣)
Theorem 2
Assuming hypotheses of Theorem 1, for any K ∈ K(I RN), there exists a unique solution
K(t) = {x(t) : ˙ x(s) ∈ −∂φ(x(s)) + V(s, K(s))(x(s)), x(0) ∈ K}
RN
“ φ(y) +
1 2µ|y − x|2”
allows to prove this result.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 16 / 20
1 The heat equation in time-varying domains. 2 Morphological shape equations: a way to describe
3 The heat equation in a time-varying domain described
4 Conclusions. J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 17 / 20
The problem
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω0 9 > > = > > ; (NCP) where K(t) = Ωt satisfyies
(♣)
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 18 / 20
Main result
ut(x, t) − div (a(x)∇u(x, t)) = f (x, t), in Q u = 0,
u(x, 0) = u0(x), x ∈ Ω0 9 > > = > > ; (NCP) where K(t) = Ωt satisfyies
(♣)
Theorem 3
For any initial states u0 ∈ L2(Ω0), Ω0 ⊂ I RN a nonempty open bounded set, there exists a unique solution of the problem, i.e. a tube K(·) ∈ C0,1(0, T; K(I RN)) and a map u ∈ L2(0, T; H1
0(Ωt)) ∩ C([0, T]; L2(Ωt))
satisfying the heat equation in a weak sense.
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 18 / 20
Morphological shape equations provide time-evolving families of sets in a more general way than the “flows of velocities”. It is possible to consider (and solve) noncylindrical problems associated with these type of set evolutions for the heat equation, even when a subdifferential perturbation is considered. This scheme could be appropriate for problems involving different kind of PDEs. A challenging task is to consider evolutions associated to set-valued maps, where the topology of domains could be modified along time,
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 19 / 20
Morphological shape equations provide time-evolving families of sets in a more general way than the “flows of velocities”. It is possible to consider (and solve) noncylindrical problems associated with these type of set evolutions for the heat equation, even when a subdifferential perturbation is considered. This scheme could be appropriate for problems involving different kind of PDEs. A challenging task is to consider evolutions associated to set-valued maps, where the topology of domains could be modified along time,
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 19 / 20
Morphological shape equations provide time-evolving families of sets in a more general way than the “flows of velocities”. It is possible to consider (and solve) noncylindrical problems associated with these type of set evolutions for the heat equation, even when a subdifferential perturbation is considered. This scheme could be appropriate for problems involving different kind of PDEs. A challenging task is to consider evolutions associated to set-valued maps, where the topology of domains could be modified along time,
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 19 / 20
Morphological shape equations provide time-evolving families of sets in a more general way than the “flows of velocities”. It is possible to consider (and solve) noncylindrical problems associated with these type of set evolutions for the heat equation, even when a subdifferential perturbation is considered. This scheme could be appropriate for problems involving different kind of PDEs. A challenging task is to consider evolutions associated to set-valued maps, where the topology of domains could be modified along time,
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 19 / 20
Morphological shape equations provide time-evolving families of sets in a more general way than the “flows of velocities”. It is possible to consider (and solve) noncylindrical problems associated with these type of set evolutions for the heat equation, even when a subdifferential perturbation is considered. This scheme could be appropriate for problems involving different kind of PDEs. A challenging task is to consider evolutions associated to set-valued maps, where the topology of domains could be modified along time,
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 19 / 20
Morphological shape equations provide time-evolving families of sets in a more general way than the “flows of velocities”. It is possible to consider (and solve) noncylindrical problems associated with these type of set evolutions for the heat equation, even when a subdifferential perturbation is considered. This scheme could be appropriate for problems involving different kind of PDEs. A challenging task is to consider evolutions associated to set-valued maps, where the topology of domains could be modified along time,
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 19 / 20
J.A. Murillo (UPCT) The heat equation in non-cylindrical domains, PICOF’12 20 / 20