Proof of Theorem 6.1.4

Z t s

⟨DE_r²(u(r)), u^′(r)⟩_V^∗×Vdr

= lim

h→0−

Z t s

⟨DE_r²(u₀+ (KS_hu^′)(r)),(S_hu^′)(r)⟩_V^∗_×V dr

= lim

h→0

E_s²(u₀+ (KS_hu^′)(s))− E_t²(u₀+ (KS_hu^′)(t))

=E_s²(u(s))− E_t²(u(t)) (6.4.5)

for all s, t∈[0, T]. Second, it has been shown in Emmrich & Thalhammer [77, Lemma 6] that

−

Z t 0

⟨u^′′(r) + E(u(r)), u^′(r)⟩_V^∗×V dr

≤ 1

2|v0|²− 1

2|u^′(t)|² +E¹(u₀)− E¹(u(t))

for almost everyt ∈(0, T). The latter inequality together with (6.4.5) implies (6.4.3), which completes the proof.

6.5 Proof of Theorem 6.1.4

Letu₀ ∈U, v₀ ∈H and (τ_n)_n∈N be a vanishing sequence of positive step sizes. Let (u^k₀)k∈N ⊂ U ∩V and (v₀^k)k∈N ⊂ V be such that u^k₀ → u₀ in U and v^k₀ → v₀ in H as k → ∞. We let k ∈ N be fixed and we denote the interpolations associated with the initial data u^k₀ andv^k₀ again by (5.3.1)-(5.3.3) and (6.3.1). Henceforth, we suppress the dependence of the interpolations on k for simplicity. By the previous lemma, there exists a subsequence (relabeled as before) of the interpolations and limit

functionsu∈C_w([0, T];U)∩W^1,∞(0, T;H)∩W^1,q(0, T;V^∗)∩W^2,r∗(0, T;U^∗+V^∗) (notice that u^k₀ ∈ U ∩V) and u(0) = u^k₀ in U and u^′(0) = v₀^k in H such that the convergences (6.4.1) hold, where we again suppress the dependence of the limit functions onk. First, we prove that the inclusion (6.1.3) holds. To do so, we note that the Euler–Lagrange equation (6.3.7) reads

Vb_τ^′_n(t) +η_τn(t) + DE_tτn(t)(U_τn(t)) +S_τn(t) = 0 inU^∗+V^∗,

η_n(t)∈∂_VΨ_Uτn(t)(V_τn(t)) (6.5.1)

for allt∈(0, T), where S_τn(t) = B(t_τn(t), V_τn(t), U_τn(t))−f_τn(t), t∈[0, T]. Testing equation (6.5.1) with w∈L^max{2,q}(0, T;U ∩V), we obtain

Z T 0

⟨V^b_τ^′_n(r) +η_τn(r) + DE_tτn(s)(U_τn(s)) +S_τn(r), w(r)⟩_(U^∗_+V^∗)×(U∩V)dr= 0.

Then, with the aid of the convergences (6.4.1), we are allowed to pass to the limit in the weak formulation obtaining

Z T 0

⟨u^′′(r) +η(r) + DE_s(u(s)) +B(t, u(r), u^′(r))−f(r), w(r)⟩_(U^∗_+V^∗_)×(U∩V)dr= 0 for allw∈L^max{2,q}(0, T;U∩V). Then, by a density argument and the fundamental lemma of calculus of variations, we deduce

u^′′(t) +η(t) + DE_t(u(t)) +B(t, u(t), u^′(t)) =f(t) in U^∗+V^∗

for a.e. t ∈ (0, T). We shall identify the weak limit η as subgradient of the dissipation potential almost everywhere, i.e, η(t) ∈∂_VΨ_u(t)(u^′(t)) for almost every t∈(0, T). For that purpose, we will employ Lemma 2.4.4 with f_n(t, v) =Ψ_U

τn(t)(v) andf(t, v) = Ψ_u(t)(v) for all v ∈X =V andn ∈N. Assumption (2.4.1) is already fulfilled by Condition (6.Ψc). Hence, it remains to show that

lim sup

n→∞

Z T 0

⟨η_n(t), V_τn(t)⟩_V^∗×V dt ≤

Z T 0

⟨η(t), u^′(t)⟩_V^∗×V dt. (6.5.2) In order to show the latter limes superior estimate, we use the fact thatη_τn can be expressed through the remaining terms of the Euler–Lagrangeequation (6.5.1).

Therefore, we will split the integral on the left-hand side of (6.5.2) and note first that

−

Z t 0

⟨V^b_τ^′_n(r), V_τn(r)⟩_V^∗×V dr

=−

Z t 0

⟨V^b_τ^′_n(r),V^b_τn(r)⟩_V^∗×V dr+

Z t 0

⟨V^b_τ^′_n(r),V^b_τn(r)−V_τn(r)⟩_V^∗×V dr

= 1

2|v₀|²− 1

2|V^b_τn(t)|²+

Z t 0

⟨V^b_τ^′_n(r),V^b_τn(r)−V_τn(r)⟩_V^∗×V dr

≤ 1

2|v0|²−1

2|V^bτn(t)|²,

where we used the fundamental theorem of calculus for the absolutely continuous

We continue with the term involving the derivative of the energy functional and start with the linear part:

−

which can be shown in the same way as above by using the strong positivity of E.

As for the nonlinear part, we obtain by employing the λ-convexity of E_t² that

−

+ t_m−t

Now, we want to make use of the inequality (6.4.3). However, the aforementioned inequality only holds true for almost everyt ∈(0, T). Therefore, we take a sequence of increasing values (β_l)_l∈N,β_i ∈(0, T) for all i∈N, converging toT for which (6.4.3) holds true. Then, choosingt =β_l, we obtain with the convergences (6.4.1b), (6.4.1h), (6.4.1e), and (6.4.1g), the sequential weak lower semicontinuity of E_t¹ and | · | and

the continuity ofE_t¹, the limes superior condition and growth condition (6.Ed) on

∂_tE_t² and Fatou’s Lemma that

Then, in view of the convergences (6.4.1m) and (6.4.1n), the Euler–Lagrange equation (6.5.1), we obtain

u satisfying the inclusion (6.1.3), and the initial values u(0) = u^k₀ ∈ U ∩V and u^′(0) = v^k₀ ∈ V. Denote with (u_k)k∈N the sequence of solutions to the associated sequence of initial values, and with (η_k)_k∈N the subgradients of Ψ_uk(t)(u^′_k(t)). In the last step, we want to show that there exists a limit functionu which satisfies (6.1.3) and (6.1.4) as well as the intial valuesu(0) =u₀ in U andu^′(0) = v₀ inH. We recall that u^k₀ →u₀ in U and v₀^k → v₀ in H as k → ∞. As in Chapter 5, the next steps are the following.

1. We derive a priori estimates based on the energy-dissipation inequality (6.1.4),

2. We show compactness of the sequences (u_k)k∈N and (η_k)k∈N in appropriate Condition (6.Ed), (6.Bb), and (6.Bb), we obtain with the lemma of Gronwall (Lemma A.1.1)

Ad 2. With the same reasoning as for the interpolations, we obtain the convergences u_k⇀ u^∗ in L^∞(0, T;U), (6.5.3a) u_k−u^k₀ ⇀ u^∗ −u₀ in L^∞(0, T;V), (6.5.3b) uk(t)⇀ u(t) in U for all t∈[0, T], (6.5.3c) u_k(t)−u^k₀ ⇀ u(t)−u₀ in V for all t∈[0, T], (6.5.3d) u_k →u in L^r(0, T;W^f) for any r≥1, (6.5.3e) u_k(t)→u(t) in W^f for all t∈[0, T], (6.5.3f) u^′_k(t)⇀ u^{∗ ′} in L^q(0, T;V)∩L^∞(0, T;H), (6.5.3g) u^′_k(t)→u^′ in L^p(0, T;H) for all p≥1, (6.5.3h) u^′_k(t)⇀ u^′(t) in H for all t∈[0, T], (6.5.3i)

η_τ^k

n ⇀ η in L^q∗(0, T;V^∗), (6.5.3j) Eu_k⇀Eu in L²(0, T;U^∗), (6.5.3k) DE_t²(u_k)→DE_t²(u) in L^r(0, T;U^∗) for any r≥1, (6.5.3l) u^′′_k ⇀ u^′′ in L^min{2,q∗}(0, T;U^∗+V^∗), (6.5.3m) B(·, u_k, u^′_k)→B(·, u, u^′) in L^r∗(0, T;V^∗), (6.5.3n) and if Ψ_u satisfies (6.Ψd), then

u^′_k→u^′ in L^max{2,q}(0, T;U), u_k →u in C([0, T];U).

Ad 3. Therefore, u ∈ C_w([0, T];U)∩W^1,∞([0, T];H)∩W^2,q∗(0, T;U^∗ +V^∗) with u−u₀ ∈W^1,q(0, T;V) and η∈L^q∗(0, T;V^∗) satisfies the initial conditions u(0) = u₀ inU and u^′(0) =v0 in H. Along the same lines as for the interpolations, we obtain with Condition (6.Ψc) and Lemma 2.4.4 that u andη satisfy the inclusions (6.1.3).

It remains to show the energy-dissipation balance (6.1.4). The inequality 1

2|u^′(t)|²+E_t(u(t)) +

Z t 0

Ψ_u(r)(u^′(r)) +Ψ_u(r)^∗ (S(r)−DE_r(u(r))−u^′′(r))dr

≤ 1

2|v₀|²+E₀(u₀) +

Z t 0

∂_rE_r(u(r)) dr+

Z t 0

⟨S(r), u^′(r)⟩_V^∗×V dr,

for all t ∈ [0, T] with S(r) =f(r)−B(r, u(r), u^′(r)) is obtained by passing to the limit ask → ∞while taking into account the convergences (6.4.1). Then, employing again (6.4.3) and theFenchel–Young inequality, we obtain

Z t 0

Ψ_u(r)(u^′(r)) +Ψ_u(r)^∗ (S(r)−DE_r(u(r))−u^′′(r))dr

≤ 1

2|v₀|²− 1

2|u^′(t)|²+E₀(u₀)− E_T(u(t)) +

Z t 0

∂_rE_r(u(r)) dr +

Z t 0

⟨S(r), u^′(r)⟩_V^∗_×V dr

≤

Z t 0

⟨DE_r(u(r))−u^′′(r), u^′(r)⟩_V^∗×V dr+

Z t 0

⟨S(r), u^′(r)⟩_V^∗×V dr

=

Z t 0

⟨S(r)−DE_r(u(r))−u^′′(r), u^′(r)⟩_V^∗×V dr

≤

Z t 0

Ψ_u(r)(u^′(r)) +Ψ_u(r)^∗ (S(r)−DE_r(u(r))−u^′′(r))dr

for almost everyt∈(0, T). Now, if V ,→U, then the inequality (6.4.3) indeed holds as equality for allt ∈[0, T] by the classical integration by parts formula. This shows (6.1.4), and hence the completion of the proof.

Remark 6.5.1 The proof of Theorem 6.1.4 reveals that one can consider dissipation potentials that depend on a parameter ε. In this case, the Condition (6.Ψa) is assumed to hold for every ε≥ 0 while Condition (6.Ψb) holds uniformly in ε≥ 0.

Condition (6.Ψc) can either be replaced with the Mosco-convergence Ψ_u^ε_nⁿ −→^M Ψ_u⁰ for every sequenceu_n ⇀ uas ε↘0, or with a more general liminf estimate (2.4.1).

Applications

In this section, we want to apply the abstract results on linear and nonlinear inertial systems developed and proven in Chapter 5 and 6, respectively, to concrete examples.

We will give a sufficient number of examples to cover the range of applications from the abstract results. Since our main results are established in a nonsmooth setting, the examples with nonsmooth functionals, in particular, can not be cast into the framework of the existing results. Those nonsmooth functionals correspond to multi-valued equations or nonlinear constraints. We first start with examples for linearly damped inertial systems and continue with examples for nonlinearly damped inertial systems. We assume the same notation and function spaces as in Chapter 4.

7.1 Differential inclusion I A

In the first example, we consider a system, which can be treated in the Case (a) of the linearly damped intertial system, where the dissipation potential is given by the Dirichletenergy and the energy functional is a nonsmoothλ-convex function which to the best of the authors’ knowledge can not be treated with the abstract results known thus far. More precisely, we consider the initial-boundary value problem

(P1)























∂_ttu−∆∂_tu−∆_pu+ (|u|²−1)u− ∇ ·p+b(x, t,u, ∂_tu) =f inΩ_T, p(x, t)∈Sgn(∇u(x, t)) a.e. in Ω_T,

u(x,0) =u₀(x) onΩ, u^′(x,0) = v₀(x) onΩ,

u(x, t) = 0 on ∂Ω×[0, T],

where Sgn : R^d×m ⇒ R^d×m is the sign function defined in (4.0.1), f : Ω → R^m, b : Ω ×[0, T]×R^m ×R^m → R^m a Carathéodory function in the sense that b(x,·,·,·) is continuous for almost every x∈Ω and b(·, t,y,z) is measurable for all t∈[0, T] and y,z ∈R^m. Furthermore, b is assumed to satisfy the following growth condition: there exists a constantC_b >0 and numbersq, r > 1 such that

|b(x, t,u,v)| ≤Cb(1 +|u|^q−1+|v|^r−1) for a.e. x∈Ω, t∈[0, T] and all u,v ∈R^m. Here, p, q, r≥1 are to be chosen in accordance with the assumptions.

Choosing the spaces U = W^1,p₀ (Ω)^m∩L⁴(Ω)^m, V = H¹₀(Ω)^m, W = L^max{2,q}(Ω)^m andH = L²(Ω)^m equipped with the standard norms, we assume f ∈L²(0, T;V^∗).

The energy functional E :V →(−∞,+∞] and the dissipation potential Ψ :V →R are given by

E(u) =





 R

Ω

1

p|∇u(x)|^p+|∇u(x)|+¹₄(|u(x)|²−1)²dx if u∈dom(E), +∞ otherwise,

and

Ψ(v) = 1 2

Z

Ω

|∇v(x)|²dx,

respectively, whereas the perturbationB : [0, T]×W ×H →V^∗ is defined by

⟨B(t,u,v),w⟩_V^∗×V =⟨B(t,u,v),w⟩_W^∗×W =

Z

Ω

b(x, t,u(x),v(x))·w(x) dx.

The Legendre–Fenchel transformation Ψ^∗ : H⁻¹(Ω)^m → R of Ψ is obviously given byΨ^∗(ξ) = ¹₂∥ξ∥²_−1,2. Furthermore, it is readily seen that the energy functional is not Gâteaux differentiable and its effective domain is given by dom(E) = W₀^1,p(Ω)^m∩L⁴(Ω)^m. The valuesp, q, r ≥1 are to be chosen such that all assumptions are fulfilled. We can choose, e.g.,

d= 1, p∈(1,+∞), r∈[1,2], q∈[1, p/2 + 1], d= 2, p∈(1,+∞), r∈[1,2], q∈







[1, pd/(p−d))∩[1, p/2 + 1] if p∈(1,2), [1, p/2 + 1] if p≥2,

d≥3, p∈(1,+∞), r∈[1,2], q ∈







[1, q^∗) if p∈(1,2), [1, p/2 + 1] if p≥3,

where q^∗ = min{^d(p+2)_2(d−p),^3d+4_d , p+ 1}. Then, by the Sobolev embedding theorem and theRellich–Kondrachov theorem,U and V are densely, continuously and compactly embedded in W and H, respectively. We will verify for illustration the assumptions for the case d≥3. Since the dissipation potential is state-independent, it is induced by the bilinear form a:V ×V →R,

a(v,w) := 1 2

Z

Ω

∇v· ∇wdx,

and therefore satisfies all conditions. The conditions (5.Eb)-(5.Ed) are obviously fulfilled by the energy functional. In order to verify (5.Ea), we note that every convex and lower semicontinuous functional on a Banach space is weakly lower semicontinuous. Taking the latter into account, we observe that foru∈dom(E), the

energy functional V. The lower semicontinuity of W on V follows immediately from the converse of the dominated convergence theorem (see, e.g., Brézis [35, Theorem 4.9, p. 94]) and Fatou’s lemma. Further, due to the compact embedding of V in H, the concave function is continuous on V with respect to the weak topology. This implies E to be weakly lower semicontinuous onV. In fact, the convex part of the energy is perturbed by the negativeHilbert space norm ofH squared which by the parallelogram law and the embedding V ,→ H leads to the λ-convexity of E with λ := C being the constant of the very same embedding. Now, we show the closedness property (5.Ee).

First, we note that for each u ∈ D(∂E), there holds ξ ∈ ∂_UE(u) = ∂_UW(u)−u

andΛu =∇u. We note thatΛ is linear and bounded and has as adjoint operator Λ^∗ : L^∞(Ω)^d×m → W^−1,p∗(Ω)^m,A 7→ −∇ ·A the divergence operator. Let u ∈ dom(∂_UW), then by the variational sum rule (Lemma2.2.7) and Lemma 2.2.8, there holds

Second, we express ∂_XW₂(Λu) with the aid of Lemma 2.3.1 equivalently through the equation

⟨p, Λu⟩_X^∗_×X =W₂(Λu) +W₂^∗(p). (7.1.1) Third, by Ekeland & Temam [69, Proposition 1.2, p. 87], the conjugate W₂^∗ is given by

W₂^∗(B) =

Z

Ω

ı_B

Rm×d(0,1)(B(x)) dx, with the indicator function ı_B

Rm×d(0,1) → {0,+∞} defined by ı_B

Rm×d(0,1)(A) =







0 if |A| ≤1 +∞ otherweise.

This implies

W₂^∗(B) =







0 if |B(x)| ≤1 a.e. in Ω +∞ otherweise.

Inserting the latter expression into the equality (7.1.1), we obtain

Z

Ω

p(x) :∇u(x) dx=

Z

Ω

|p(x)|dx

and |p(x)| ≤ 1 a.e. in Ω. Since p(x) :∇u(x) ≤ |p(x)| by theFenchel–Young (or Cauchy–Schwarz) inequality, we deduce

p(x) :∇u(x) =|p(x)| a.e. in Ω.

Therefore, p(x) ∈ B_R^m×d(0,1) if |∇u(x)| = 0 and p(x) = _|∇u(x)|^∇u(x) otherwise. We obtainp(x)∈Sgn(∇u(x)) a.e. inΩ. Now let u_n ⇀^∗ u in L^∞(0, T;U)∩H¹(0, T;V), u_n → u in L²(0, T;V), and ξ_n ⇀^∗ ξ in L²(0, T;U^∗) as n → ∞ such that ξ_n(t) ∈

∂E(u_n(t)) for almost every t∈(0, T), sup_{n∈N,t∈[0,T]}E(u_n(t))≤C₂, and lim sup

n→∞

Z T 0

⟨ξ_n(t),u_n(t)⟩_U^∗×Udt ≤

Z T 0

⟨ξ(t),u(t)⟩_U^∗×Udt. (7.1.2) We note that we can decomposeξ =ζ−u∈V^∗ with ζ ∈∂W(u). Then, defining ζ_n:=ξ_n+u_n, there holdsζ_n∈∂W(u_n) and ζ_n⇀ζ :=ξ+uin L²(0, T;U^∗). By the Lions–Aubin lemma, we obtain the strong convergence ofu_n →u in C([0, T];H).

Thus, in view of (7.1.2), we deduce lim sup

n→∞

Z T 0

⟨ζn(t),un(t)⟩U^∗×Udt≤

Z T 0

⟨ζ(t),u(t)⟩U^∗×Udt.

Since W is convex, by Theorem 2.3.7, there holds ζ(t) ∈ ∂_UW(u(t)) in U^∗ and W(u_n(t)) → W(u(t)) as n → ∞ a.e. in (0, T), whence ξ(t) ∈ ∂_UE(u(t)) a.e. in (0, T) and E(u_n(t))→ E(u(t)) as n→ ∞ a.e. in (0, T). We proceed with showing

the control of the subgradient of E, i.e., Condition (5.Eg). Let u ∈ D(∂E) and ξ∈∂_UE(u). Then, byHölder’s andYoung’s inequality, the Sobolevembedding theorem, we obtain uniformly bounded, from which (5.Eg) follows. Finally, we verify the assumptions on the perturbationB. The continuity condition (5.Ba) can easily be checked with the dominated convergence theorem.

Ad (5.Bb). Let u ∈ dom(E) and v, w ∈ V. Then, by the Hölder & Young inequalities as well as the Sobolev embedding theorem, there holds

⟨B(u,v),w⟩_V^∗×V = to (P1) in the sense that

Z T

with p(x, t)∈Sgn(∇u(x, t)) a.e. in Ω_T, and the energy-dissipation inequality

Im Dokument On the Nonsmooth Analysis of Doubly Nonlinear Evolution Inclusions of First and Second Order with Applications (Seite 144-156)