Nonsmooth analysis of doubly nonlinear second-order evolution equations with nonconvex energy functionals

1.1 Problem setting

In the present work, we aim to prove the existence of strong solutions to the following abstract Cauchy problem

where Ψ is the dissipation potential, ℰ t the energy functional, B the perturbation, and f the external force. The functionals and operators are defined on suitable spaces, which are specified below. Here, the main assumptions are that the leading part of Ψ is defined by a strongly positive, symmetric, and bounded bilinear form a , the energy functional ℰ t is λ -convex, and the perturbation B is a strongly continuous perturbation of ∂ Ψ and ∂ ℰ t . Within the aforementioned class of dissipation potentials, we consider the following two cases separately: in the first case (Case (a)), we assume that Ψ ( v ) = a ( v , v ) , and in the second case (Case (b)), we assume that Ψ = Ψ 1 + Ψ 2 , where Ψ 1 ( v ) = a ( v , v ) and Ψ 2 is a strongly continuous, convex, and nonsmooth perturbation that. Furthermore, we will specifically consider the case when ℰ t is convex. We note that the energy functional and the dissipation potential are, in general, defined on different spaces. An illustrative example that satisfies all the aforementioned assumptions is given by

where p , q > 1 are to be chosen suitably, ν ≥ 0 , A : R d → R d is a linear, symmetric, and elliptic operator, W : R → R is a double-well potential given by W ( u ) = 1 4 ( u 2 − 1 ) 2 , b : R → R a lower order perturbation, and f : R → R an external force. The energy functional and the dissipation potential are given by

and

and the perturbation is (formally) given by

We note that both functionals ℰ and Ψ are nonsmooth and are, therefore, not eligible for the abstract results known in the literature. Moreover, the evolution equations of the second-order, where the operators acting on u and u ′ are multivalued, have not been studied earlier, even for concrete equations. Note that if ν = 0 , we are in Case (a), and if ν > 0 , we are in Case (b). More, in particular, multivalued applications will be discussed in Sections 5.1 and 5.2.

Evolution equations of second-order occur in many applications, in particular in physics where many phenomena are modeled as such, e.g., the equations in peridynamics, the equations in visco-elastodynamics, the nonlinear Klein-Gordon equation from quantum mechanics and other nonlinear wave equations arising in mechanics and quantum mechanics, equations for describing a vibrating membrane [16,22,23,25,40,41]. Multivalued evolution equations (evolution inclusions) occur in many applications, e.g., physical phenomena where rate-independent responses of the body are typical, such as in plasticity [37], in ferromagnetic hysteresis [36,43], or in elasto-viscoplasticity [39]. Applications are also found in optimal control theory [5] or nonsmooth dynamical systems [31].

1.2 Literature review

Evolution inclusions of second-order of the form (1.1) have been studied by Rossi and Thomas in [40], where ∂ Ψ = A : V → V * is a single-valued, linear, bounded, strongly positive, and symmetric operator defined on a reflexive and separable Banach space V , i.e., possessing a convex and quadratic potential. In their framework, ∂ ℰ t is the subdifferential of a λ -convex functional with effective domain in a reflexive and separable Banach space U ⊂ V . In the framework of the Gelfand quintuplet

and under the assumption that ℰ t satisfies a chain rule and ∂ ℰ t satisfies a closedness condition, they showed the existence of a strong solution. While this work is covered with the result presented here, we allow further a strongly continuous nonmonotone and nonvariational perturbation that depends on u and u ′ as well as a nonlinear monotone perturbation of A of variational type. We also do not assume the rather restrictive assumption that U ↪ d V . Furthermore, the strong closedness condition of ∂ ℰ t assumed in [40] excludes the application to nonlinear elastodynamics where the operator satisfies the so-called Andrews–Ball type condition [23]. Emmrich and Šiška [23] developed an abstract theory in the smooth setting with the application to nonlinear elastodynamics. They prove the existence of strong solutions for the case where A : V A → V A * is linear, bounded, strongly positive, and symmetric, and B : V B → V B * is supposed to be demicontinuous and a bounded potential operator. In addition, B satisfies an Andrews–Ball-type condition, meaning that ( B + λ A ) : V → V * is monotone, where V ≔ V A ∩ V B is densely and continuously embedded into the separable and reflexive Banach spaces, V A and V B , for which we assume not that either of the two spaces is continuously embedded in the other one. Since we allow a more general nonsmooth functional ℰ , this result is also covered by our main result. Another motivation for this work is to complement the results obtained in B. [7] where the principal part of the operator acting on u ′ is nonlinear and multivalued, and the principal part of the operator acting on u is linear, symmetric, and positive [6]. Hence, our contribution to the literature concerns the following:

• We allow the functionals that are acting on u and u ′ to be nonsmooth, hence generating multivalued subdifferentials in the differential inclusion. This has not even been considered in the nonabstract setting.

• We allow the multivalued operators to live on different spaces.

• We allow nonvariational and nonmonotone perturbations of the subdifferential operators.

• We provide in the applications an existence result to concrete problems for which there are no results known and cannot be obtained by other abstract results.

• The existence result is based on a numerical scheme, which can be used to obtain numerical approximations.

Doubly nonlinear evolution equations where the leading parts of A and B are both nonlinear and contain the same order of spatial derivatives in the applications are unfortunately not covered by our result. However, in the Hilbert space setting, the well posedness of doubly nonlinear evolution equations of second-order under a Lipschitz condition on the operator B has been shown in [9]. In certain other situations, this can also be demonstrated by exploiting the special structure of the operators [15,16,28,38].

For further results on nonlinear evolution equations, we refer to Leray [32], Dionne [18], Emmrich and Thalhammer [25,26], Emmrich and Šiška [22] for stochastic perturbations, Emmrich et al. [24] for a numerical analysis, Emmrich et al. [27] and Ruf [42] for results on Orlicz spaces, and the monographs Lions [33], Lions and Magenes [34, Chapitre 3.8], Barbu [11, Chapter V], Wloka [44, Chapter V], Zeidler [46, Chapter 33], Roubíček [41, Chapter 11] and the references therein.

The list of literature presented in this section is not intended to be exhaustive.

1.3 Organization of the article

The article is organized as follows. In Section 2, we set the analytical framework and briefly introduce some notions and results from the theory of subdifferential calculus and convex analysis. In Section 3, we present and discuss the assumptions on the dissipation potential Ψ , the energy functional ℰ , and the perturbation B as well as the external force f , and we state the main result. Section 4 is devoted to the proof, and in Section 5, we show some examples that does not fit into the abstract framework of the cited authors.

2.1 Function space setting

In the following, let ( U , ‖ ⋅ ‖ U ) , ( V , ‖ ⋅ ‖ V ) , ( W , ‖ ⋅ ‖ W ) , and ( W ˜ , ‖ ⋅ ‖ W ˜ ) be real, separable, and reflexive Banach spaces such that the dual space W * is uniformly convex. Furthermore, let ( H , ∣ ⋅ ∣ , ( ⋅ , ⋅ ) ) be a Hilbert space with norm ∣ ⋅ ∣ induced by the inner product ( ⋅ , ⋅ ) . Then, we assume the dense, continuous, and compact embeddings

and if the perturbation does not explicitly depend on u or u ′ , then we do not need to assume U ↪ c W ˜ or V ↪ c W , respectively, but instead that V ↪ c H . We stress that we neither assume U ↪ V nor V ↪ U . The spaces can coincide if a certain embedding is not assumed to be compact. For instance, the cases V = U , W ˜ = H or W = H are admissible. Introducing the spaces W and W ˜ allows us to make use of the finer structure of the spaces, which enables us to treat additional nonlinearities of lower order. As examples for the appearing spaces, we can think of the Sobolev spaces U = W k , p ( Ω ) , V = H l ( Ω ) and the Lebesgue spaces W = L q ( Ω ) and H = L 2 ( Ω ) or U = W k , p ( Ω ) , V = W s , p ( Ω ) , W = H l ( Ω ) and H = L 2 ( Ω ) for suitably chosen numbers k , l ∈ N and real values s , p > 0 , q > 1 .

2.2 Facts from convex analysis

Before we present the precise assumptions on the functionals and the operators, we recall some facts and results from functional and convex analysis. For a proper functional F : X → ( − ∞ , + ∞ ] defined on a Banach ( X , ‖ ⋅ ‖ X ) space, the (Fréchet) subdifferential of F is given by the multivalued map ∂ F : X ⇉ X * with

where ⟨ ⋅ , ⋅ ⟩ denotes the duality pairing between the Banach space X and its topological dual space X * . The elements of the subdifferential are also called subgradients. We denote the effective domain of F and the domain of its subdifferential ∂ F by dom ( F ) ≔ { v ∈ X ∣ F ( v ) < + ∞ } and dom ( ∂ F ) ≔ { v ∈ X : ∂ F ( v ) ≠ ∅ } , respectively. If the set of subgradients of F at a given point u is nonempty, we say that F is subdifferentiable at u . The following lemma states that the subdifferential of the sum of a subdifferential function and a Gâteaux differentiable function equals the sum of the subdifferentials of both functions. More precisely, the following result holds:

Since we are dealing with convex and λ -convex functions, we would like to express the subdifferential in an equivalent way that is easier to handle. The function F is called λ -convex with parameter λ ∈ R if

for all u , v ∈ D ( F ) and t ∈ ( 0 , 1 ) . If λ < 0 and λ = 0 , then the function is called strongly convex and convex, respectively. Employing the definition of a subdifferential and the λ -convexity, it is easy to show that for a λ -convex and proper function F , the subdifferential of F is equivalently given by

It follows immediately that

for all ξ ∈ ∂ F ( u ) and ζ ∈ ∂ F ( v ) . Hence, for λ < 0 , the subdifferential of a λ -convex function F is strongly monotone. This gives rise to the following lemma.

Let us now introduce an important tool from the theory of convex analysis. For a proper, lower semicontinuous, and convex function F : X → ( − ∞ , + ∞ ] , we define the so-called convex conjugate (or Legendre–Fenchel transform) F * : X * → ( − ∞ , + ∞ ] by

By definition, we directly obtain the Fenchel–Young inequality

It can be checked that the convex conjugate itself is proper, lower semicontinuous and convex, see, e.g., Ekeland and Témam [20]. If, in addition, we assume F ( 0 ) = 0 , then F * ( 0 ) = 0 holds as well. We may ask how the convex conjugate of the sum of two functions can explicitly be expressed in terms of the two functions.

For an illustration of the lemma, we consider

The following lemma shows that there is, in fact, a relation between the subgradient of a function and its convex conjugate.

The next result shows a certain chain rule for the subdifferential.

Finally, we recall that the space U ∩ V equipped with the norm ‖ ⋅ ‖ U ∩ V = ‖ ⋅ ‖ U + ‖ ⋅ ‖ V is a separable and reflexive Banach space and the dual space is given by ( U ∩ V ) * = U * + V * with the norm ‖ ξ ‖ U * + V * = inf ξ = ξ 1 + ξ 2 ξ 1 ∈ U * , ξ 2 ∈ V * max { ‖ ξ 1 ‖ U * , ‖ ξ 2 ‖ V * } , see Example 2.4. Furthermore, the duality pairing between U ∩ V and U * + V * is given by

for all v ∈ U ∩ V and any partition f = f 1 + f 2 with f 1 ∈ U and f 2 ∈ V . Second, for any p ∈ [ 1 , + ∞ ] , there holds L p ( 0 , T ; U ) ∩ L p ( 0 , T ; V ) = L p ( 0 , T ; U ∩ V ) , where the measurability immediately follows from the Pettis theorem, see, e.g., Diestel and Uhl [17, Theorem 2, p. 42]. And third, for the Banach spaces X , Y ∈ { U ∩ V , U , V , W , W ˜ , H } satisfying the embedding X ↪ Y , there holds

See, e.g., Brézis [14, Remark 3, p. 136] and Gajewski et al. [29, Kapitel 1, §5].

3.1 Assumptions on the functionals and operators

In this section, we summarize all the assumptions regarding the functionals Ψ and ℰ , and the operators B and f . Since the subdifferential of the main part of Ψ is linear, we refer henceforth to the inclusion (1.1) in the given framework as linearly damped inertial system ( U , V , W , W ˜ , H , ℰ , Ψ , B , f ) . The assumptions we make for the linearly damped inertial system resemble the structure of those we made for the perturbed gradient system in [10] where the same evolution inclusion has been investigated after neglecting the inertial term u ″ ( t ) . Involving inertia makes the situation much more delicate. As a consequence, we will impose, in general, stronger conditions on the functionals and operators to ensure the solvability of the problem. Hereinafter, we collect the assumptions for the dissipation potential Ψ and remind the reader that we distinguish two cases (a) and (b).

• (3.Ψ) Dissipation potential.

• Case (a): we assume that there exists a strongly positive, symmetric, and continuous bilinear form a : V × V → R such that Ψ ( v ) = 1 2 a ( v , v ) , i.e., there is a constant μ > 0 such that

  (3.1) μ ‖ v ‖ V 2 ≤ Ψ ( v ) for all v ∈ V .

  Case (b): we assume that Ψ = Ψ 1 + Ψ 2 , where Ψ 1 ( v ) = 1 2 a ( v , v ) with the bilinear form a : V × V → R as above and Ψ 2 : W → R to be a lower semicontinuous and convex functional with Ψ 2 ( 0 ) = 0 satisfying the following growth condition: there exists a positive number q > 1 and constants c ˆ , C ˆ > 0 such that

  (3.2) c ˆ ( ‖ v ‖ W q − 1 ) ≤ Ψ 2 ( v ) ≤ C ˆ ( ‖ v ‖ W q + 1 ) for all v ∈ W .

  In addition, we assume for the subgradients of Ψ 2 the following growth condition: for all R > 0 , there exists a positive real constant C R > 0 such that

  (3.3) ‖ η ‖ W * ≤ C R ( 1 + ‖ v ‖ W q − 1 ) for all η ∈ ∂ Ψ 2 ( v ) and v ∈ W with ∣ v ∣ ≤ R .

We proceed with collecting the assumptions for the energy functional ℰ . To do so, we define V λ = U if λ = 0 and V λ = U ∩ V if λ > 0 . We make this distinction because for the convex case, i.e., when λ = 0 , we will obtain a stronger result meaning that the initial value u 0 can be chosen to be in dom ( ℰ t ) instead of dom ( ℰ t ) ∩ V as in the λ -convex case with λ ≠ 0 , and that the subgradient of ℰ t is in U * instead of U * + V * , see Theorem 3.4.

1. Lower semicontinuity. For all t ∈ [ 0 , T ] , the functional ℰ t : U → ( − ∞ , + ∞ ] is proper and sequentially weakly lower semicontinuous with time-independent effective domain D ≔ dom ( ℰ t ) ⊂ U for all t ∈ [ 0 , T ] . Furthermore, the set D ∩ V is dense in D in the topology of U , and if ℰ t is convex, the interior of D is nonempty.

2. Bounded from below. ℰ t is bounded from below uniformly in time, i.e., there exists a constant C 0 ∈ R such that

   ℰ t ( u ) ≥ C 0 for all u ∈ U and t ∈ [ 0 , T ] .

   Since the potential of a function is, if it exists, unique up to a constant, we assume without loss of generality, C 0 = 0 .

3. Coercivity. For every t ∈ [ 0 , T ] , ℰ t has bounded sublevel sets in U .

4. Control of the time derivative. For all u ∈ U , the mapping t ↦ ℰ t ( u ) is in C ( [ 0 , T ] ) ∩ C 1 ( 0 , T ) , and its derivative ∂ t ℰ t is controlled by the function ℰ t , i.e., there exists C 1 > 0 such that

   ∣ ∂ t ℰ t ( u ) ∣ ≤ C 1 ℰ t ( u ) for all t ∈ ( 0 , T ) and u ∈ U .

5. λ -convexity. There exists λ ≥ 0 such that for every t ∈ [ 0 , T ] , the energy functional ℰ t is λ -convex on V (by extending ℰ on V ), i.e., for all u , v ∈ U ∩ V and ϑ ∈ ( 0 , 1 ) , there holds

   (3.5) ℰ t ( ϑ v + ( 1 − ϑ ) u ) ≤ ϑ ℰ t ( v ) + ( 1 − ϑ ) ℰ t ( u ) + λ ϑ ( 1 − ϑ ) ‖ v − u ‖ V 2 .

6. Closedness of Gr ( ∂ ℰ ) . For all sequences of measurable functions ( t n ) n ∈ N with t n : [ 0 , T ] → [ 0 , T ] , n ∈ N , ( u n ) n ∈ N , ( ξ n ) n ∈ N , and measurable functions u , ξ satisfying

   • (a) t n ( t ) → t for a.e. t ∈ ( 0 , T ) , as n → ∞ ,

   • (b) ∃ C 2 > 0 : sup n ∈ N , t ∈ [ 0 , T ] ℰ t ( u n ( t ) ) ≤ C 2 ,

   • (c) ξ n ( t ) ∈ ∂ V λ ℰ t n ( t ) ( u n ( t ) ) a.e. in ( 0 , T ) , n ∈ N ,

   • (d) u n − u ˜ 0 ⇀ * u − u ˜ 0 in L ∞ ( 0 , T ; V λ ) and u n − u ˜ 0 → u − u ˜ 0 in L 2 ( 0 , T ; V ) for any u ˜ 0 ∈ D and ξ n ⇀ ξ in L 2 ( 0 , T ; V λ * ) as n → ∞ . In addition, there exists a constant C 3 > 0 such that for sufficiently small h > 0 , there holds

   • Case (a):

     (3.6) sup n ∈ N ‖ σ h u n − u n ‖ L 2 ( 0 , T − h , V ) ≤ C 3 h .

     Case (b):

     (3.7) sup n ∈ N ‖ σ h u n − u n ‖ L 2 ( 0 , T − h , V ) ∩ L r ( 0 , T − h , W ) ≤ C 3 h ,

     where σ h v ≔ χ [ 0 , T − h ] v ( ⋅ + h ) for any function v : [ 0 , T ] → V ,

   • (e) limsup n → ∞ ∫ 0 T ⟨ ξ n ( t ) − ξ ( t ) , u n ( t ) − u ( t ) ⟩ V λ * × V λ d t ≤ 0 ,

   we have the relations

   ξ ( t ) ∈ ∂ V λ ℰ t ( u ( t ) ) ⊂ V λ * , ℰ t n ( t ) ( u n ( t ) ) → ℰ t ( u ( t ) ) as n → ∞ and limsup n → ∞ ∂ t ℰ t ( u n ( t ) ) ≤ ∂ t ℰ t ( u ( t ) ) for a.e. t ∈ ( 0 , T ) .

7. Control of the subgradient. There exist constants C 4 > 0 and σ > 0 such that

   ‖ ξ ‖ V λ * σ ≤ C 4 ( 1 + ℰ t ( u ) + ‖ u ‖ V λ ) ∀ t ∈ [ 0 , T ] , u ∈ D ( ∂ V λ ℰ t ) , ξ ∈ ∂ V λ ℰ t ( u ) .