Differential Equations in Metric Spaces with Applications

This paper proves the local well posedness of differential equations in metric spaces under assumptions that allow to comprise several different applications. We consider below a system of balance laws with a dissipative non local source, the Hille-Yosida Theorem, a generalization of a recent result on nonlinear operator splitting, an extension of Trotter formula for linear semigroups and the heat equation.


Introduction
This paper is concerned with differential equations in a metric space and their applications to ordinary and partial differential equations. More precisely, we modify the structure of quasidifferential equations introduced in [23,24], see also [4], and prove existence, uniqueness and continuous dependence under conditions weaker than those therein. A general estimate on the difference between any Euler polygonal approximation and the exact solution is also provided.
Following the model theory of ordinary differential equations, we provide a result that comprises also other entirely different examples: the heat equation, the Hille-Yosida theorem, the Lie-Trotter product formula and a balance law with a dissipative non local source. In particular, the latter result is an extension of [12] that was announced in [10]. Furthermore, the construction below weakens the assumptions introduced in [9] on the non linear operator splitting technique in metric spaces, as well as those introduced in [21] on linear Lie-Trotter products. Remark that the commutativity condition adopted here, namely (2.7), is optimal, in the sense of Paragraph 2.1.
Following [4,23,24], for any u in the metric space X, the tangent space T u X to X at u is the quotient of the set γ ∈ C 0,1 [0, 1]; X : γ(0) = u of continuous curves exiting u modulo the equivalence relation of first order contact, i.e. γ 1 ∼ γ 2 if and only if lim t→0 1 τ d γ 1 (τ ), γ 2 (τ ) = 0. Any map v: [0, T ] × X → u∈X T u X, such that v(t, u) ∈ T u X for all t and u then defines both a vector field on X and the (quasi)differential equatioṅ u = v(t, u) . Within this framework, we prove that if the vector field v can be defined through a suitable local flow, then the Cauchy problem (1.1) is well posed globally in u o ∈ X and locally in t ∈ R. The proof is constructive and based on Euler polygonal, similarly to [23,24], but with somewhat looser assumptions. This approach to differential equations, being sited in metric spaces, does not rely on any linearity assumption whatsoever. It is therefore particularly suited to describe truly nonlinear or even discontinuous models. In this connection, we refer to the example in Paragraph 3.2 below and to [23,Theorem 3.1], which is contained in the present framework.
The next section is the core of this paper: it presents the definition of local flow and Theorem 2.5, which shows that a local flow generates a global process. We also give an example showing the necessity of the assumptions in Theorem 2.5. Section 3 presents several applications. The first one deals with a recent new result about balance laws with a diagonally dominant non local source. The final Section 4 provides the various technical details.

Notation and Main Result
Throughout, (X, d) denotes a complete metric space. In view of the applications in Section 3, we need to slightly extend the basic definitions about differential equations in metric spaces in [4], see also [2,23,24].
To explain the notation, consider the case of a Banach space X and let D ⊆ X. If v: I × D → X is a vector field defining the ordinary differential equationu = v(t, u), then a local flow (generated by whenever it is defined.
On the other hand, an elementary computation shows that so that the limit ε → 0 is possible only when Lip (F ) ≤ 1. In the case of conservation laws, for instance, the usual procedure to prove the well posedness is based on this key point: the Lipschitz dependence of F ε on u is achieved through the introduction of a functional (see [5,Chapter 8] and [13]) or a metric (see [3,6]) equivalent to the L 1 distance d and according to which the Euler polygonals turn out to be non expansive. Therefore, in the theorem below, the Lipschitzeanity of F ε on u is explicitly required.
With the same notation of Definition 2.1, we introduce what provides a solution to the Cauchy problem (1.1).

Definition 2.4 Consider a family of sets D to ⊆ D for all t o ∈ I, and a set
A global process on X is a map P : A → X such that, for all t o , t 1 , t 2 , u satisfying t 1 , t 2 ≥ 0, t o , t o + t 1 + t 2 ∈ I and u ∈ D to , satisfies To state the main theorem, we consider the following sets for any t o ∈ I Theorem 2.5 The local flow F is such that there exists 1. a non decreasing map ω: whenever τ ∈ [0, δ], k ∈ N and the left hand side above is well defined; 2. a positive constant L such that whenever ε ∈ ]0, δ], u, w ∈ D, t ≥ 0, t o , t o + t ∈ I and the left hand side above is well defined.
Then, there exists a family of sets D to and a unique a global process (as defined in Definition 2.1) P : A → X with the following properties: c) P is tangent to F in the sense that for all u ∈ D to , for all t such that t ∈ ]0, δ] and t o + t ∈ I: The proof is deferred to Section 4. Observe that in the general formulation of Theorem 2.5, the set A where P is defined could be empty. However, in the applications, the following stronger condition, equivalent to [22,Condition 2.], is often satisfied: (D) There is a familyD to of subsets of D satisfying F (t, t o )D to ⊆D to+t for any t ∈ [0, δ] and t o , t o + t ∈ I If (D) holds, then obviously one hasD to ⊆ D 3 to ⊆ D to for any t o ∈ I, providing a lower bound for the set where the process is defined. If (D) does not hold, then a lower bound on A needs to be found exploiting specific information on the considered situation, see Paragraph 3.4. Observe that, by (2.9), the curve τ → P (τ, t o )u represents the same tangent vector as τ → F (τ, t o )u, for all u.
We remark that assumption 1 is satisfied, for instance, when 2. F is defined combining two commuting Lipschitz semigroups through the operator splitting algorithm.
Once the global process is built and its properties proved, the following well posedness result for Cauchy problems in metric spaces is at hand.
The proof is essentially as that of [4, Corollary 1, § 6] (see also the proof of [5, Theorem 2.9]) and is omitted.
Here, moreover, we show that all Euler polygonals of F converge to P , providing the following estimate on the speed of convergence.
Proposition 2.7 With the same assumptions of Theorem 2.5, fix u ∈ D to . Let F E be defined as in (2.2). If F E (s, t o ) ∈ D to+s for any s ∈ [0, t] and ∆ = max h (τ h+1 − τ h ) ∈ ]0, δ], then the following error estimate holds:

A Condition Weaker than (2.7) Is Not Sufficient
We now show that requiring (2.7) for k = 1 only: does not allow to prove the semigroup property in Theorem 2.5.

Proposition 2.8 Let the hypotheses of Theorem 2.5 hold with (2.7)
replaced by (2.10). Assume that (D) holds. Then, there exists a map P as in Theorem 2.5, but with the "semigroup" condition (2.5) replaced by the weaker: and only continuous (not necessarily Lipschitz) with respect to (t, t o ).
The proof is deferred to Section 4. Note that P is not necessarily a process. Indeed, we show by an example that this property may fail, due to the fact that condition (2.7) may fail for k ≥ 2.
In the metric space X = [0, 1] with the usual Euclidean distance we construct an example showing that the assumptions in Proposition 2.8, namely (2.10) instead of (2.7), do not suffice to show that P is a process.
Let ϕ be any non constant function satisfying . Define now The following lemma, listing the main properties of f and F above, is immediate and its proof omitted.
Lemma 2.9 With the notation above, f ∈ C 0,1 [0, +∞[ ; R and f (2 n t) = f (t) 2 n for all t ≥ 0 and n ∈ N. Choosing D t = X for all t > 0, the map F satisfies Definition 2.1, in particular so that F is non expansive in u and the Euler polygonals F ε trivially satisfy (2.8). Finally, F satisfies (2.10).
We now prove that (2.7) is strictly stronger than (2.10) and that the latter assumption may not guarantee the existence of a gobal process. Since the local flow F does not depend on t o , we write F (t) for F (t, t o ).

Proposition 2.10
For any fixed u ∈ X, there does not exist any Lipschitz semigroup P tangent to F at u in the sense of (2.9).
Proof of Proposition 2.10. By contradiction, let P be a Lipschitz semigroup tangent to F in the sense of (2.9).
Since with P (t)u we denote the action of P on the real number u ∈ [0, 1], to avoid confusion, we explicitly denote with the dot any product between real numbers.
First, we verify that P must be multiplicative, i.e. , P (t)u = P (t)1 · u (remember that P (t)1 is the action of P on the number 1 ∈ X = [0, 1]). By [5, Theorem 2.9] applied to the Lipschitz curve τ → w(τ ) = P (τ )1 · u where we have used the tangency condition (2.9) and the equality We may now denote p(t) = P (t)1 so that P (t)u = p(t)·u. The invariance of [0, 1] implies p(t) = P (t)1 ∈ [0, 1] for all t and, by (2.9), Moreover, for any n ∈ N, applying Lagrange Theorem to the map x → x 2 n on the interval between f (2 −n t) 2 n and p(2 −n t) 2 n ,

Applications
In the autonomous case, we often omit the initial time is satisfied in all but one case.

Balance Laws
Consider the following nonlinear system of balance laws: where f : Ω → R n is the smooth flow of a nonlinear hyperbolic system of conservation laws, Ω is a non empty open subset of R n and G:  allows to prove also 2. The details are found in [11], where the present technique is applied to the diagonally dominant case [14, formula (1.14)], yielding a process defined globally in time.

Constrained O.D.E.s -The Stop Problem
Let X be a Hilbert space and fix a closed convex subset C ⊆ X. For a positive T , let f : [0, T ] × C → X. Consider the following constrained ordinary differential equation: The stop problem above was considered, for instance, in [20], see also [19,Section 4.1].
We denote by Π: X → C the projection of minimal distance, i.e. Π(x) = y if and only if y ∈ C and x − y = d(x, C). Recall that Π is non expansive.
This problem fits in the present setting, for instance when f and C satisfy the following conditions: As usual, we denote B(C, d) = x ∈ X: d(x, C) < d . By [17,Theorem 2], if X = R n and C is a compact convex set with C 3 boundary, then (C) holds.
is a local flow that satisfies (D) and the hypotheses of Theorem 2.5.
Proof. The continuity of F , as well as its Lipschitzeanity in t and u, is immediate.
Consider (2.7). Fix τ, τ ′ ∈ [0, δ] and compute: Consider the two summands above separately. Using (C), the latter one is estimated in Lemma 4.10. The former is estimated as follows: Summing up the two bounds for the two terms we obtain We note that condition (C) can be slightly relaxed. Indeed, Proposition 3.1 essentially requires that (4.5) is satisfied. This bound can be proved also with only the C 1,α regularity of Π.

Nonlinear Operator Splitting in a Metric Space
The construction in Theorem 2.5 generalizes that in [9]. Indeed, consider the case of two Lipschitz semigroups S 1 , S 2 : R + × X → X and assume that they commute in the sense of [9, (C)], i.e.
for a suitable map ω: [0, δ/2] → R. In this framework, let D t = X for all t ∈ R + , so that A = R + × R + × X. Introduce Note that F is a local flow in the sense of Definition 2.1. The next lemma shows that (3.2) implies 1. in Theorem 2.5.
The following corollary of Theorem 2.5 that extends [9, Theorem 3.8] is now immediate. Corollary 3.3 Let S 1 , S 2 be Lipschitz semigroups satisfying (3.2). Suppose also that the two semigroups are Trotter stable, i.e. for any t ∈ [0, T ], n ∈ N \ {0} and u, w ∈ X: Then, the Euler ε-polygonals with F as in (3.3) converge to a unique product semigroup P tangent to the local flow F (τ )u = S 1 τ S 2 τ u.
Here we do not require the strong commutativity condition [9, (C * )] as in [9, Theorem 3.8], but only its weaker version (3.2). Note also that the assumption [9, (S3)] requiring d(S t u, S t w) ≤ e Ct d(u, w) is stronger then the Trotter stability requirement (3.4) as shown in the following lemma.

Trotter Formula for Linear Semigroups
The present non linear framework recovers, under slightly different assumptions, the convergence of Trotter formula [25]   (c) the commutator condition is satisfied for all u ∈ Y and t ∈ [0, δ] with some δ > 0, and for a suitable ω: (II) all Euler polygonals with initial data in Y , as defined in (2.2) and in (2.3) with F (t) = S 1 t S 2 t , converge to the orbits of P .

Remark 3.6 With respect to [21, Theorem 3], the regularity assumptions on the semigroups are stronger due to (a). On the other hand, (c) is weaker and we further obtain (I) and (II).
In the theorem above, standard techniques allow to relate the generators of S 1 and S 2 with that of P , see [ In the setting of Theorem 2.5, consider the metric space X, · X and define where the closure is meant with respect to the X norm. It is straightforward to show that F is a Lipschitz local flow on D. The stability (with respect to the X norm) of the Euler ε-polygonals implies hypothesis 2. Computations similar to those in the proof of Lemma 3.2 show that also hypothesis 1. is satisfied. Hence, Theorem 2.5 yields the existence of a process tangent to the local flow F . The stability (in the Y norm) of the Euler ε-polygonals ensures that clD * M ⊆ D 3 to for any t o ∈ I. Finally, the arbitrariness of M allows us to extend P to all Y and, by density, to all X.

Hille-Yosida Theorem
This section is devoted to show that Theorem 2.5 comprehends the constructive part of Hille-Yosida Theorem, see [

Theorem 3.7 Let X be a Banach space and A be a linear operator with domain D(A). If A, D(A)
is closed, densely defined and for all λ > 0 one has λ ∈ ρ(A) and λ R(λ, A) ≤ 1, then A generates a strongly continuous contraction semigroup. A and R(λ, a) is the resolvent. Note that we consider only the contractive case and refer to [15,Theorem 3.8, Chapter II] to see how the general case can be recovered. Proof of Theorem 3.7. Fix a positive M and define

Above, as usual, ρ(A) is the resolvent set of
It is easy to see that F (t)D ⊆ D, so that (D) holds. Moreover, the stability condition 2. in Theorem 2.5 trivially holds in the contractive case. Recall the usual identities for the resolvent (see [15, Chapter IV]) λ R(λ, A)u = u + R(λ, A) A u for any u ∈ D(A) Written using the local flow F , these identities become Now we first show that F is Lipschitz in t and u. The Lipschitz continuity with respect to u is a straightforward consequence of the bound on the resolvent norm. Concerning the variable t, we take u ∈ D(A) with Au ≤ M and consider two cases: Since F (t) is a bounded operator, the Lipschitz continuity with constant M extends to all D. We are left to prove hypothesis 1. in Theorem 2.5. For Again, the boundedness of F (t) allows us to extend the inequality to all u ∈ D. Therefore, letting t = kτ , s = τ , Theorem 2.5 applies with ω(τ ) = 3M τ . Due to the arbitrariness of M and the density of D(A 2 ), the resulting semigroup can be extended to all X. Standard computations, see [16, p. 362-363], show that A is the corresponding generator.
Note that, by Proposition 2.7, we also provide the convergence of all polygonal approximation.

The Heat Equation
Let T , δ be positive and X = C 0 B (R; R) be the set of continuous and bounded real functions defined on R equipped with the distance d(u, w) = u − w C 0 , where u C 0 = sup x∈R u(x) . Fix a positive M and for all t o ≥ 0 let D to = D be the subset of X consisting of all twice differentiable functions whose second derivative u ′′ satisfies max u ′′ , Lip (u ′′ ) ≤ M . For t ∈ [0, δ], t o ∈ R and u ∈ X, using the numerical algorithm [18, Chapter 9, § 4], define Proposition 3.8 F is a local flow satisfying (2.7) and (2.8).
Consider now 1. and write s = kt for k ∈ N. Then showing that F is non expansive in u and, hence, that 2. holds.

Technical Details
Occasionally, for typographical reasons, we write d u, w for d(u, w).
In this section, we use the following definition Observe that one obviously has D 3 to ⊆ D 2 to . Proof of Lemma 2.3.
Let k = [t/ε] and h = [s/ε] + 1. The case h > k is recovered by the previous computation. If h ≤ k, then by (2.3) to , the following holds: where Proof. The Definition (4.1) of D 2 to implies that we can apply the triangle inequality and hypothesis 1. in Theorem 2.5 to obtain completing the proof. Then, for all h, k, t o , u and ε satisfying t o ∈ I, h, k ∈ N, kε ∈ ]0, δ], t o +hkε ∈ I and u ∈ D 2 to , the following holds: Proof. The Definition (4.1) of D 2 to implies that we can apply the triangle inequality. Then, by the assumptions in Theorem 2.5, applying Lemma 4.1 and with computations similar to the ones in [21], we have completing the proof. Then, for all m, n, t, t o and u satisfying t > 0, t o , t o + t ∈ I, m, n ∈ N, n ≥ m, t2 −m ∈ ]0, δ] and u ∈ D 2 to , the following holds: Proof. Applying Lemma 4.2 with ε = t2 −(j+1) , k = 2, h = 2 j and t = hkε, converges uniformly to a continuous map (t, t o , u) → P (t, t o )u.
Proof. Observe that the maps in the sequence above are continuous, since  On the other hand, to prove these lemmas for k = 2, it is enough to assume 1. of Theorem 2.5 only for k = 1. Assumption 1. in Theorem 2.5 for all k ∈ N is needed to show that all the sequence of Euler approximates F ε converges (Proposition 4.6) and, hence, that the limit is indeed a process (Lemma 4.8). The necessity of hypothesis 1. for all k ∈ N is shown in Paragraph 2.1 Proposition 4.6 Let F be a local flow satisfying the assumptions of Theorem 2.5. Then, for all t o , t, u with t ≥ 0, t o , t o + t ∈ I and u ∈ D 2 to , the following limit holds: with P defined as in Corollary 4.4. Proof. Fix t o , t, u as above. Let n ε = [t/ε]. Observe first that as ε → 0, due to the Lipschitz continuity of F and the continuity of P . With computations similar to the ones found in [21], for l ∈ N such that T 2 −l < δ write: Consider the three terms separately. Estimating the former one, apply first Lemma 4.2 with h = n ε 2 j , k = 2 and ε substituted by ε2 −(l+1) and then treat the summation as in the proof of Lemma 4.3: → 0 as ε → 0 + uniformly in l.
Finally, also the latter term in (4.2) vanishes as l → +∞ by Corollary 4.4. The proof is completed passing in (4.2) to the limits first l → +∞ and, secondly, ε → 0.
Remark 4.7 Observe that, since F ε (t, t o )u is uniformly Lipschitz in (t, u), P (t, t o )u is also uniformly Lipschitz in (t, u) with the same constant as F ε . Lemma 4.8 There exists a family of sets D to ⊆ D, for t o ∈ I such that i) D 3 to ⊆ D to ⊆ D 2 to for any t o ∈ I; ii) The map P defined in Corollary 4.4 and restricted to the set A = (t, t o , u): t ≥ 0, t o , t o + t ∈ I, u ∈ D to is a global process according to Definition 2.4 and Lipschitz in all its variables.
Proof. Define Obviously, D to ⊆ D 2 to . Take now u ∈ D 3 to , we want to show that u also belongs to D to . By (2.6) for any ε, ε 1 , ε 2 ∈ ]0, δ] and t, t 1 , t 2 ≥ 0 such that t o + t + t 1 + t 2 ∈ I. Since u ∈ D 2 to , D is close and F ε 1 , F ε 2 are Lipschitz, we let ε → 0 and obtain for any ε 1 , ε 2 ∈ ]0, δ] and t 1 , to for all t ≥ 0 such that t o + t ∈ I and hence u ∈ D to . For t o ∈ I and u ∈ D 2 to ⊃ D to one trivially has P (0, t o )u = u. We are left to prove the "semigroup properties" (2.4) and (2.5). We first show (2.5) for any u ∈ D to . If t 1 or t 2 vanishes, property (2.5) is trivial. Fix t 1 > 0 and take t 2 > 0 such that t o + t 1 + t 2 ∈ I and t 2 t 1 ∈ Q so that there exist two integers h, k ∈ N \ {0} which satisfy t 1 Since u ∈ D to ⊆ D 2 to one has that F εν (t 2 +t 1 , t o )u converges to P (t 2 +t 1 , t o )u as ν → +∞ (see Proposition 4.6). Moreover the definition of D to implies that P (t 1 , t o )u ∈ D 2 to+t 1 and therefore F εν ( Hence taking the limit as ν → +∞ in (4.3) we obtain (2.5) for any t 2 with t 2 t 1 ∈ Q. The continuity of P concludes the proof of (2.5). Now, if u ∈ D to then, by definition, P (t 1 , t o )u ∈ D 2 to+t 1 . But (2.5) implies also that P (t, t o + t 1 ) • P (t 1 , t o )u = P (t + t 1 , t o )u ∈ D 2 to+t+t 1 for any t ≥ 0 such that t o + t + t 1 ∈ I, therefore P (t 1 , t o )u ∈ D to+t 1 proving (2.4). Finally, the Lipschitz continuity with respect to t and u follows from Remark 4.7, while the Lipschitz continuity with respect to t o is a direct consequence of the semigroup property. Indeed, take 0 ≤ t 1 ≤ t 2 ≤ T , u ∈ D t 1 ∩ D t 2 and use the Lipschitz continuity with respect to (t, u): d P (t, t 1 ) u, P (t, t 2 ) u = d P (t, t 2 ) • P (t 2 − t 1 , t 1 ) u, P (t, t 2 ) u ≤ L · d P (t 2 − t 1 , t 1 ) u, u ≤ L · Lip (F ) · (t 2 − t 1 ) .
The following Lemma concludes the proof of Theorem 2.5. Proof. When t ∈ ]0, δ] one has F t (t, t o )u = F (t, t o )u. Hence the Lemma is proved taking t ∈ ]0, δ], m = 0 and n → +∞ in Lemma 4.3.
Proof of Proposition 2.7. Use the Lipschitz continuity of P and (2.9): Proof of Proposition 2.8. Since we assume that (D) is satisfied, we have no problem with the domains, i.e.
F (t, t o )D to ⊆ D to+t , for t ∈ [0, δ] . (4.4) By Remark 4.5, F t2 −m (t, t o ) u converges uniformly to a map P (t, t o ) u which is continuous in (t, t o ) and uniformly Lipschitz in u. With m = 0 n → ∞ in Lemma 4.3 we get immediately the tangency condition. Hence, because of (4.4) P has all the properties of the process in Theorem 2.5 except the Lipschitz continuity with respect to (t, t o ) and the "semigroup" property (2.5). Finally, since as n → +∞ we get (2.11).

Lemma 4.10
If C is a closed and convex subset of the Hilbert space X that satisfies (C), then for all u ∈ C, v ∈ X and τ, τ ′ ≥ 0, Proof. Assume first that v = 1.
If u + τ v ∈ C, the right hand side above vanishes and the inequality trivially holds.