PARTIAL DIFFERENTIAL INCLUSIONS OF TRANSPORT TYPE WITH STATE CONSTRAINTS

. The focus is on the existence of weak solutions to the quasilinear ﬁrst-order partial diﬀerential inclusion ∂ t f ∈ − div x (cid:0) G ( t,f ) f (cid:1) + U ( t,f ) · f + W ( t,f ) with values in L p ( R N ) for p ∈ (1 , ∞ ). The solution is to satisfy state constraints in addition, i

1. Introduction. A large class of mathematical models is formulated in terms of a continuity equation or, more generally, a transport equation in divergence form Whenever the models take a form of feedback into consideration, the coefficient functions g, u, w do not depend just on time t and space x, but also on the current state described as f = f (t, x). Clearly, this leads to nonautonomous and nonlinear partial differential equations of first order. Although these analytical criteria make an inevitable impression, they open the door to various and extended theories of hyperbolic equations.
In this article, the motivation is based on hyperbolic models for cancer cell migration (like in [66]) and traffic flow (such as [58]). This context justifies the conceptual aspects that nonlocal dependence (in space) is taken into consideration. Indeed, drivers of vehicles on a highway can watch the vehicles in their respective neighborhoods (and strictly speaking, they are even obliged to do so). The resulting relationships between vehicles are usually described in terms of convolution operators instead of Nemytskii operators. Nonlocal models of hyperbolic type have already been investigated thoroughly by Colombo, Goatin, Quincampoix and collaborators, for example (see [1,18,28,29,30,27,31,45,67] and references therein). From the analytical point of view, similar problems have occurred in models for migrating cancer cells. Indeed, each cell has a small, but positive extension and the cells are communicating with each other by means of signaling substances. More details about chemotaxis and adhesion are described in, e.g., [32,36,37,38,39,40,44,53,56,57,66,71]. These model classes motivate our interest in nonlinear hyperbolic problems with functional dependence, i.e., with given coefficient functions of appropriate regularity for p ∈ (1, ∞) and q := p p−1 ∈ (1, ∞). Strictly speaking, we focus on renormalized solutions (in the sense of Di Perna and Lions [35,60]) and so, the coefficients G, U will even be assumed to have their values in the Sobolev spaces W 1,∞ (R N , R N ), W 1,∞ (R N ) respectively. This class of partial differential equations has already been investigated in, e.g., [58] (see also [63, § § 2.5, 3.8] and [65,66]).
The new contribution of this article is motivated by uncertainty: The coefficient functions G, U and W are not single-, but set-valued. This mathematical change is required whenever the relationships between state function f (t) ∈ L p (R N ) and the coefficients are not known precisely or if the coefficients depend on an openloop control in addition. We prefer set-valued maps to probabilistic approaches for preserving the deterministic character of the mathematical concept rather than neglecting a class of events from the very beginning. This leads to a so-called partial differential inclusion of first order for a function f : [0, T ] −→ L p (R N ) with the set-valued coefficient mappings given. Our main result concerns sufficient conditions on the set-valued coefficients G, U, W and a set of state constraints V ⊂ L p (R N ) such that every function f 0 ∈ V initializes at least one renormalized solution f : [0, T ] −→ L p (R N ) with all its values in V. Then the subset V ⊂ L p (R N ) is usually called weakly invariant or viable w.r.t. the corresponding differential inclusion (see, e.g., [8,11,13,23,26,41,52,74,75,76] and references therein). The criterion is based on a tangential condition -adapting Bouligand's contingent cone to L p (R N ), but supplied with an appropriate metric d L p (closely related to its weak topology as we will specify in Appendix A.2). This aspect is not really surprising because many other types of differential inclusions have already been investigated in regard to viability since Nagumo's first publication [70] about ordinary differential equations in 1942. Indeed, most viability results in the literature so far concern ordinary differential inclusions (usually for values in the Euclidean or a Banach space, see, e.g., [8,23,24,26,42,41,48,49,61,77] and references therein) or, the authors consider semilinear evolution inclusions in Banach spaces (see, e.g., [19,22,23,52,74,76] and references therein). There are, however, much less results available considering nonautonomous evolution inclusions with nonlinear semigroups, whose generators are usually hemicontinuous monotone w.r.t. state and assumed to satisfy a coercivity-type condition (see, e.g., [52, § I.4] for details and [55,68,81,82,83] as well as references therein). To the best of our knowledge, initial value problem (1) does not belong to the classes of evolution problems covered in the literature about viability theory so far: • We focus on nonautonomous and nonlinear transport inclusions of first order in L p (R N ). They cannot be reformulated as a semilinear evolution inclusion by means of a semigroup in an obvious way and hence, this problem is not covered by the results concerning evolution inclusions (like in, e.g., [23,69,74,76]). • The influence of the respective state f (t) ∈ L p (R N ) on the coefficient mappings G, U, W in inclusion (1) is of functional type and so, various forms of nonlocal dependence in space R N can be taken into consideration. • There are no assumptions about the set-valued coefficient mappings G, U imitating the gist of monotonicity or parabolicity. • The compositions of the coefficient mappings with f (t) like G t, f (t) : [0, T ] ⇒ W 1,∞ (R N , R N ) have their set values in the bounded and Lipschitz continuous functions of space, but this regularity (in space) is not restricted to a pointwise and continuous dependence on f (t) : R N −→ R (as suggested for quasilinear conservation laws in, e.g., [16, § § 3.3, 5.5]). The main result is formulated in subsequent Theorem 2.4. Briefly speaking, the tangent condition is related to the so-called contingent cone by Bouligand as introduced for subsets of finite-dimensional vector spaces in, e.g., [8,11,12,14], formulated for Banach spaces in, e.g., [19,23,26,33] and finally extended to metric spaces in, e.g., [9,10,63,64]. In more detail, for Lebesgue-almost every (L 1 -a.e.) t ∈ [0, T ) and every element φ ∈ V ⊂ L p (R N ), we assume (at least) one tuple g, u, w ∈ G(t, φ) × U(t, φ) × W(t, φ) such that the renormalized solution ϑ g,u,w (·, φ) := f : [0, 1] −→ L p (R N ) of the autonomous linear initial value problem Here the choice of the metric d L p on L p (R N ) proves to play a key role. On the one hand, it should be sufficiently "weak" so that the asymptotic condition (3) is not too restrictive. On the other hand, it specifies the topology on the domain of the set-valued coefficient mappings Based on earlier results about transport equations in [58,63,66], the state space L p (R N ) is now supplied with the metric It metrizes the weak topology on (norm-) bounded and tight subsets of L p (R N ) as formulated in more detail in Appendix A.2 below.
In other words, the main theorem, Theorem 2.4, specifies an existence result for solutions under state constraints. The proof follows essentially the same steps of constructing approximate solutions and extracting appropriately converging subsequences as they are known for ordinary differential inclusions (see, e.g., [8,12]) and as they have also been applied to set-valued states in, e.g., [62,63,64]. This approximating approach is reported to go back to Haddad [48,49].
Here we consider nonautonomous differential inclusions though and, we would like to keep the regularity assumptions w.r.t. time rather weak so that the typical situation of open-loop control problems is still covered. In this context, the standard choice of regularity is based on Lebesgue measurability. The contingent hypothesis (3) is formulated in terms of the flow ϑ g,u,w : [0, 1]×L p (R N ) −→ L p (R N ) induced by the autonomous linear initial value problem (2). Hence, it suggests itself to construct the approximate solutions by means of autonomous linear auxiliary problems in a way piecewise in time (see Subsection 3.2 below). In particular, we can benefit from earlier results about transport equations proved in [58] and summarized in Appendix B. After selecting appropriately converging subsequences of both the approximate solutions and the coefficient mappings, we need a form of closed graph property w.r.t. time and state for verifying that the limits lead to a solution of the original partial differential inclusion (1).
This step is based on an extension of the Scorza-Dragoni theorem. Originally established for real-valued functions of two real variables in [73], the statement about "almost continuity" has been extended to set-valued maps in various regards. Our focus of interest is on restrictions with closed graphs -and in the general situation of metric spaces like (L p (R N ), d L p ). There are several generalizations in these directions available in the literature (such as, e.g., [54,72]), but they usually provide just a further set-valued map whose closed graph is contained in the given graph and whose set values might be empty ([72, Theorem 1], for example, is cited in subsequent Lemma A.1). At first glance, it is not clear at all how to draw any conclusions about this additional set-valued mapping in regard to contingent hypothesis (3).
Motivated by the general interest in Scorza-Dragoni-type results, we investigate in more detail which conclusions can be drawn from the established assumptions in [54,72]. In Corollary A.3 below, the (joint) measurability of the set-valued coefficient mappings is identified as an appropriate assumption for guaranteeing the "almost" closed graph (as required for the proof of main Theorem 2.4). It might be worth mentioning that this result is a special case of [15, Theorem 2], but we give a different proof (by means of other auxiliary functions) and formulate it here for the sake of a self-contained and clarifying presentation.
The article has the following structure. In Section 2, we give the definition of a weak solution to the first-order partial differential inclusion (1) and specify the metric d L p on the basic set L p (R N ). This lays the foundations for formulating our main result, i.e., Theorem 2.4. The detailed proofs concerning Theorem 2.4 can be found in Section 3. Several analytical tools are collected in Appendices A and B. Indeed, Section A.1 provides extensions of the Scorza-Dragoni theorem to set-valued maps between metric spaces -with special focus on the closed graph property (not on continuity or lower semicontinuity as frequently formulated in the literature). In Appendix A.2, we collect various results about the metrics d L p ,ȇ L p used on tight subsets of the state space L p (R N ). Then, Section A.3 provides the metric δ L q ,loc of local L q norm convergence in R N which proves to be suitable for the values of coefficient mappings G, U. In Appendix B, we collect several properties of the renormalized solutions to nonautonomous linear transport equations. Their proofs can be found in [58], for example.
2. Main results. On our way to the main result, the first step consists in the analytical characterization which type of solution we are interested in. In particular, we choose the state space L p (R N ) deliberately because some forms of singularities (w.r.t. space) are still mathematically admissible. This supplementary advantage has already been pointed out in the article [66] about cancer cell migration.
General Hypothesis. Fix p ∈ (1, ∞) arbitrarily and q := p p−1 ∈ (1, ∞). Furthermore, L q loc (R N ) and L q loc (R N , R N ) are supplied with the topology of L q norm convergence on each compact subset of R N . L 1 and L N abbreviate the Lebesgue measure on R and R N respectively.
for any 0 ≤ t 1 < t 2 ≤ T and ϕ ∈ C 1 c (R N ). In the case of more regularity w.r.t. space, solutions to the corresponding transport equations prove to be even renormalized and, this leads to the following class (investigated in [58] in more detail): be defined by means of the renormalized (and thus, unique weak) solution to the autonomous linear transport equation (as introduced by Di Perna and Lions [35], see, e.g., [3,4,5,58,60,66] for more details).
(iii) (Locally uniform choice of function dominating values of W) For every radius r > 0, there exist w ∈ L p (R N ) and a compact set , · L p are measurable with nonempty, closed and convex values.
(v) At L 1 -a.e. time instant t ∈ [0, T ], the graphs of the following set-valued maps Let V be a nonempty subset of L p (R N ) with the following properties: (vi) V ⊂ L p (R N ) is norm-bounded and weakly closed.
(viii) For L 1 -a.e. t ∈ [0, T ) and every element φ ∈ V ⊂ L p (R N ), there exists a tuple g, u, w ∈ G(t, φ) × U(t, φ) × W(t, φ) with Then, every function f 0 ∈ V initializes a solution f : [0, T ] −→ L p (R N ) of the following partial differential inclusion with state constraint 3. Proof of main theorem 2.4. In a word, we follow a concept of approximate solutions well-established in the literature of viability theory (see, e.g., [8,19,23,47]) and then extract appropriately converging subsequences (see § § 3.2 -3.4 below). The limit curve in L p (R N ) proves to be a renormalized solution of a transport equation ( § 3.5). Finally, the Scorza-Dragoni-type result in Corollary A.3 lays the foundations for concluding indirectly that the limits of the coefficient curves are related to selections of the set-valued coefficient mappings L 1 -a.e. in [0, T ] ( § 3.6). Hence, the limit curve in L p (R N ) is a viable solution of the partial differential inclusion (in the sense of Definition 2.1).

3.1.
A priori bounds for solutions to the PDI with state constraints.
This uniform a priori bound results directly from Proposition B.1 (1.) due to hypothesis 2.4 (i). In particular, it provides the radius r max : For all the considerations from now on, hypothesis 2.4 (ii) provides a priori bounds γ a , γ b for the time-dependent coefficients as required in Proposition B.1 (i), i.e., for all t ∈ [0, T ] and every φ ∈ According to hypotheses 2.4 (vii), the subset |φ| p φ ∈ V ⊂ L 1 (R N ) is tight in Using the modulus of tightness ω(·) specified on the basis of the a priori bounds ) for all ρ > 0 is tight, convex and norm-bounded with V ⊂ L. Hence, the metric space (L, d L p ) is compact due to Corollary A.12 and the lower semicontinuity of L p norms w.r.t. weak convergence. It is worth mentioning that all subsequent approximate solutions of the PDI will have their values in L.

Constructing approximate solutions of PDI with state constraints.
We adapt the construction of approximate solutions which was developed by Haddad [47,48] for functional differential inclusions and is presented for the case of ordinary differential inclusions in [8], for example. There is a modification though: We do not use the projections on the constraint set V ⊂ L p (R N ) (i.e., minimizers of the d L p distance to V) because their existence results from local compactness of the underlying basic set, which is usually regarded as a rather restrictive requirement. Instead, we prefer their approximations as a separate component of the tuplesimilarly to what Bothe suggested in [20].
ε and such that this contingent condition (i.e., equation (6) in Theorem 2.4) holds for every t ∈ J ε . Now we formulate the approximate solutions and their relevant features for each ε ∈ (0, 1).
is a renormalized solution of the nonautonomous linear transport equation holds for all t ∈ [0, T ] and ρ > 0, (g) P ε is piecewise constant "to the left" in (0, T ), i.e., in the sense that for each Its proof is based on Zorn's lemma applied to the set A ε (f 0 ) of all tuples τ , f (·), P (·), τ (·), g(·), u(·), w(·) consisting of a scalar τ ∈ [0, T ] and functions (h') there exists an (at most) countable family [a j , b j ) j∈J P of pairwise disjoint intervals whose union is [0, τ ) and which satisfies for all j ∈ J P and t ∈ [a j , b j ) P has at most finitely many points of discontinuity in aj +bj A ε (f 0 ) is supplied with the order relation motivated by extension, i.e., We present the further steps of the proof in the following lemmata. Finally, an indirect conclusion from Lemma 3.3 reveals τ = T for the maximal element of A ε (f 0 ), .
Proof. There are two cases excluding each other: In a word, we follow the proof strategy of [64,Lemma 4.15], but for states in L p (R N ) (rather than compact subsets of a Hilbert space). Due to the con- JustP remains to be defined on ( τ , τ + h 1 ]. Furthermore we are going to specify some sufficiently small ρ ∈ (0, h 1 ) such that the respective restrictions to [0, τ + ρ] fulfill the conditions of A ε (f 0 ). This will be in form of ρ := h m0 with an index m 0 ∈ N. Then the index set J P of τ , f , P , τ , g, u, w ∈ A ε (f 0 ) mentioned in condition (h') is extended by a new indexj and, we set aj := τ , bj := τ + ρ. For every index m ∈ N, we conclude from Lemma B.3 i.e., the last inequality of condition (b') is also satisfied at time t = τ + h m for every index m ∈ N.
Next, the first inequality of condition (b') is verified for each m ∈ N sufficiently large. The functions are continuous w.r.t.ȇ L p according to Proposition B.1 (2.). Due to the tightness property in Proposition B.1 (6.), Corollary A.11 guarantees their continuity w.r.t.
The remaining parts of condition (b') result from the construction ofP andτ .
Finally, we focus on property (h'). For every t ∈ τ , τ + h m0 , there exists a unique index m ≥ m 0 with t − τ ∈ (h m+1 , h m ] and, we conclude from Remark A.7 and Proposition B.1 (2.) Moreover, we obtain T ] as in the first case and, set is extended by a new indexj again and, we set aj := τ , bj In regard to the assumptions of Zorn's lemma (see, e.g., [80, § 0.I]), subsequent Lemma 3.4 implies the following statement: Every totally ordered subset of A ε (f 0 ) has an upper bound w.r.t. .
for each k ∈ N, there exists a tuple τ , f , P , τ , g, u, w ∈ A ε (f 0 ) with τ k , f k , P k , τ k , g k , u k , w k τ , f, P, τ, g, u, w for every k ∈ N.
Proof. We adapt the notions proving [64,Lemma 4.16] to renormalized solutions in L p (R N ) (again). Assuming that ( τ k , f k , P k , τ k , g k , u k , w k ) k∈N is monotone w.r.t. , the sequence ( τ k ) k∈N is non-decreasing in [0, T ] and so, it tends to some τ ∈ [0, T ]. Next, we specify candidates for the wanted functions f , P , τ , g, u, w in [0, τ ) by choosing the index k ∈ N sufficiently large: For each t ∈ [0, τ ), there exists some k 0 ∈ N with t < τ k ≤ τ for every k ≥ k 0 and, set f (t) := f k (t), P (t) := P k (t), Due the Λ-Lipschitz continuity of each f k (k ∈ N) in condition 3.2 (e), f ( τ k ) k∈N is a Cauchy sequence w.r.t.ȇ L p . The uniform tightness property formulated in Corollary B.2 implies in addition that f ( τ k ) = f k ( τ k ) (k ∈ N) induce a tight sequence and so, they converge w.r.t. both d L p andȇ L p according to Corollary A.10 and A.11. Its limit is called f ( τ ) ∈ L p (R N ) and coincides with the corresponding weak limit. Hence, the lower semicontinuity of the norm w.r.t. weak convergence (see, e.g., [21,Prop. 3.5 (iii)]) leads to f ( τ ) p L p (R N \Bρ) ≤ e κ τ ω(ρ) for every ρ > 0. Similarly, we specify P ( τ ) ∈ V ⊂ L p (R N ): Each function P k : [0, τ k ] −→ L p (R N ) is related to an (at most) countable family of pairwise disjoint subintervals [a k,j , b k,j ), j ∈ J P k , as specified in condition (h'). The monotonicity w.r.t. implies for every k ∈ N Then the family [a k,j , b k,j ) for any k ∈ N, j ∈ J P k is countable and has the property that any two subintervals are either disjoint or their left boundary points coincide. This induces a countable family of pairwise disjoint subintervals [a j , b j ), j ∈ J P , whose union is [0, τ ) and which satisfy for any j ∈ J P and t ∈ P has at most finitely many points of discontinuity in aj +bj These features result from property (h') of P k with any index k = k(j) ∈ N sufficiently large such that either There are two cases excluding each other: In the first case, there is no sequence (t ) ∈N in [0, τ ) which converges to τ and consists of points of discontinuity of P . There exist some δ > 0 and an index k 0 = k 0 (δ) ∈ N instead such that for all k ≥ k 0 , τ k > τ − δ is satisfied and the restriction P ( τ −δ, τ k ] is constant. Due to the assumed monotonicity w.r.t. , these restrictions have some unique value P 0 ∈ V ⊂ L p (R N ) in common and, we define P ( τ ) := P 0 ∈ V.
In the second case, there exists a sequencet ↑ τ consisting of points of discontinuity of P . Then, there is a sequence j ∈N of indices in J P such that t ∈ a (j ) , b (j ) holds for every ∈ N and a (j ) converges to τ for → ∞ (as we conclude from the last property in condition (h') indirectly). Hence, P (t ) ∈N proves to be a Cauchy sequence w.r.t. the metricȇ L p sincȇ holds for all , m ∈ N ( ≤ m). Furthermore, |P (t )| p ∈N is tight in R N due to hypothesis 2.4 (vii). As a consequence of Corollary A.10 and A.11, we obtain the joint limit P ( τ ) ∈ L p (R N ) w.r.t. bothȇ L p and d L p . (In particular, this limit does not depend on the sequence (t ) ∈N as an indirect standard conclusion reveals.) This construction of f ( τ ), P ( τ ) ∈ L p (R N ) preserves conditions (b'), (h') and (d) -(g) because V is assumed to be closed in L p (R N ), d L p . In particular, P : 3.3. Aspects of compactness for coefficient functions of time. Suppose the hypotheses of Theorem 2.4 and for ε ∈ (0, 1), consider any sequence tending to 0. Then, Lemma 3.2 provides sequences with the following properties: is a renormalized solution of the nonautonomous linear transport equation [79,Proposition 7]). Let (Ω, Σ, µ) be a finite measure space and X an arbitrary real Banach space. For any weakly compact and convex subset W ⊂ X, the subset h ∈ L 1 (µ, X) h(ω) ∈ W for µ-a.e. ω ∈ Ω is relatively weakly compact in L 1 (µ, X). Lemma 3.6. There exist both a sequence k ∞ of indices and three functions g Proof. As mentioned in § 3.1, assumptions 2.4 (i), (ii) imply global a priori bounds γ a , γ b > 0 (depending on p, N , T and f 0 L p (R N ) ) such that hold for every t ∈ [0, T ) and each index k ∈ N. In particular, {w k (t) k ∈ N, t ∈ [0, T ) is contained in a closed ball in L p (R N ) which is weakly compact due to reflexivity (1 < p < ∞) and the theorem of Kakutani (see, e.g., [21, Theorems 3.17, 4.10]). Preceding Lemma 3.5 guarantees a subsequence of (w k ) k∈N converging weakly to some w ∈ L 1 0, T ; L p (R N ) . Next, let us fix the radius ρ ∈ N arbitrarily and, consider all the restrictions to the open ball B ρ ⊂ R N (in space). The L q norms are uniformly bounded, i.e., we obtain for all t ∈ [0, T ), k ∈ N

Hence, the values of g k (t)
Bρ and u k (t) Bρ for all t ∈ [0, T ), k ∈ N are contained in sufficiently large closed balls in L q which are also weakly compact. As a consequence of Lemma 3.5, the sets g k | Bρ k ∈ N ⊂ L 1 0, T ; L q (B ρ , R N ) and u k | Bρ k ∈ N ⊂ L 1 0, T ; L q (B ρ ) are relatively weakly compact. Their weakly converging subsequences depend on the radius ρ ∈ N though.
Cantor's diagonal method w.r.t. ρ ∈ N leads to the claimed sequence k ∞ of indices such that in addition to the weak convergence of w k ∈N , the sequences g (k ) Bρ ∈N , u (k ) Bρ ∈N are converging weakly in L 1 0, T ; L q (B ρ ) for each radius ρ ∈ N. Their limits induce two further functions . It remains to verify both g(t) ∈ W 1,∞ (R N , R N ) and u(t) ∈ W 1,∞ (R N ) for L 1 -a.e. t ∈ [0, T ). We focus on the scalar function u(t). For the radius ρ > 0 fixed arbitrarily, u Bρ : [0, T ) −→ L q (B ρ ) is the strong L 1 limit of a sequence of convex combinations of u (k ) Bρ ∈N as a consequence of Mazur's lemma (see, e.g., [ In combination with j → ∞, the standard condition on weak derivatives implies that v ρ is the weak gradient of u(t) Bρ and so, u(t) Bρ belongs to W 1,q (B ρ ) for every ρ > 0 and L 1 -a.e. t ∈ [0, T ). Due to the uniform L ∞ bound of (ν ρ,t,j ) j∈N , the convergence L N -a.e. in B ρ ⊂ R N guarantees u(t) Bρ ∈ L ∞ B ρ and u(t) Bρ L ∞ (Bρ) ≤ γ a for every ρ > 0, i.e., u(t) ∈ L ∞ (R N ) and u(t) L ∞ (R N ) ≤ γ a hold for L 1 -a.e. t ∈ [0, T ).
Next, we apply the corresponding arguments to the weak gradient ∇ x u(t) Bρ as the weak L q limit of ∇ x ν ρ,t,j Bρ j∈N . This leads to Essentially the same arguments used componentwise reveal g(t) ∈ W 1,∞ (R N , R N ) and g(t) W 1,∞ ≤ γ a + γ b for L 1 -a.e. t ∈ [0, T ).

3.4.
A uniformly converging subsequence of continuous approximate solutions. In regard to the sequence of approximate solutions (f k ) k∈N , the wellknown theorem of Arzelà-Ascoli proves to be useful. Continuous functions between metric spaces can be handled by the generalizations in [6, Theorem 4.4.3], [46], for example: Lemma 3.7 (Arzelà-Ascoli in metric spaces, [6,46]). Let (X, d X ), (Y, d Y ) be two metric space such that X is compact. Whenever a sequence (ϕ k ) k∈N of continuous functions X −→ Y fulfills • (ϕ k ) k∈N is equicontinuous, i.e., for every ε > 0, there exists δ = δ(ε) > 0 such that all x 1 , x 2 ∈ X with d X (x 1 , x 2 ) < δ and every k ∈ N satisfy d Y ϕ k (x 1 ), ϕ k (x 2 ) < ε and, • there exists a compact set C ⊂ Y such that for every ε-neighborhood B ε (C) ⊂ Y of C, the inclusion ϕ k (X) ⊂ B ε (C) holds for all k ∈ N sufficiently large, then it has a uniformly converging subsequence.
Proof. Consider the set L ⊂ L p (R N ) specified in Subsection 3.1, i.e., ) for all ρ > 0 . It is relatively compact w.r.t.ȇ L p due to Corollary A.10. Moreover, it contains all the values f k (t) for k ∈ N, t ∈ [0, T ]. As each f k (k ∈ N) is Λ-Lipschitz continuous w.r.t.ȇ L p , we conclude the existence of a (w.r.t.ȇ L p ) uniformly converging subsequence from preceding Lemma 3.7. Finally, the pointwise convergence w.r.t. d L p results from Corollary A.11.

Hypotheses 2.4 (vi),(vii) ensure that all P k (t)
p ∈ L 1 (R N ) for k ∈ N and t ∈ [0, T ] are norm-bounded and tight. Hence, we conclude from the sequence properties (b), (h) and Corollary A.11 successively: Corollary 3.9. At every time instant t ∈ [0, T ], the following convergences for

Limits induce a renormalized solution of a transport equation.
The sequences (f k ) k∈N , (P k ) k∈N , (τ k ) k∈N , (g k ) k∈N , (u k ) k∈N , (w k ) k∈N specified at the beginning of § 3.3 lead to subsequences and limit functions f , g, u, w as formulated in Lemma 3.6 and Corollaries 3.8, 3.9.
In regard to this nonautonomous linear transport equation with the Lebesgue measurable coefficients g, u, w given in Lemma 3.6, we conclude from Proposition B.1 (5.) immediately: Proof of Lemma 3.10. For an arbitrary test function ϕ ∈ C 2 c (R N ) and any 0 ≤ t 1 < t 2 ≤ T , we have to verify

Sequence property (d) (formulated at the beginning of § 3.3) guarantees for every
According to Corollaries A.9 and A.11, the convergence of f (k ) (t 2 ) ∈N to f (t 2 ) ∈ L p (R N ) w.r.t. d L p implies its weak convergence and so, we obtain in regard to the left-hand side Next, we focus on the following convergence for → ∞ in detail since essentially the same arguments can be applied to the remaining integrals of The difference of the integrals is bounded in the following way: is bounded uniformly w.r.t. ∈ N and s ∈ [0, T ] due to hypotheses 2.4 (i),(ii) and the compact support of ϕ in R N . Hence, Lebesgue's theorem of dominated convergence ensures that the first integrals (of the upper bounds) tend to 0 for → ∞. Finally, choose ρ > 0 sufficiently large such that the ball B ρ ⊂ R N contains the compact support of ϕ ∈ C 2 c (R N ). Then the weak convergence of u (k ) ∈N to u in

Limits induce a solution of the partial differential inclusion (PDI).
The links with the corresponding partial differential inclusion remains to be verified: Proof. As a consequence of preceding Lemma 3.10, we are to prove for the functions g, u, w specified in Lemma 3.6 that g(t) Otherwise, the set of all t ∈ [0, T ] with g(t) ∈ G t, f (t) , u(t) ∈ U t, f (t) or w(t) ∈ W t, f (t) has positive outer Lebesgue measure denoted λ e > 0. Consider the superset L ⊂ L p (R N ) of V specified in Subsection 3.1. Then, the metric space (L, d L p ) is complete due to Proposition A.14. Moreover, it is separable since so is

have closed graphs. There always exists an open subset
3 λ e because R \ I is open and so, its boundary consists of (at most) countably many real numbers.
Moreover, there exist some relatively compact subsets G ⊂ L q loc (R N , R N ), δ L q ,loc , U ⊂ L q loc (R N ), δ L q ,loc and W ⊂ L p (R N ), weakly containing all the values of G, U, W respectively. Indeed, using the a priori bounds specified in Subsection 3.1 and fixing the radius ρ > 0 arbitrarily, the sets are compactly embedded in L q (B ρ , R N ) and L q (B ρ ) respectively according to the embedding theorem by Rellich and Kondrachov (see, e.g., [21,Theorem 9.16] ≤ γ a which is weakly compact as a consequence of Kakutani's theorem (see, e.g., [21,Theorem 3.17]).
Sequence properties (b), (c), (f) (formulated at the beginning of § 3.3) state for every t ∈ [0, T ) and Fix ρ ∈ N arbitrarily. Lemma 3.6 guarantees for → ∞ For a moment, we focus on the coefficients u (k ) , u as an example. Mazur's lemma provides a sequence u ρ, Next, this construction of sequences is repeated inductively for every radius ρ ∈ N. Finally, Cantor's diagonal method leads to a sequence u ∈ N with the following properties In particular, Now we are going to conclude from the closed graph of U I×L that for every where the closure is understood w.r.t. the metric δ L q ,loc because it leads to a contradiction: Inclusion (7) and Corollary 3.9 ensure u(t) ∈ U t, f (t) for L 1 -a.e. t ∈ I. The corresponding arguments also imply g(t) ∈ G t, f (t) and w(t) ∈ W t, f (t) for L 1 -a.e. t ∈ I, i.e., in a subset of [0, T ] of Lebesgue measure > T − 2 3 λ e -contracting the initial choice of the outer Lebesgue measure λ e > 0.
Similarly to the proof of [63, Lemma 5.20], we introduce the auxiliary set for arbitrary δ, ρ > 0 It is closed w.r.t. δ L q ,loc and convex since so is U t, f (t) ⊂ W 1,∞ (R N ) ⊂ L q loc (R N ) by assumption 2.4 (iv).

THOMAS LORENZ
For all t ∈ I, δ > 0 and ρ > 0, there exists a radius ε = ε(t, δ, ρ) > 0 with Indeed, we can easily adapt the indirect conclusion concerning the well-known step from the closedness of graph of a set-valued mapping to its upper semicontinuity (a.k.a. outer semicontinuity) (see, e.g., [ where the closed ball mentioned last, i.e., B δ U(t, f (t)) ⊂ L q loc (R N ), refers to the metric δ L q ,loc (introduced in Definition A.15). According to hypothesis 2.4 (iv), U has closed values in L q loc (R N ), δ L q ,loc and so, we obtain the claimed inclusion (7): Essentially the same arguments can be used for W I×L : I × L, d L p ⇒ L p (R N ), · L p (and its closed graph) because all values are convex and so, we are free to supply L p (R N ) with the weak topology in combination with Mazur's lemma.
Appendix A. Tools from metric spaces and set-valued analysis.
A.1. An extension of Scorza-Dragoni theorem to metric spaces and closed graphs. There are many versions of the Scorza-Dragoni theorem available in the literature (see, e.g., [24,50,51,54,78]). Inspired by the results of Frankowska, Plaskacz and Rzeżuchowski in [41], we prefer the following formulation concerning set-valued maps with closed graphs on metric spaces:  (iii) For any ε > 0, there is a closed subset J ε ⊂ [0, T ] with L 1 [0, T ] \ J ε < ε such that the set-valued restriction F Jε×X : J ε × X ⇒ Y has a closed graph.
In regard to the Viability Theorem 2.4 and its set-valued coefficient maps G, U, W, however, there are three obstacles of the auxiliary map F. First, the set values of F might be empty. Second, it is not guaranteed that its set values are convex if so are the values of F. But then we are free to consider the pointwise convex hull of F instead. The third obstacle concerns contingent condition 2.4 (viii). It is not clear in which form this property holds for the "modified" coefficients (i.e., after the Scorza-Dragoni argument is applied to G, U and W).
Hence, we prefer the following extensions of Lemma A.1. Indeed, the arguments proving the established formulations of Scorza-Dragoni-type theorems in, e.g., [51,54,72] imply some more properties than usually listed.
Then, F has all the properties of F specified in Proposition A.2. In particular, for each ε > 0, there exists a closed subset J ε ⊂ [0, T ] with L 1 [0, T ] \ J ε < ε such that the set-valued restriction F Jε×X : J ε × X ⇒ Y has a closed graph.
Proof of Proposition A.2. For constructing F and verifying the claimed properties (i) -(iii), we essentially follow the arguments presented by Rzeżuchowski in [72] (similarly to the proof of [51,Ch. 2,Proposition 7.20]).
First, the set-valued map F : [0, T ] × X ⇒ Y is constructed. The product X × Y supplied with the sum of the componentwise metrics of X and Y is separable. Let {a k ∈ X × Y k ∈ N} denote an arbitrary dense subset of X × Y and consider ϕ k : F has the claimed properties (the first three ones are already proved in [72]): (ii) Consider any Lebesgue measurable set J ⊂ [0, T ] and measurable functions u : Hence, for every index k ∈ N, the function d X×Y a k , z(·) : J −→ R is Lebesgue measurable with ϕ k (t) ≤ d X×Y a k , z(t) for L 1 -a.e. t ∈ J. The choice of ϕ k implies ϕ k ≤ d X×Y a k , z(·) L 1 -a.e. in J and so, we conclude u(t), v(t) = z(t) ∈ Graph F(t, · ) for L 1 -a.e. t ∈ J.
In particular, the graph of the restriction F Jε×X : J ε × X ⇒ Y is closed. In other words, it contains all its accumulation points since the construction of F is based on the sequences of scalar functions d X×Y (a k , · ) : X × Y −→ R and ϕ k Jε : J ε −→ R (k ∈ N) each of which is (sequentially) continuous.
(iv) Now we benefit from assumption (e) (i.e., the measurability of Select any τ ∈ B ∩ E. By definition of E, there exists an element x τ ∈ X with F(τ, x τ ) = ∅. The closed graph of F A×X provides a radius ρ = ρ(τ, The set-valued map F( · , x τ ) : [0, T ] ⇒ Y is assumed to be Lebesgue measurable with nonempty closed values. Hence, the well-known selection theorem of Kuratowski and Ryll-Nardzewski [59] guarantees a Lebesgue measurable selection v : [0, T ] −→ Y of F(·, x τ ) (see, e.g., [7,Theorem 8.1.4] or [25,Theorem III.6]). The tuple consisting of the constant curve u(·) := x τ : A −→ X and the restriction v| A : A −→ Y satisfies the conditions in property (ii) and so, we obtain v(t) ∈ F t, u(t) = F(t, x τ ) for every t ∈ A -contradicting the empty values of F( · , x τ ) concluded before.
The next step formulated in Corollary A.3 is based on the following consequence of joint measurability: Lemma A.4. Under the assumptions of Corollary A.3 about X,Y and F, the Proof of Lemma A.4. The assumed joint measurability of F implies its so-called graph measurability, i.e., that the graph of F belongs to the σ-algebra L 1 ([0, T ]) ⊗ B(X) ⊗B(Y ) (see, e.g., [25,Proposition III.13]  Finally, for arbitrary ξ ∈ X × Y , we conclude the claimed Lebesgue measurability of [0, T ] −→ R, t −→ dist ξ, Graph F(t, ·) from the same well-known theorems about measurable set-valued maps (see, e.g., [25,Theorem III.9] complementarily).
Proof of Corollary A.3. Consider a dense sequence (a k ) k∈N in X × Y and the func- N) as in the proof of Proposition A.2. According to Lemma A.4, each ϕ k is Lebesgue measurable. Hence, essentially the same conclusions as in the proof of Proposition A.2 can now be drawn for ϕ k (instead of ϕ k ). In particular, we obtain property A.2 (iii) for F: For any ε > 0, there is a closed subset J ε ⊂ [0, T ] with L 1 [0, T ] \ J ε < ε such that the set-valued restriction F Jε×X : J ε × X ⇒ Y has a closed graph. A.2. The metrics d L p , e L p of the state space L p (R N ). Now we specify the metrics of the state space L p (R N ) and summarize their essential features for the sake of a self-contained presentation. The details are verified in, e.g., [58, § 4.1]. In a word, the metric d L p introduced in Definition 2.3 is usually used for comparing two states in L p (R N ) at the same time instant whereas the further metrics e L p ,ȇ L p are preferred for describing the regularity of weak solutions with respect to time.
Definition A.6 ([58, Definition 10]). Fix 1 < p < ∞ and q > 1 with 1 p + 1 q = 1 and defineȇ L p , e L p : Remark A.7. Obviously the following inequalities hold for all f, g ∈ L p (R N ) and ϕ ∈ C 1 c (R N ) using the abbreviations α 1 := max ϕ W 1,q , ϕ W 1,∞ and Remark A.8. The metrics d L p andȇ L p are constructed in a very similar way, namely in terms of a supremum for all test functions in a unit ball. What differs, however, is the class of test functions and the norm underlying the unit ball. From a more general point of view, they both modify the L p norm. Indeed, the well-known Hahn-Banach theorem implies for every f, g ∈ L p (R N ) since L q (R N ) represents the dual space of L p (R N ) and C 1 c (R N ) is dense in the Banach space L q (R N ). The main difference concerns the class of test functions (for more details see, e.g., [58,63,66]).
Lemma A.9 ([58, Lemma 13]).ȇ L p metrizes the weak topology on norm-bounded tight balls in L p (R N ) in the following sense: Suppose f ∈ L p (R N ) and let (f k ) k∈N be any sequence in L p (R N ) such that |f k | p is tight in R N , i.e., Then,  . Let (f k ) k∈N and (g k ) k∈N be two bounded sequences in L p (R N ) such that both |f k | p k∈N and |g k | p k∈N are tight in R N . Then the following equivalence holds Proposition A.13 ([58, Proposition 17]). This equivalence holds for any sequence A.3. The metrics of local L q norm convergence for coefficient functions of space. Several conclusions in Section 3 concern sequences of coefficient and their converging subsequences due to an appropriate form of compactness. The standard compactness results, however, concern bounded domains with sufficiently smooth boundaries. We are going to restrict our considerations to open balls with center 0 and arbitrary radius ρ > 0. This notion and hypotheses 2.4 (iv), (v) motivate the following notation: and define δ L q ,loc : L q loc (R N , R N ) × L q loc (R N , R N ) −→ [0, ∞) by the corresponding formula (just applied to vector-valued functions).
Remark A.16. (1.) δ L q ,loc is a Fréchet metric on L q loc (R N ) and L q loc (R N , R N ) respectively (as characterized in [2, § 2.7], for example). In particular, the triangle inequality results from the same arguments as in [80, page 27]. (2.) A sequence (u j ) j∈N in L q loc (R N ) converges to u ∈ L q loc (R N ) w.r.t. δ L q ,loc if and only if for every radius ρ > 0, the restrictions u j Bρ : B ρ −→ R (j ∈ N) tend to u Bρ w.r.t. the L q (B ρ ) norm. Hence, assumption 2.4 (v) can be (re-) formulated in the following way: For L 1 -a.e. t ∈ [0, T ], the graphs of the set-valued maps G(t, · ): L p (R N ), d L p ⇒ L q loc (R N , R N ), δ L q ,loc , U(t, · ): L p (R N ), d L p ⇒ L q loc (R N ), δ L q ,loc , W(t, · ): L p (R N ), d L p ⇒ L p (R N ), · L p . are closed.
This estimate can be regarded as the key purpose why we assume all coefficient values g(t), u(t) to be in L q . Indeed, the hypotheses g(t) ∈ W 1,∞ (R N , R N ), u(t) ∈ W 1,∞ (R N ) imply that they all belong to L q loc anyway. A further step of approximation will lay the foundations for dispensing with the additional L q assumptions: Proof. The spatial regularity of the coefficients implies that every weak solution of transport equation (8) is even a renormalized solution (in the sense of Di Perna and Lions). This is formulated in [58,Proposition 19] and results from smoothing arguments presented in [35,60].
Choose any smooth cut-off function θ ∈ C ∞ (R, [0, 1]) with θ = 1 in (−∞, 1] and support in (−∞, 2). For each radius ρ > 0, we consider the modified coefficient functions whose values have compact support in B 2 ρ (0) ⊂ R N . Proposition B.1 ensures a renormalized solution f ρ : [0, T ] −→ L p (R N ) of the nonautonomous linear transport equation Its L p norm is bounded uniformly w.r.t. ρ > 0 due to Proposition B.1 (1.). As a consequence of Corollary B.2, the subset |f ρ (t)| p t ∈ [0, T ], ρ > 0 ⊂ L 1 (R N ) is tight. Corollaries A.11 and A.12 imply that f ρ (t) t ∈ [0, T ], ρ > 0 ⊂ L p (R N ) is relatively compact w.r.t. both d L p andȇ L p . Furthermore, there is a constant Λ = Λ(N, p, γ a , f 0 L p , sup t g(t) L ∞ (R N ,R N ) ) such that for every ρ > 0, the function This function f proves to be a weak solution of initial value problem (8). Indeed, fix the test function ϕ ∈ C ∞ c (R N ) arbitrarily and select a radius r = r(ϕ) > 0 with supp ϕ ⊂ B r ⊂ R N . By construction, we have g ρ = g and u ρ = u in B r for all ρ > r and thus, the weak solution property of f ρ states for all t 1 , t 2 ∈ [0, T ] (t 1 < t 2 ) and ρ > r For ρ = ρ k (k ∈ N) in particular, the limit for k → ∞ reveals t1 R N f g · ∇ x ϕ(x) + f u + w ϕ(x) dx ds due to g · ∇ x ϕ, u ϕ ∈ L q ∩ W 1,∞ with compact support, the pointwise convergence of (f ρ k ) k∈N to f w.r.t. d L p and Remark A.7.
Properties B.1 (1.) -(3.), (6.) of f result from its approximation by (f ρ k ) k∈N and the equivalence between d L p and weak convergence in L p (R N ) (specified in Lemma A.9).