On a condition of strong precompactness and the decay of periodic entropy solutions to scalar conservation laws

We propose a new sufficient non-degeneracy condition for the strong precompactness of bounded sequences satisfying the nonlinear first-order differential constraints. This result is applied to establish the decay property for periodic entropy solutions to multidimensional scalar conservation laws.


Introduction
Let Ω be an open domain in R n . We consider the sequence u k (x), k ∈ N, bounded in L ∞ (Ω), which converges weakly- * in L ∞ (Ω) to some function u(x): u k ⇀ k→∞ u. Now let ϕ(x, u) ∈ L 2 loc (Ω, C(R, R n )) be a Caratheodory vector-function (i.e. it is continuous with respect to u and measurable with respect to x) such that the functions α M (x) = max |u|≤M |ϕ(x, u)| ∈ L 2 loc (Ω) ∀M > 0 (1.1) (here and below | · | stands for the Euclidean norm of a finite-dimensional vector). By θ(λ) we shall denote the Heaviside function: Suppose that for every p ∈ R the sequence of distributions div x [θ(u k − p)(ϕ(x, u k ) − ϕ(x, p))] is precompact in W −1 d,loc (Ω) (1.2) for some d > 1. Recall that W −1 d,loc (Ω) is a locally convex space of distributions u(x) such that uf (x) belongs to the Sobolev space W −1 d for all f (x) ∈ C ∞ 0 (Ω).
The topology in W −1 d,loc (Ω) is generated by the family of semi-norms u → uf W −1 d , f (x) ∈ C ∞ 0 (Ω). If the distributions div x ϕ(x, k) are locally finite measures on Ω for all k ∈ R, then the notion of entropy solutions (in Kruzhkov's sense) of the equation div ϕ(x, u) + ψ(x, u) = 0 (1.3) (with a Caratheodory source function ψ(x, u) ∈ L 1 loc (Ω, C(R))) is defined, see [15] and [16] (in the latter paper the more general ultra-parabolic equations are studied). As was shown in [16], assumption (1.2) is always satisfied for bounded sequences of entropy solutions of (1.3).
Our first result is the following strong precompactness property.
Theorem 1.1. Suppose that for almost every x ∈ Ω and all ξ ∈ R n , ξ = 0 the function λ → ξ · ϕ(x, λ) is not constant in any vicinity of the point u(x) (here and in the sequel "·" denotes the inner product in R n ). Then u k (x) → k→∞ u(x) in L 1 loc (Ω) (strongly). Theorem 1.1 extends the results of [15], where the strong precompactness property was established under the more restrictive non-degeneracy condition: for almost every x ∈ Ω and all ξ ∈ R n , ξ = 0 the function λ → ξ · ϕ(x, λ) is not constant on nonempty intervals.
The proof of Theorem 1.1 is based on a new localization principle for Hmeasure (with "continuous" indexes) corresponding to the sequence u k , see Theorem 3.5 and its Corollary 3.6 below.
As usual, condition (1.5) means that for all non-negative test functions f = f (t, x) ∈ C 1 0 (Π) As was shown in [13] (see also [14]), an e.s. u(t, x) always admits a strong trace u 0 = u 0 (x) ∈ L ∞ (R n ) on the initial hyperspace t = 0 in the sense of relation In the sequel we will always assume that this property is satisfied.
Suppose that the initial function u 0 is periodic with a lattice of periods L, i.e., u 0 (x + e) = u 0 (x) a.e. on R n for every e ∈ L (we will call such functions L-periodic). Denote by T n = R n /L the corresponding n-dimensional torus, and by L ′ the dual lattice L ′ = { ξ ∈ R n | ξ · x ∈ Z ∀x ∈ L }. In the case under consideration when the flux vector is merely continuous the property of finite speed of propagation for initial perturbation may be violated, which, in the multidimensional situation n > 1, may even lead to the nonuniqueness of e.s. to Cauchy problem (1.4), (1.7), see examples in [8,9]. But for a periodic initial function u 0 (x), an e.s. u(t, x) of (1.4), (1.7) is unique (in the class of all e.s., not necessarily periodic) and space-periodic, the proof can be found in [12]. It is also shown in [12] that the mean value of e.s. over the period does not depend on time: where dx is the normalized Lebesgue measure on T n . The following theorem generalizes the previous results of [3,17].
is not affine on any vicinity of I. (1.10) Moreover condition (1.9) is necessary and sufficient for the decay property (1.10).
In the case ϕ(u) ∈ C 2 (R, R n ) Theorem 1.4 was proved in [3]. As was noticed in [3, Remark 2.1], decay property (1.10) holds under the the weaker regularity requirement ϕ(u) ∈ C 1 (R, R n ) but under the more restrictive assumption that for each ξ ∈ L ′ I is not an interior point of the closure of the union of all open intervals, over which the function ξ · ϕ ′ (u) is constant. Let us demonstrate that condition (1.9) is less restrictive than this assumption even in the case ϕ(u) ∈ C 1 (R, R n ). Suppose that n = 1, ϕ(u) ∈ C 1 (R) is a primitive of the Cantor function, so that ϕ ′ (u) is increasing, continuous, and maximal intervals, over which it remains constant, are exactly the connected component of the complement R \ K of the Cantor set K ⊂ [0, 1]. Since K has the empty interior the assumption of [3] is never satisfied while (1.9) holds for each I ∈ K.

Preliminaries
We need the concept of measure valued functions (Young measures). Recall (see [4,20]) that a measure-valued function on Ω is a weakly measurable map x → ν x of Ω into the space Prob 0 (R) of probability Borel measures with compact support in R.
The weak measurability of ν x means that for each continuous function g(λ) the function x → ν x , g(λ) .
Measure-valued functions of the kind ν x (λ) = δ(λ−u(x)), where u(x) ∈ L ∞ (Ω) and δ(λ−u * ) is the Dirac measure at u * ∈ R, are called regular. We identify these measure-valued functions and the corresponding functions u(x), so that there is a natural embedding of L ∞ (Ω) into the set MV(Ω) of bounded measure-valued functions on Ω.
Measure-valued functions naturally arise as weak limits of bounded sequences in L ∞ (Π) in the sense of the following theorem by L. Tartar [20].
Theorem 2.1. Let u k (x) ∈ L ∞ (Ω), k ∈ N, be a bounded sequence. Then there exist a subsequence (we keep the notation u k (x) for this subsequence) and a bounded measure valued function ν x ∈ MV(Ω) such that We will essentially use in the sequel the variant of H-measures with "continuous indexes" introduced in [10]. This variant extends the original concept of H-measure invented by L. Tartar [21] and P. Gerárd [5] and it appears to be a powerful tool in nonlinear analysis.
Let F (u)(ξ), ξ ∈ R n , be the Fourier transform of a function u(x) ∈ L 2 (R n ), S = S n−1 = { ξ ∈ R n | |ξ| = 1 } be the unit sphere in R n . Denote by u → u, u ∈ C the complex conjugation. The (2.2) (ii) For any p 1 , . . . , p l ∈ E the matrix {µ p i p j } l i,j=1 is Hermitian and nonnegative definite, that is, for all ζ 1 , . . . , ζ l ∈ C the measure l i,j=1 We call the family of measures {µ pq } p,q∈E the H-measure corresponding to the subsequence u r (x) = u kr (x).
As was demonstrated in [10], the H-measure µ pq = 0 for all p, q ∈ E if and only if the subsequence u r (x) converges as r → ∞ strongly (in L 1 loc (Ω)). Since |U k (x, p)| ≤ 1, it readily follows from (2.2) and Plancherel's equality that pr Ω |µ pq | ≤ meas for p, q ∈ E, where meas is the Lebesgue measure on Ω, and by |µ| we denote the variation of a Borel measure µ (this is the minimal of nonnegative Borel measures ν such that |µ(A)| ≤ ν(A) for all Borel sets A). This implies the representation µ pq = µ pq x dx (the disintegration of H-measures). More exactly, choose a countable dense subset D ⊂ E. The following statement was proved in [11,Proposition 3], see also [15,Proposition 3].
There exists a family of complex finite Borel measures µ pq x ∈ M(S) in the sphere S with p, q ∈ D, x ∈ Ω ′ , where Ω ′ is a subset of Ω of full measure, such that µ pq = µ pq x dx, that is, for all Φ(x, ξ) ∈ C 0 (Ω × S) the function is Lebesgue-measurable on Ω, bounded, and We choose a non-negative function K(x) ∈ C ∞ 0 (R n ) with support in the unit ball such that K(x)dx = 1 and set K m (x) = m n K(mx) for m ∈ N. Clearly, the sequence of K m converges in D ′ (R n ) to the Dirac δ-function ( that is, this sequence is an approximate unity ). We define Φ m (x) = (K m (x)) 1/2 . As was shown in [11,Remark 4] (see also [15,Remark 2(b)] ), the measures µ pq x can be explicitly represented by the relation , and Φ(y) ∈ L 2 loc (Ω) be an arbitrary function such that x is its Lebesgue point.

By Young's inequality for any positive constant c and all Borel sets
µ qq x is nonnegative Borel measure, it follows from this inequality that the variation |µ pq x | ≤ µ. This implies that It is easily computed that and (2.6) follows from (2.7).
3 Localization principles and the strong precompactness property Moreover, Var µ pq± ≤ 1 and for every Borel set A ⊂ S and each p i ∈ R, i = 1, . . . , l the matrices {µ Then, in view of (2.3) and the equality µ qp By the Cauchy criterion, this implies that there exists a limit µ pq+ x in M(S) as (p ′ , q ′ ) → (p, q), p ′ , q ′ ∈ D, p ′ > p, q ′ > q. Similarly, for each p 1 , q 1 , p 2 , q 2 ∈ D such that p 2 < p 1 < p, q 2 < q 1 < q In the limits as p ′ i → p i ± this implies (3.1).
Proof. Relations (3.2) follow from (3.1) in the same way as in the proof of inequality (2.6) above.
Remark 3.3. By continuity of µ pq x with respect to variables p, q ∈ D, we see that If the both indices p, q ∈ D, then evidently µ pq± x = µ pq x . Now we suppose that f (y, λ) ∈ L 2 loc (Ω, C(R, R n )) is a Caratheodory vectorfunction on Ω × R. In particular, Since the space C(R, R n ) is separable with respect to the standard locally convex topology generated by seminorms · M,∞ , then, by the Pettis theorem (see [6], In particular (see [6], Chapter 3), the set Ω f of common Lebesgue points of the maps F (x), |F (x)| 2 has full measure. As was demonstrated in [15], for Suppose that x ∈ Ω ′′ , p ∈ R, H + , H − are the minimal linear subspaces of R n , containing supports of the measures µ pp+ x , µ pp− x , respectively. We fix q ∈ D and introduce for p ′ ∈ D the function Proposition 3.4. Assume that q > p and f (x, λ) ∈ H ⊥ + for all λ ∈ R. Then for all ψ(ξ) ∈ C(S). Analogously, if q < p and f (x, λ) ∈ H ⊥ − ∀λ ∈ R, then ∀ψ(ξ) ∈ C(S) Here Φ m = Φ m (x − y) = K m (x − y) and I r (y, p ′ ), U r (y, p ′ ) are functions of the variable y ∈ Ω.
Proof. Note that starting from some index m the supports of the functions Φ m (x− y) lie in some compact subset B of Ω. Without loss of generality we can assume that supp Φ m ⊂ B for all m ∈ N. Let Here we take account of the equality From the above estimate and (3.4) it follows that Observe that the functionf (λ) = f (x, λ) ∈ C(R, H ⊥ + ) is continuous and does not depend on y. Therefore for any ε > 0 there exists a piece-wise constant vector- Moreover, by the density of D, we may suppose that p i ∈ D for i > 1. We define Using again Plancherel's identity and the fact that for all ψ(ξ) ∈ C(S). Since where p ′ i = max(p i , p ′ ) ∈ D, it follows from (2.4) with account of Remark 3.3 that The last equality is a consequence of the inclusion supp µ p i p+ x ⊂ supp µ pp+ x ⊂ H + (because of Corollary 3.2) combined with the relation v i ⊥H + . By (3.8), (3.9) and (3.10), we have and it suffices to observe that ε > 0 can be arbitrary to complete the proof of (3.6). The proof of relation (3.7) is similar to the proof of (3.6) and is omitted.
The proof is complete. Proof. First, note that since x ∈ Ω ′′ ⊂ Ω ′ is a Lebesgue point of the functions u 0 (·, p) for all p ∈ D while D is dense, the distribution function u 0 (x, λ) = ν x ((λ, +∞)) is uniquely defined by the relation u 0 (x, λ) = sup p∈D,p>λ u 0 (x, p). In particular, the measure ν x is well-defined at the point x.
The statement that the function λ → ξ · ϕ(x, λ) is constant in a vicinity of p 0 for all ξ ∈ H + ∩ H − , ξ = 0 readily follows from the assertion of Theorem 3.5. Hence, we only need to show that supp where we use that µ pp x (A) ≤ µ pp x (S) ≤ 1 for all p ∈ D and every Borel set A ⊂ S, see (2.3). It follows from the obtained estimate and Lemma 3.1 that On the other hand, by (2.4) µ pq In the limit as m → ∞ this yields Here we take into account that x is a Lebesgue point of the functions u 0 (y, p), u 0 (y, q). By (3.27), (3.28) we find since a < p 0 < b and [a, b] is the minimal segment containing supp ν x . The obtained contradiction implies that S + ∩ S − = ∅ and completes the proof. Now we are ready to prove Theorem 1.1.
Proof. Let u r = u kr be a subsequence of u k chosen in accordance with Proposition 2.2. In particular, this subsequence converges to a measure-valued function ν x ∈ MV(Ω). In view of (2.1) for a.e. x ∈ Ω u(x) = λdν x (λ). (3.29) We define the set of full measure Ω ′′ ⊂ Ω and the minimal segment [a(x), b(x)], containing supp ν x , x ∈ Ω ′′ . In view of (3.29) u(x) ∈ (a(x), b(x)) whenever a(x) < b(x). By Corollary 3.6 the function ξ·ϕ(x, ·) is constant in a vicinity of u(x) for some vector ξ = 0. But this contradicts to the assumption of Theorem 1.1. Therefore, a(x) = b(x) = u(x) for a.e. x ∈ Ω. This means that ν x (λ) = δ(λ − u(x)). By Theorem 2.1 the subsequence u r → u as r → ∞ in L 1 loc (Ω). Finally, since the limit function u(x) does not depend of the choice of a subsequence u r , we conclude that the original sequence u k → u in L 1 loc (Ω) as k → ∞. The proof is complete.

Decay property
This section is devoted to the proof of Theorem 1.4. Suppose that u(t, x) is a unique e.s. to problem (1.4), (1.7) with the periodic initial data u 0 (x). By Remark 1.3 we can assume that u(t, x) ∈ C([0, +∞), L 1 (T n )) (after possible correction on a set of null measure). We consider the sequence u k (t, x) = u(kt, kx), k ∈ N, consisting of e.s. of (1.4). As was firstly shown in [2], the decay property (1.10) is equivalent to the strong convergence u r (t, x) → r→∞ I = const in L 1 loc (Π) of a subsequence u r = u kr (t, x). As follows from [17, Lemma 3.2(i)], u r ⇀ u * , where u * = u * (t) is a weak- * limit of the sequence a 0 (k r t), where a 0 (t) = T n u(t, x)dx.
Since u(t, x) is an e.s. of (1.4), this function is constant: a 0 (t) ≡ I = T n u 0 (x)dx, in view of (1.8). Therefore, u r ⇀ I as r → ∞ (actually, the original sequence u k ⇀ I as k → ∞).
Let µ pq , p, q ∈ E, be the H-measure corresponding to a subsequence u r = u kr (t, x). Recall that µ pq = µ pq (t, x, τ, ξ) ∈ M loc (Π × S), where is a unit sphere in the dual space R n+1 (the variable τ corresponds to the time variable t).
Example. Let n = 1, ϕ(u) = |u|. Let u = u(t, x) be an e.s. of the problem u t + (|u|) x = 0, u(0, x) = u 0 (x), (4.4) where u 0 (x) ∈ L ∞ (R) is a nonconstant periodic function with a period l (for a constant u 0 ≡ c the e.s. u ≡ c and the decay property is evident). Notice that no previous results [2,3,17] can help to answer the question whether the decay property is satisfied. However, as follows from Theorem 1.4, if I = 1 l l 0 u 0 (x)dx = 0, then the decay property holds: l 0 |u(t, x)|dx → 0 as t → ∞. Actually, the condition l 0 u 0 (x)dx = 0 is also necessary for the decay property (1.10). Indeed, u(t, x) = u 0 (x ∓ t) if ±u 0 (x) ≥ 0 (then ±I > 0), and the decay property is evidently violated. In the remaining case when u 0 changes sign we define the functions u + (t, x) = v + (x−t), u − (t, x) = v − (x+ t), where v + (x) = max(u 0 (x), 0) ≥ 0, v − (x) = min(u 0 (x), 0) ≤ 0. Note that this functions take zero values on sets of positive measures. By the construction, v − (x) ≤ u 0 (x) ≤ v + (x) and u ± (t, x) are e.s. of (4.4) with initial data v ± (x). In view of the known property of monotone dependence of e.s. on initial data u − (t, x) ≤ u(t, x) ≤ u + (t, x) a.e. on Π. These inequality can be written in the form (4.5) Assuming that u(t, x) satisfies the decay property, we find, with the help of xperiodicity of u(t, ·), that l 0 |u(t, x ± t) − I|dx = l 0 |u(t, x) − I|dx → 0 as t → +∞, that is, the functions u(t, x±t) → t→+∞ I in L 1 ([0, l]). Passing to the limit as t → +∞ in (4.5), we find that v − (x) ≤ I ≤ v + (x) for a.e. x ∈ R. The latter is possible only if I = 0. We conclude that the decay property holds only in the case I = 0.
Remark 4.1. Theorem 1.4 can be extended to more general case of almost periodic initial data (in the Besicovitch sense [1]). Repeating the arguments of [18], we arrive at the following analogue of Theorem 1.4. Here − R n v(x)dx denotes the mean value of an almost periodic function v(x) (see [1] ).