Scalar conservation law in a bounded domain with strong source at boundary

We consider a scalar conservation law with source in a bounded open interval $\Omega\subseteq\mathbb R$. The equation arises from the macroscopic evolution of an interacting particle system. The source term models an external effort driving the solution to a given function $\varrho$ with an intensity function $V:\Omega\to\mathbb R_+$ that grows to infinity at $\partial\Omega$. We define the entropy solution $u \in L^\infty$ and prove the uniqueness. When $V$ is integrable, $u$ satisfies the boundary conditions introduced in [F. Otto, C. R. Acad. Sci. Paris 1996], which allows the solution to attain values at $\partial\Omega$ different from the given boundary data. When the integral of $V$ blows up, $u$ satisfies an energy estimate and presents essential continuity at $\partial\Omega$ in a weak sense.


Introduction
In this paper, we study the following initial-boundary value problem for a quasilinear scalar balance law in the bounded interval (0, 1) ⊆ R given by ∂ t u + ∂ x [J(u)] + G = 0, t > 0, x ∈ (0, 1), u(0, •) = u 0 , u(•, 0) = α, u(•, 1) = β. (1.1) where the source term G = G(t, x, u) reads and J, V , ̺ are nice functions defined respectively on R, (0, 1) and R + × (0, 1).Since the weak solution to (1.1) is not unique, we need to consider the entropy solution obtained through the vanishing viscosity limit.The entropy solution presents discontinuities both inside (0, 1) and at the boundaries.In particular, the values of u at {0, 1} can be different from the prescribed boundary data (α, β), so the boundary conditions are a priori formal.The first definition of the entropy solution is given in [2] for smooth u 0 and homogeneous boundary (α, β) ≡ (0, 0).It is then generalized in [15,14] to the case with u 0 , α and β being L ∞ functions, see also [13,Section 2.6].These definitions provide a set of possible boundary values, reflecting the formulation of boundary layer during the vanishing viscosity limit.We refer to [5,7,16,4] and [6, Section 6.9] and references therein for more details and recent development.Suppose that V (x) > 0, then G = G(t, x, u) satisfies that ∂ u G > 0 and G(•, •, ̺) ≡ 0, i.e., G acts as a source (resp.sink) when u is less (resp.greater) than ̺.When ̺ is a constant, (1.1)-(1.2) can be viewed as a conservation system with relaxation introduced in [11], with the first component degenerated to a stationary solution.In this paper, we aim at understanding the effect on the boundary discontinuities caused by extremely strong perturbation.Roughly speaking, suppose that V → ∞ as x → 0, 1 and choose ̺ that is compatible to the boundary data: ̺| x=0 = α, ̺| x=1 = β.We define the L ∞ entropy solution and prove the well-posedness.We then investigate its behavior near the boundaries and show that the appearance of discontinuity is dependent on the integrability of V .Generally speaking, • If V is integrable, the boundary condition provides a set of possible values for u at x = 0 (resp.x = 1) which can be different from α (resp.β).The compatibility conditions are not necessary here.
• If the integral of V is divergent at x ∈ {0, 1}, u satisfies an energy estimate which prescribes the boundary values in a weak sense, and one always observes continuous flux at the boundaries.
1.1.Physical motivation.The equation studied in this paper arises naturally from the hydrodynamic limit for asymmetric exclusion process with open boundaries [1,19,18,20].
It is an open interacting particle system that describes the dynamics of stochastic lattice gas with hard core repulsion.Observed at properly chosen macroscopic space-time scale, the particle density evolves with a balance law with boundary conditions.Consider the one-dimensional finite lattice Λ N = {1, . . ., N − 1}.A variable η i is assigned to each site i ∈ Λ N , with η i = 0 if the site is empty and η i = 1 if it is occupied by a particle.The configuration is denoted by η = (η 1 , . . ., η N −1 ) ∈ {0, 1} ΛN . (1. 3) The dynamics is described as following.If there is a particle at site i, it waits for a random time τ distributed as P (τ > t) = e −t and jumps to another vacant site i ′ > i on its right with probability p γ (i ′ − i), where Consider the Markov process {η(t) = η N (t); t ≥ 0} generated by The factor N means that the dynamics of exclusion on Λ N is accelerated to the hyperbolic scale N t.Meanwhile, N γ corresponds to a different scale for the reservoirs, for which the reason will be clarified later.Assume some u 0 ∈ L ∞ ((0, 1)), such that in probability, which precisely means that lim for any δ > 0 and continuous function g.The hydrodynamic limit corresponds to the convergence that for almost every t > 0, in probability.Since γ > 1, p γ possesses finite first moment: p γ := k>0 kp γ (k) < ∞.Hence, without considering the effects of reservoirs, u is the entropy solution to (see [17]): To investigate the effect of the left reservoirs, observe that (1.12) The factor N γ is chosen to get the non-trivial limit (1.13) Similar argument works for the right reservoir.Putting them together, we obtain formally the following hydrodynamic equation for x ∈ (0, 1), with the natural initial and boundary conditions The source term can be written as V (x)(u − ̺(x)), where Conservation law with general V and ̺ can be modelled by exclusion process with Glauber dynamics, see [20] for details.Note that in (1.8), the total variation of the initial empirical density u N 0 can grow in order O(N ).For this reason, we focus on constructing the entropy solution in L ∞ space, rather than in the space of bounded-variation functions.
Remark 1.1.Assume some t 0 > 0 such that (1.14) has a classical solution for t < t 0 .Using the method of characteristics, one obtains the characteristic equation associated to (1.14): (1.17) where V (x) and ̺(x) are functions given by (1.16).Hence, we formally obtain the second-order ordinary differential equation for the characteristic: The classical solution is then determined by (1.18) along these lines.
Our first aim is to define the unique entropy solution to (1.1)-(1.2) in L ∞ (Σ).The concept of Lax entropy-flux pair plays a central role.
As mentioned before, the properties of the entropy solution rely heavily on the integrability of V .Hereafter, we distinguish two cases.

Integrable case.
The source G is called integrable when V belongs to L 1 ((0, 1)).In this case, we begin with Otto's definition of boundary entropy and the corresponding flux [15].
The definition of entropy solution to (1.1) for the integrable case is similar to the case without V (see, e.g., [13  (2.5) The pair (η, ξ) is called the Kruzhkov entropy-flux pair.
Proposition 2.5.Assume V ∈ L 1 ((0, 1)).The entropy solution is equivalently defined as u ∈ L ∞ (Σ) such that ) Using the methods in [13, Section 2.7 & 2.8], we obtain the well-posedness of u and an explicit expression for the boundary conditions.Proposition 2.6.Assume that V ∈ L 1 ((0, 1)), then (1.1) admits a unique entropy solution u ∈ L ∞ (Σ).Proposition 2.7.Let u be as in Definition 2.3.For all 0 < s < t and boundary entropy-flux pairs (F, Q), (2.8) 2.2.Non-integrable case.The source G is called non-integrable when the integral of V is infinite.In this case, the singular points of the integral of V can only be {0, 1}.We will see later in Remark 2.10 that, when the integral of V is divergent at only one of them, the equation can be treated as a mixed boundary problem with one side integrable and the other side non-integrable.Hence, we assume without loss of generality that (2.9) Also assume the compatibility conditions: for all T > 0 lim (2.10) Notice that (2.10) is generally true in the integrable case, since ̺, α and β are essentially bounded.
Similarly to the integrable case, we can define the entropy solution using the Kruzhkov entropy instead.
We are now ready to state our main results.
Observe that for any δ > 0 and y ∈ (0, 1), dx. (2.16) Taking y → 0 and choosing δ arbitrarily small, (2.15) suggests that the boundary conditions in (1.1) hold in the sense of space-time average: and similarly for β(t).The convergence in (2.17) can be significantly improved under extra conditions.
Theorem 2.13.Let α(t) ≡ α, β(t) ≡ β be almost everywhere constants and u satisfy (EB) and (EI).Assume (2.15) and for all T > 0 that Then, for all 0 ≤ s < t and Lax entropy-flux pairs (f, q), (2.19) Corollary 2.14.Assume the same conditions as in Theorem 2.13 and that J is convex (or concave), then for all 0 ≤ s < t, (2.20) Finally, the existence of the entropy solution for non-integrable source with smooth coefficients and boundary data is established below.
) and (2.10).Moreover, suppose that for each T > 0, there is a family of functions {̺ ε ; ε > 0} such that the following conditions are fulfilled.
Remark 2.17.The method presented for integrable V can be extended to scalar balance laws in spatial dimensions d ≥ 2, see, e.g., [15,14].For the multi-dimensional non-integrable case, one can construct an entropy solution satisfying an energy estimate similar to (2.11) via the standard vanishing viscosity limit.However, the corresponding uniqueness remains open.

Organization of the paper.
The arguments for the integrable case are largely the same as those used in [13, Section 2.7, 2.8], see also [5].Hence, we only summarize the ideas briefly.The focus is the non-integrable case.In Section 4, we prove Theorem 2.12 exploiting Kruzhkov's doubling of variables technique.In Section 5, we prove Theorem 2.13 via an L 1refinement of the energy bound.In Section 6, we prove Theorem 2.15 with vanishing viscosity method.Proposition 2.11 and some preliminary results are proved in Section 3.
2.4.Notations.For a measure space (X; µ) and p ≥ 1, let For p = ∞, L ∞ (X; µ) stands for the space of essentially bounded measurable functions and • L ∞ (X;µ) is the essential supremum norm.When X ⊆ R d and µ is the Lebesgue measure, we use the abbreviations L p (X) and • L p (X) .

Preliminary results
First, we verify the alternative definitions of the entropy solution with (2.6) and (2.14).The integrability of V is irrelevant here.
3) holds for all Lax entropy-flux pairs if and only it holds for the Kruzhkov entropy-flux pair and all k ∈ R.
Suppose that (2.3) holds for all Lax entropy-flux pairs (f, q).To get (2.6), it suffices to take (F ε , Q ε )(•, k) in (2.3) and let ε → 0. On the other hand, assume (2.6) for all k ∈ R. Since k can be chosen smaller than − u L ∞ (Σ) , (2.3) is true for the linear entropy f = u and the corresponding flux q = J.Then, one only needs to use the fact that any Lax entropy-flux pair (f, q) is contained in the convex hull of (η, ξ)(•, k) and (u, J).
The proofs of Proposition 2.5 and 2.11 are standard.Below we assume the non-integrable case and prove Proposition 2.11 as an example.

Uniqueness of the entropy solution
In this section, we first prove the uniqueness of the entropy solution in the non-integrable case, then briefly summarize the difference when V is integrable.In the non-integrable case, the uniqueness is a direct consequence of the stability below.
Theorem 4.1.Assume (2.10), (2.15) and let u, v satisfy Definition 2.8 with the same (α, β) and different (u 0 , ̺), (v 0 , ̺ * ), respectively.Then, for almost all t > 0, The next Kruzhkov-type lemma plays a key role in the proof.Since we choose the test function ϕ to be compactly supported in Σ, the integrability of V is indeed irrelevant to either the statement or the proof.
Proof of Theorem 4.1.First, observe that the estimate is trivial if the integral of Fix an arbitrary φ ∈ C 2 c ((0, T )) such that φ ≥ 0 and recall the function ψ ε given by (3.12).Using Lemma 4.
Taking ε → 0, we have Notice that the convergence of the second term follows from ̺ − ̺ * ∈ L 1 (Σ T ; ν).We are left with the integral of ξ(u, v)ψ ′ ε φ.From the construction of ψ ′ ε , this term is identically 0 for x ∈ [ε, 1 − ε].Using the same argument as in Lemma 3.2, for any δ > 0, with a constant C depending on φ.From (2.15), the second term vanishes as δ → 0, uniformly in ε.Also observe that Putting these estimates together, is an essentially decreasing function of t.It suffices to apply the L 1 -continuity of the entropy solution at t = 0.
The proof goes in the same line as that of Lemma 4.2, with the boundary terms treated with the argument used in [13,Theorem 2.7.28].The only difference is that, when estimating G 2 , the support of ϕ contains boundary points.Observe that |G 2 | is bounded from above by As V is integrable, almost every point in (0, 1) is a Lebesgue point of V .This assures that the integral in (4.15) vanishes when ε → 0.

Flux at boundary
This section is devoted to the identification of the behavior of the entropy solution at the boundaries.For the integrable case, (2.8) follows from (2.6) and exactly the same argument as used in [13,Theorem 2.7.31], so we focus on the non-integrable case and prove Theorem 2.13.Hereafter, always assume (2.15) and that α and β are almost everywhere constant functions.
To continue, we make use of the condition (2.18) to refine the energy bound (2.11) to the following L 1 -integrability. (5.8) Proof.Thanks to (2.18), it suffices to prove for y ∈ (0, 1) that and the similar bound for β.Recall the functions (F ε , Q ε )(•, k) defined in (3.1) and observe that (F ε , Q ε ) forms a boundary entropy-flux pair for fixed ε.Pick some T * > T and y * ∈ (y, 1), the previous lemma yields that Fix such a ϕ and decompose (5.12) is bounded from above uniformly in ε.Therefore, From the construction of Then, for each fixed ε, and in consequence, sup ε>0 Σ Monotonic convergence theorem then yields that (5.17 The proof is concluded by choosing ϕ such that ϕ| (0,T )×(0,y) ≡ 1.
Remark 5.3.Assume the conditions in Theorem 2.13.Due to Proposition 5.2, (2.12) in Definition 2.3 can be generalized to for all Lax entropy-flux pairs (f, q) and ϕ ∈ C 2 c (R 2 ) such that ϕ ≥ 0. The same generalization works for (2.14).Now we can state the proof of Theorem 2.13.

5.19)
There is a constant C = C(f, φ), such that Due to the integrability proved in Proposition 5.2, is well-defined as a measurable function on (0, 1).If ψ(1) = 0, This holds for all nonnegative ψ ∈ C 2 c ((−∞, 1)), so F q,φ is non-increasing after possible modification on a Lebesgue null subset of (0, 1).Hence, exists for all φ ∈ C ∞ c ((0, T )) such that φ ≥ 0. For all (s, t) ⊆ (0, T ), the result extends to φ = 1 (s,t) with standard argument.The first equation in (2.19) then follows from (5.9) and the fact that V is not integrable on any neighbor of 0. The second one is proved similarly.
When J is convex or concave, more information can be extracted from (2.19) by exploiting the idea in [12,8].

Existence of the entropy solution
In this section, we fix some T > 0 and construct an entropy solution on Σ T via the vanishing viscosity limit.With the uniqueness proved in Theorem 2.12, we obtain an entropy solution on Σ.As before, we focus on the non-integrable case and then summarize the argument for the integrable case.
For the non-integrable case, assume that the conditions in Theorem 2.15 hold.For each ε > 0, the viscosity problem is constructed as where Some useful properties of u ε are collected in Appendix A.
Theorem 6.1.Along proper subsequence of ε → 0, u ε converges to some u ∈ L ∞ (Σ T ) with respect to the weak-⋆ topology of L ∞ (Σ T ).Furthermore, the limit point satisfies (EB) for the given T and (EI) for all Lax entropy-flux pairs (f, q) and ϕ ∈ C 2 ((−∞, T )×R) such that ϕ ≥ 0. Recall that a Young measure µ = {µ t,x ; (t, x) ∈ Σ T } is a family of probability measures on R such that (t, x) → µ t,x (A) is a measurable map from Σ T to [0, 1] for any Borel subset A of R. For continuous function h, define In view of Lemma A.1, u ε L ∞ (ΣT ) is uniformly bounded.According to the fundamental theorem of Young measure, we obtain a µ = {µ t,x ; (t, x) ∈ Σ T } as a subsequential limit point of u ε in the following sense: for all h ∈ C(R) and ϕ ∈ We also have µ t,x ([−m, m]) = 1, where m = sup ε>0 u ε L ∞ (ΣT ) .