Uniqueness and Weak-BV Stability for $$2\times 2$$ Conservation Laws

Arch. Rational Mech. Anal. Digital Object Identifier (DOI) https://doi.org/10.1007/s00205-022-01813-0 Uniqueness and Weak-BV Stability for 2 × 2 Conservation Laws Geng Chen, Sam G. Krupa & Alexis F. Vasseur Communicated by A. Bressan Abstract Let a 1-d system of hyperbolic conservation laws, with two unknowns, be endowed with a convex entropy. We consider the family of small BV functions which are global solutions of this equation. For any small BV initial data, such global solutions are known to exist. Moreover, they are known to be unique among BV solutions verifying either the so-called Tame Oscillation Condition, or the Bounded Variation Condition on space-like curves. In this paper, we show that these solutions are stable in a larger class of weak (and possibly not even BV ) solutions of the system. This result extends the classical weak-strong uniqueness results which allow comparison to a smooth solution. Indeed our result extends these results to a weak-BV uniqueness result, where only one of the solutions is supposed to be small BV , and the other solution can come from a large class. As a consequence of our result, the Tame Oscillation Condition, and the Bounded Variation Condition on space-like curves are not necessary for the uniqueness of solutions in the BV theory, in the case of systems with 2 unknowns. The method is L 2 based, and builds up from the theory of a-contraction with shifts, where suitable weight functions a are generated via the front tracking method. 1. Introduction We consider 1-d system of hyperbolic conservation laws with two unknowns u t + ( f (u))x = 0 t > 0, x ∈ R, (1.1) where (t, x) ∈ R+ × R are time and space, and u = (u 1 , u 2 ) ∈ V0 ⊆ R2 is the unknown. The set of states V0 is supposed to be bounded, and we denote V its interior. Then f = ( f 1 , f 2 ) ∈ [C(V0 )]2 ∩ [C 4 (V)]2 is the flux function, and is assumed to be continuous on V0 and C 4 on V. For any g ∈ C 1 (V), let us denote the vector valued function g  = Dg. Then, we denote the eigenvalues and Geng Chen, Sam G. Krupa & Alexis F. Vasseur associated right eigenvectors of f  on V as λ1 , r1 and λ2 , r2 , corresponding to the 1 and 2 characteristic families respectively. Throughout the paper, we will make the following general assumptions on the system: Assumption 1.1. Assumptions on the system (a) For any u ∈ V: λ1 (u) < λ2 (u). (b) For any u ∈ V, and i = 1, 2: λi (u) · ri (u) = 0. (c) There exists a strictly convex function η ∈ C(V0 ) ∩ C 3 (V) and a function q ∈ C(V0 ) ∩ C 3 (V) such that q  = η f  , on V. (1.2) (d) For any b ∈ V, and any left eigenvector  of f  (b): the function u →  · f (u) is either convex or concave on V. (e) There exists L > 0 such that for any u ∈ V and i = 1, 2: |λi (u)|  L. (f) For u L ∈ V, we denote s → Su1L (s) the 1-shock curve through u L defined for s ∈ [0, su L ) for a constant su L > 0 (possibly su L = +∞). We choose the parametrization such that s = |u L − Su1L (s)|. Therefore, (u L , Su1L (s), σu1L (s)) is the 1-shock with left hand state u L , right hand state Su1L (s), velocity σu1L (s) and strength s. Similarly, we denote s → Su2R (s) to be the 2-shock curve defined for s ∈ [0, su R ) for a constant su R > 0 (possibly su R = +∞). Thus, (Su2R (s), u R , σu2R (s)) is the 2-shock with right hand state u R , left hand state Su2R , velocity σu2R (s) and strength s. We assume that these curves are defined globally in V for every u L ∈ V and u R ∈ V. (g) (for 1-shocks) If (u L , u R ) is an entropic Rankine–Hugoniot discontinuity with shock speed σ , then σ > λ1 (u R ) (for the definition of entropic Rankine– Hugoniot discontinuity, see (1.9)). (h) (for 1-shocks) For any d ∈ V, there exists ε > 0 such that if (u L , u R ) (with u L ∈ Bε (d)) is an entropic Rankine–Hugoniot discontinuity with shock speed σ verifying, σ  λ1 (u L ), then u R is in the image of Su1L . That is, there exists su R ∈ [0, su L ) such that Su1L (su R ) = u R (and hence σ = σu1L (su R )). (i) (for 2-shocks) If (u L , u R ) is an entropic Rankine–Hugoniot discontinuity with shock speed σ , then σ < λ2 (u L ). (j) (for 2-shocks) For any d ∈ V, there exists ε > 0 such that: If (u L , u R ) (with u R ∈ Bε (d)) is an entropic Rankine–Hugoniot discontinuity with shock speed σ verifying, σ  λ2 (u R ), then u L is in the image of Su2R . That is, there exists su L ∈ [0, su R ) such that Su2R (su L ) = u L (and hence σ = σu2R (su L )). Uniqueness and Weak-BV Stability d (k) For u L ∈ V, and for all s > 0, η(u L |Su1L (s)) > 0 (the shock “strengthens" ds with s) (for the definition of the relative entropy, see (3.1)). Similarly, for u R ∈ d V, and for all s > 0, η(u R |Su2R (s)) > 0. Moreover, for each u L , u R ∈ V and ds s > 0, dds σu1L (s) < 0 and dds σu2R (s) > 0. These assumptions are fairly general. The first one corresponds to the strict hyperbolicity of the system in V. The second one means that both characteristics families of the system are genuinely nonlinear in V in the sense of Lax [31]. The third assumption is related to the second law of thermodynamics. The function η is called an entropy of the system, and q is called the entropy flux associated with η. The next two assumptions are less classical. Assumption (d) ensures a contraction property on rarefaction waves (see Section 4). Assumption (e) provides a global bound on the speeds of propagation. Assumptions (f) to (k) are now standard for the a-contraction theory. It was showed in [33] that they are verified for a large family of systems, including the Full Euler system and the isentropic Euler system. A typical example of systems verifying the Assumptions 1.1 is the system of isentropic Euler equations for γ > 1: t > 0, x ∈ R, ρt + (ρv)x = 0, t > 0, x ∈ R. (ρv)t + (ρv 2 + ρ γ )x = 0, (1.3) This is endowed with the physical entropy η(u) = ρv 2 /2 + ρ γ /(γ − 1), where u = (ρ, ρv). For any fixed constant C > 0, we can define the space of states as the invariant region V0 = {u = (ρ, ρv) ∈ R+ × R : −C < w1 (u) = v − c1 ρ  w2 (u) = v + c1 ρ γ −1 2 < C}, γ −1 2 (1.4) √ 2 γ where c1 := γ −1 . Note that V0 = V ∪ {(0, 0)} where (0, 0) is the vacuum state. It justifies the precise distinction of V and V0 (see [33,46]). The fact that (1.3) verifies the three first assumptions of Assumption 1.1 is well-known (see Serre [39] for instance). We prove in Section 4 that this also verifies assumptions (d) and (e). We will consider only entropic solutions of (1.1), that is, solutions which verify additionally (η(u))t + (q(u))x  0, t > 0, x ∈ R. (1.5) More precisely, we ask that for all φ ∈ C0∞ ([0, ∞) × R) verifying φ  0,  ∞ ∞  φt (t, x)η(u(t, x)) 0 −∞  ∞  + φx (t, x)q(u(t, x)) dxdt + φ(0, x)η(u 0 (x)) dx  0, (1.6) −∞ where u 0 : R → R is the prescribed initial data for the solution u. Geng Chen, Sam G. Krupa & Alexis F. Vasseur For σ ∈ R and u L , u R ∈ V0 , the function u : R+ × R → V0 defined by  u(t, x) := uL, u R, if x < σ t, if x > σ t, (1.7) is a weak solution to (...truncated)