Uniqueness Domains for $$\textbf{L}^\infty $$ Solutions of $$2\times 2$$ Hyperbolic Conservation Laws

Arch. Rational Mech. Anal. (2025) 249:70 Digital Object Identifier (DOI) https://doi.org/10.1007/s00205-025-02147-3 Uniqueness Domains for L∞ Solutions of 2 × 2 Hyperbolic Conservation Laws Alberto Bressan , Elio Marconi & Ganesh Vaidya Communicated by C. Dafermos Abstract For a genuinely nonlinear 2 × 2 hyperbolic system of conservation laws, assuming that the initial data have a small L∞ norm but a possibly unbounded total variation, the existence of global solutions was proven in a classical paper by Glimm and Lax (1970). In general, the total variation of these solutions decays like t −1 . Motivated by the theory of fractional domains for linear analytic  semigroups, we consider here solutions with a faster decay rate: Tot.Var. u(t, ·) ≤ Ct α−1 . For these solutions, a uniqueness theorem is proven. Indeed, as the initial data range over a domain of functions with ūL∞ ≤ ε1 small enough, solutions with a fast decay yield a Hölder continuous semigroup. The Hölder exponent can be taken arbitrarily close to 1 by further shrinking the value of ε1 > 0. An auxiliary result identifies a class of initial data whose solutions have rapidly decaying total variation. 1. Introduction Consider a strictly hyperbolic system of conservation laws in one space dimension u t + f (u)x = 0, (1.1) u(0, x) = u(x). (1.2) with initial data It is well known that (1.1) generates a Lipschitz continuous semigroup of Liuadmissible weak solutions [5,9–12,14,19,22,28], on a domain D ⊂ L1 (R; Rn ) of functions with suitably small total variation. In the papers [13,32–34], a semigroup was constructed on a domain of functions with large but finite total variation. In 70 Page 2 of 37 Arch. Rational Mech. Anal. (2025) 249:70 view of the recent uniqueness theorems proven in [15,17], these results show that, for a wide class of systems, the Cauchy problem (1.1)–(1.2) has a unique solution which depends continuously on the initial data, as long as the total variation remains bounded. A major open problem is to understand whether the semigroup generated by (1.1) can be extended to a domain of L∞ functions, possibly with unbounded total variation. At present, this is known in the scalar case [21,29], and for some special systems of Temple class [16]. In addition, unique solutions with L∞ initial data were constructed in [18] as limit of vanishing viscosity approximations, for a class of 2 × 2 systems of conservation laws in triangular form, not necessarily strictly hyperbolic. Even for 2×2 strictly hyperbolic, genuinely nonlinear systems, where existence of L∞ solutions can be proved by a compensated compactness argument [23], it is not known whether the vanishing viscosity limit yields unique solutions. In particular, the uniqueness of the weak solutions constructed by Glimm and Lax in [25] has remained an elusive open problem for over 50 years. The aim of this paper is to provide some results in this direction. Our approach is motivated by the theory of parabolic equations and fractional domains for analytic semigroups [27,36], which we briefly review. Consider a bounded open set  ⊂ Rn with smooth boundary. Call u(t, ·) = et u the solution to Cauchy problem for the linear heat equation  u t − u = 0 on , (1.3) u(0, x) = u(x). u = 0 on ∂ , For any initial data u ∈ L2 (), it is well known that this solution satisfies the decay property   u(t, ·) L2 ≤ C uL2 . t (1.4) Moreover, if the initial data lies in the domain of a fractional power of the Laplace . operator, say ū ∈ X α = Dom(−)α for some 0 < α < 1, then a faster decay rate is achieved:   u(t, ·) L2 () ≤ C t 1−α u X α . (1.5) Relying on (1.5), one can show that the semilinear parabolic Cauchy problem  u t − u = F(x, u, ∇u on , (1.6) u(0, x) = u(x), u = 0 on ∂ , with F Lipschitz continuous, is well posed for initial data u ∈ X α , for a suitable 0 < α < 1 depending on the dimension n of the space. For all of the details we refer the reader to [27]. We shall pursue a similar approach in connection with a 2×2 genuinely nonlinear, strictly hyperbolic system of conservation laws, as considered in the classical Arch. Rational Mech. Anal. (2025) 249:70 Page 3 of 37 70 paper [25]. We recall that, for any bounded measurable initial data u : R → R2 , if the norm uL∞ ≤ ε1 (1.7) is sufficiently small, then the Cauchy problem (1.1)–(1.2) has a weak solution, global in time; its L∞ norm and its total variation satisfy bounds of the form    u(t, ·) ∞ ≤ M0 ūL∞ , (1.8) L      b−a (1.9) ūL∞ + Tot.Var. u(t, ·) ; [a, b] ≤ C0 · t for some constants M0 , C0 , any interval [a, b] and every time t ∈ ]0, 1]; see also [6] for a simpler proof, based on front tracking approximations. Analogously to (1.5), we introduce the domain  . D̃α = u ∈ L∞ (R; R2 ) ; the Cauchy problem (1.1) − (1.2) has a weak (1.10)   solution such that sup t 1−α · Tot.Var. u(t, ·) < +∞ . 0<t<1 We observe that ū ∈ Dα implies  u(τ ) − uL1 = O(1) · τ α τ ∈ [0, 1]. Indeed, by Theorem 4.3.1 in [22], one has   u(t + h) − u(t) 1   L = O(1) · Tot.Var. u(t) . lim sup h h→0+ (1.11) (1.12) For ū ∈ Dα , the estimate (1.11) is an easy consequence of (1.12). Here and in the sequel, the Landau notation O(1) denotes a uniformly bounded quantity. Our main result shows that, for every initial data u ∈ D̃α with ūL∞ sufficiently small, the entropy weak solution which satisfies (1.11) is unique. Moreover, such solutions form a Hölder continuous semigroup w.r.t. the L1 distance on the initial data. By restricting the domain of the semigroup to functions with even smaller L∞ norm, the Hölder exponent of the semigroup can be rendered arbitrarily close to 1. The main idea is illustrated in Fig. 1. For any τ > 0, consider the positively invariant domain D(τ ) = u(t, ·) ; t ≥ τ, u is an entropy weak solution of (1.1)–(1.2) satisfying (1.9), with ū ∈ L1 (R; R2 ), ūL∞ ≤ ε1 . (1.13) here ε1 > 0 is a fixed constant, small enough so that the Glimm-Lax decay estimates (1.9) on the total variation are valid. Observing that these functions have uniformly bounded variation on bounded intervals, our first goal is to show that (1.1) generates a Lipschitz semigroup S on the domain D(τ ) . More precisely,    St ū − St v̄  1 ≤ L(τ ) · ū − v̄L1 , for all ū, v̄ ∈ D(τ ) , t ∈ [0, 1], (1.14) L 70 Page 4 of 37 Arch. Rational Mech. Anal. (2025) 249:70 Fig. 1. Sketch of the estimates (1.14)–(1.15). Fig. 2. The Lipschitz continuous dependence of front tracking approximations can be estimated by computing the strengths σi and shifts ξi of all wave fronts. for some Lipschitz constant L(τ ). Of course, since the total variation of functions ū ∈ D(τ ) becomes arbitrarily large as τ → 0, we expect that limτ →0 L(τ ) = +∞. Next, consider two solutions u 1 , u 2 of the hyperbolic system (1.1) with the same initial data ū ∈ D̃α . For any 0 < τ < t ≤ 1, by (1.14) and (1.11), their distance satisfies   u 1 (t) − u 2 (t)L1 ≤ L(τ ) · u 1 (...truncated)