Scalar conservation laws on constant and time-dependent Riemannian manifolds

Abstract

In this paper we establish well-posedness for scalar conservation laws on closed manifolds M endowed with a constant or a timedependent Riemannian metric for initial values in L ∞ (M). In particular we show the existence and uniqueness of entropy solutions as well as the L 1 contraction property and a comparison principle for these solutions. Throughout the paper the flux function is allowed to depend on time and to have non-vanishing divergence. Furthermore, we derive estimates of the total variation of the solution for initial values in BV(M), and we give, in the case of a time-independent metric, a simple geometric characterisation of flux functions that give rise to total variation diminishing estimates.

Well-posedness theory for degenerate parabolic equations on Riemannian manifolds

“…Although investigations concerning well-posedness of evolution equations on manifolds attracted a significant amount of attention recently, this problem for degenerate parabolic equations on manifolds has not been considered until now. The most closely related research is directed towards scalar conservation laws on manifolds and we mention [4,20,25] for the Cauchy problem corresponding to scalar conservation laws on manifolds, and [17,26] for the (initial)-boundary value problem on manifolds. The approach in [26] is based on the kinetic formulation as well, and Definition 3.1. from there inspired our kinetic solution concept.…”

Traces for Functions of Bounded Variation on Manifolds with Applications to Conservation Laws on Manifolds with Boundary

“…Similarly we get inf M ×(0,T ) u ǫ ≥ min{ess inf M u 0 , 0}. For the estimates of the time derivative and the total variation we proceed as in [20] and define a function S η : R → R ≥0 for η > 0 by…”

“…Uniqueness of the entropy solution. To prove uniqueness we will use Kruzkov's [17] technique of doubling the variables which was generalized by Lengeler and Müller [20] to the case of closed Riemannian manifolds. In this section we adapt their work to the case of compact Riemannian manifolds with boundary.…”

“…24)ˆT 0ˆMˆT 0ˆM |u(x, t) − v(y, s)|(∂ t φ + ∂ s φ) + q(u(x, t), v(y, s), x, t), grad x g φ g + q(v(y, s), u(x, t), y, s), grad y g φ g dv g (y) ds dv g (x) dt ≥ 0 with q(u, k, x, t) := sgn(u − k) (f (u, x, t) − f (k, x, t)) .We setφ(x, t, y, s) := ψ(t)ψ(x)ωǭ(t − s)κ ǫ (x, y), where ψ ∈ C ∞ 0 ((0, T )) with ψ ≥ 0 andψ ∈ C ∞ 0 (M ) withψ ≥ 0, ωǭ(s) := ω( s ǫ ) with ω ∈ C ∞ (R), supp(ω) ⊂ (−1, 1), ω ≥ 0 and´R ω(s)ds = 1 and κ ǫ (x, y) := 1 ǫ n κ dg(x,y) ǫ with κ ∈ C ∞ (R),supp(κ) ⊂ (−1, 1), κ ≥ 0 and R n κ(|z|)dz = 1. Thus, (4.24) yieldŝ T 0ˆM ψ ′ (t)ψ(x)ˆT 0ˆM ωǭ(t − s)κ ǫ (x, y)|u(x, t) − v(y, s)| dv g (y) ds dv g (x) dt +ˆT 0ˆM ψ(t)ˆT 0ˆM ωǭ(t − s) q(u(x, t), v(y, s), x, t), grad x g κ ǫ (x, y) gψ (x) + q(u(x, t), v(y, s), x, t), grad x gψ (x) g κ ǫ (x, y) + q(v(y, s), u(x, t), y, s), grad y g κ ǫ (x, y) gψ (x) dv g (y) ds dv g (x) dt ≥ 0.With the same argumentation as in[20,[1714][1715][1716][1717][1718] we obtain by subsequently lettingǭ and ǫ tend to zerôT 0ˆM ψ ′ψ |u − v| + ψ sgn(u − v) grad gψ , f (u) − f (v) g dv g dt ≥ 0. Settingψ := 1 − R δ we get withLemma 4.8 for δ ց 0 (4.25) MˆT 0 ψ ′ |u − v| dt dv g ≥ˆ∂ MˆT 0 ψ sgn(T u − T v) f (T u) − f (T v), N g dt dv g .…”

“…To our knowledge, conservation laws on manifolds were studied for the first time by [24] showing existence and uniqueness for a geometryindependent variant of (1.1). For the case of compact manifolds without boundary the theoretical and numerical study of conservation laws on manifolds has reached significant progress [8,2,7,3,19,11,20,13,14] during the last decade. The Dirichlet problem for a geometry-independent formulation of (1.1) was addressed by Panov using a kinetic formulation.…”