Balance laws with integrable unbounded sources

2008/09/16 by Graziano Guerra, Guerra, Graziano, Francesca Marcellini +4
Mathematics · Physics and Astronomy · #35L45 #35L60 #35L65 #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Navier-Stokes equation solutions #Nonlinear Waves and Solitons #math.AP #msc:35L45 #msc:35L60 #msc:35L65

paper · pdf · doi:10.48550/arxiv.0809.2664

26 pages, 4 figures

arxiv created 2008/09/16 · openalex publication_date 2008/09/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We consider the Cauchy problem for a n× n strictly hyperbolic system of balance laws \arrayc u_t+f(u)_x=g(x,u), x ∈ ℝ, t>0 u(0,.)=u_o ∈ L¹ ∩ BV(ℝ; ℝⁿ), | λ_i(u)| ≥ c > 0 for all i∈ \1,...,n\, ‖g(x,⋅)‖_C²≤ M(x) ∈ L1, array. each characteristic field being genuinely nonlinear or linearly degenerate. Assuming that the L¹ norm of ‖g(x,⋅)‖_C¹ and ‖u_o‖_BV(\reali) are small enough, we prove the existence and uniqueness of global entropy solutions of bounded total variation extending the result in [1] to unbounded (in L^∞) sources. Furthermore, we apply this result to the fluid flow in a pipe with discontinuous cross sectional area, showing existence and uniqueness of the underlying semigroup.

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