2022/12/22 by Gui‐Qiang Chen, Cao, Gaowei, Chen, Gui-Qiang G.
Mathematics · Physics and Astronomy · #35A02 #35D40 #35F21 #35F25 #35L65 #35L67 #Analysis of PDEs (math.AP) #Cosmology and Gravitation Theories #FOS: Mathematics #FOS: Physical sciences #Geometric Analysis and Curvature Flows #Mathematical Physics (math-ph) #Navier-Stokes equation solutions
paper · pdf · doi:10.48550/arxiv.2212.11430
openalex publication_date 2022/12/22 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We are concerned with the minimal entropy conditions for one-dimensional scalar conservation laws with general convex flux functions. For such scalar conservation laws, we prove that a single entropy-entropy flux pair (η(u),q(u)) with η(u) of strict convexity is sufficient to single out an entropy solution from a broad class of weak solutions in L^∞\rm loc that satisfy the inequality: η(u)t+q(u)x≤ μ in the distributional sense for some non-negative Radon measure μ. Furthermore, we extend this result to the class of weak solutions in Lp\rm loc, based on the asymptotic behavior of the flux function f(u) and the entropy function η(u) at infinity. The proofs are based on the equivalence between the entropy solutions of one-dimensional scalar conservation laws and the viscosity solutions of the corresponding Hamilton-Jacobi equations, as well as the bilinear form and commutator estimates as employed similarly in the theory of compensated compactness.