scholarly journals Well-posedness for vanishing viscosity solutions of scalar conservation laws on a network

2017 ◽  
Vol 37 (11) ◽  
pp. 5913-5942 ◽  
Author(s):  
Boris P. Andreianov ◽  
◽  
Giuseppe Maria Coclite ◽  
Carlotta Donadello ◽  
◽  
...  
Keyword(s):  
Conservation Laws ◽  
Viscosity Solutions ◽  
Well Posedness ◽  
Scalar Conservation Laws ◽  
Vanishing Viscosity ◽  
Scalar Conservation

scholarly journals Well-posedness theory for stochastically forced conservation laws on Riemannian manifolds

2019 ◽  
Vol 16 (03) ◽  
pp. 519-593
Author(s):  
L. Galimberti ◽  
K. H. Karlsen
Keyword(s):  
Conservation Laws ◽  
Hyperbolic Conservation Laws ◽  
Generalized Solutions ◽  
Well Posedness ◽  
Scalar Conservation Laws ◽  
Stochastic Forcing ◽  
Rigidity Result ◽  
Hyperbolic Conservation ◽  
The Cauchy Problem ◽  
Scalar Conservation

We investigate a class of scalar conservation laws on manifolds driven by multiplicative Gaussian (Itô) noise. The Cauchy problem defined on a Riemanian manifold is shown to be well-posed. We prove existence of generalized kinetic solutions using the vanishing viscosity method. A rigidity result àla Perthame is derived, which implies that generalized solutions are kinetic solutions and that kinetic solutions are uniquely determined by their initial data ([Formula: see text] contraction principle). Deprived of noise, the equations we consider coincide with those analyzed by Ben-Artzi and LeFloch [Well-posedness theory for geometry-compatible hyperbolic conservation laws on manifolds, Ann. Inst. H. Poincaré Anal. Non Linéaire 24(6) (2007) 989–1008], who worked with Kružkov–DiPerna solutions. In the Euclidian case, the stochastic equations agree with those examined by Debussche and Vovelle [Scalar conservation laws with stochastic forcing, J. Funct. Anal. 259(4) (2010) 1014–1042].

scholarly journals A NOTE ON ADMISSIBLE SOLUTIONS OF 1D SCALAR CONSERVATION LAWS AND 2D HAMILTON–JACOBI EQUATIONS

2004 ◽  
Vol 01 (04) ◽  
pp. 813-826 ◽  
Author(s):  
LUIGI AMBROSIO ◽  
CAMILLO DE LELLIS
Keyword(s):  
Conservation Laws ◽  
Viscosity Solutions ◽  
Entropy Solutions ◽  
Uniformly Convex ◽  
Scalar Conservation Laws ◽  
Open Set ◽  
Admissible Solutions ◽  
Scalar Conservation ◽  
Jacobi Equations

Let Ω⊂ℝ2 be an open set and f∈C2(ℝ) with f" > 0. In this note we prove that entropy solutions of Dtu+Dxf(u) = 0 belong to SBV loc (Ω). As a corollary we prove the same property for gradients of viscosity solutions of planar Hamilton–Jacobi PDEs with uniformly convex Hamiltonians.

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