Gauss kernel method for generalized solutions to conservation laws in heterogeneous media

2011 ◽  
Vol 22 (4-5) ◽  
pp. 247-254
Author(s):  
Jelena Aleksić
Keyword(s):  
Conservation Laws ◽  
Heterogeneous Media ◽  
Kernel Method ◽  
Generalized Solutions ◽  
Gauss Kernel

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].

Approximate generalized solutions and measure-valued solutions to conservation laws

2009 ◽  
Vol 20 (3-4) ◽  
pp. 163-170 ◽  
Author(s):  
J. Aleksić ◽  
J.-F. Colombeau ◽  
M. Oberguggenberger ◽  
S. Pilipović
Keyword(s):  
Conservation Laws ◽  
Generalized Solutions

scholarly journals Generalized Solutions to Conservation Laws

1994 ◽  
Vol 13 (1) ◽  
pp. 7-18 ◽  
Author(s):  
Michael Oberguggenberger ◽  
Ya-Guang Wang
Keyword(s):  
Conservation Laws ◽  
Generalized Solutions
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