scholarly journals A degenerate parabolic–hyperbolic Cauchy problem with a stochastic force

2015 ◽  
Vol 12 (03) ◽  
pp. 501-533 ◽  
Author(s):  
Caroline Bauzet ◽  
Guy Vallet ◽  
Petra Wittbold
Keyword(s):  
Cauchy Problem ◽  
Existence And Uniqueness ◽  
Stochastic Forcing ◽  
Hyperbolic Problem ◽  
Stochastic Force ◽  
The Cauchy Problem ◽  
Degenerate Parabolic ◽  
Entropy Formulation ◽  
Uniqueness Of A Solution ◽  
Nonlinear Degenerate

In this paper, we are interested in the Cauchy problem for a nonlinear degenerate parabolic–hyperbolic problem with multiplicative stochastic forcing. Using an adapted entropy formulation a result of existence and uniqueness of a solution is proven.

scholarly journals On the Theory of Entropy Solutions of Nonlinear Degenerate Parabolic Equations

2020 ◽  
Vol 66 (2) ◽  
pp. 292-313
Author(s):  
E. Yu. Panov
Keyword(s):  
Cauchy Problem ◽  
Entropy Solution ◽  
Quasilinear Equation ◽  
Entropy Solutions ◽  
Degenerate Parabolic Equations ◽  
The Cauchy Problem ◽  
Periodic Initial Data ◽  
Degenerate Parabolic ◽  
Diffusion Function ◽  
Nonlinear Degenerate

We consider a second-order nonlinear degenerate parabolic equation in the case when the flux vector and the nonstrictly increasing diffusion function are merely continuous. In the case of zero diffusion, this equation degenerates into a first order quasilinear equation (conservation law). It is known that in the general case under consideration an entropy solution (in the sense of Kruzhkov-Carrillo) of the Cauchy problem can be non-unique. Therefore, it is important to study special entropy solutions of the Cauchy problem and to find additional conditions on the input data of the problem that are sufficient for uniqueness. In this paper, we obtain some new results in this direction. Namely, the existence of the largest and the smallest entropy solutions of the Cauchy problem is proved. With the help of this result, the uniqueness of the entropy solution with periodic initial data is established. More generally, the comparison principle is proved for entropy suband super-solutions, in the case when at least one of the initial functions is periodic. The obtained results are generalization of the results known for conservation laws to the parabolic case.

THE CAUCHY PROBLEM FOR CONSERVATION LAWS WITH A MULTIPLICATIVE STOCHASTIC PERTURBATION

2012 ◽  
Vol 09 (04) ◽  
pp. 661-709 ◽  
Author(s):  
CAROLINE BAUZET ◽  
GUY VALLET ◽  
PETRA WITTBOLD
Keyword(s):  
Cauchy Problem ◽  
Conservation Laws ◽  
Existence And Uniqueness ◽  
Entropy Solution ◽  
Stochastic Perturbation ◽  
Nonlinear Conservation Laws ◽  
The Cauchy Problem ◽  
Entropy Formulation

We study the Cauchy problem for multi-dimensional nonlinear conservation laws with multiplicative stochastic perturbation. Using the concept of measure-valued solutions and Kruzhkov's entropy formulation, the existence and uniqueness of an entropy solution is established.

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