This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = ...This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = uo(.) inIR￣N ) where uo ∈ L∞(IR￣N ) is a given function, v is a divergence-free bounded functioll of class C1 from IR￣× x [0, T] to IR￣N, and f is a function of class C1 from IR toIR. It also gives a result of convergence of a numerical scheme for the discretization of this equation. The authors first show the existence of a 'process' solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proven; it is also proven that the entropy process solution is in fact an entropy weak solution. Hence the existence and uniqueness of the entropy weak solution are proven.展开更多
文摘This work is concerned with the proof of the existence and uniqueness of the entropy weak solution to the following nonlinear hyperbolic equation: at +div(vf(u)) = 0 inIR￣N × [0, T], with initial data u(., 0) = uo(.) inIR￣N ) where uo ∈ L∞(IR￣N ) is a given function, v is a divergence-free bounded functioll of class C1 from IR￣× x [0, T] to IR￣N, and f is a function of class C1 from IR toIR. It also gives a result of convergence of a numerical scheme for the discretization of this equation. The authors first show the existence of a 'process' solution (which generalizes the concept of entropy weak solutions, and can be obtained by passing to the limit of solutions of the numerical scheme). The uniqueness of this entropy process solution is then proven; it is also proven that the entropy process solution is in fact an entropy weak solution. Hence the existence and uniqueness of the entropy weak solution are proven.
