First order quasilinear equations with boundary conditions in the \(L^{\infty}\) framework
In this article the author deals with the following problem: \[ \begin{cases}\frac{\partial u}{\partial t}+\nabla \cdot \left( f\left( t,x,u\right) \right) +g\left( t,x,u\right) =0,& \text{on }Q_T, \\ u\left( 0,\cdot \right) =u^0, &\text{on }\Omega ,\\ ``u=u^D", & \text{on }\Sigma _{T,} \end{cases} \] where \(\Omega \subseteq \mathbb{R}^d,d\geq 1,\) bounded smooth domain, \(Q_T\equiv \left(0,T\right) \times \Omega , \Sigma _T\equiv \left( 0,T\right) \times \partial \Omega .\) The general assumptions are the following: \(f \in \left( C^2\left( \left[ 0,T\right] \times \overline{\Omega }\times \left[ a,b\right] \right) \right) ^d\), \(g \in C^2\left( \left[ 0,T\right] \times \overline{\Omega }\times \left[ a,b\right] \right)\); \(f,\nabla \cdot f\) and \(g\) are Lipschitz continuous w.r.t. \(u\), uniformly in \(( t,x) ;\) \(\left( u^0,u^D\right) \in L^\infty \left( \Omega ;\left[ a,b\right] \right) \times L^\infty \left( \Sigma _T;\left[ a,b\right] \right) ;\) \(\left( \nabla \cdot f+g\right) \left( \cdot ,\cdot ,a\right) \leq 0\) and \(\left( \nabla \cdot f+g\right) \left( \cdot ,\cdot ,b\right) \geq 0\) uniformly in \(\left( t,x\right) .\) Using an appropriate definition of weak entropy solution, the author establishes an existence and uniqueness theorem along with a maximum principle. The proofs for existence and uniqueness are adapted to the semi Kružkov entropy-flux pairs, dealing with additional terms induced by the source term and non-autonomous property of the flux. Existence is obtained by the author using vanishing viscosity method.
scientific article; zbMATH DE number 5161136