An inhomogeneous evolution equation involving the normalized infinity Laplacian with a transport term

In this paper, we prove the uniqueness and stability of viscosity solutions of the following initial-boundary problem related to the random game named tug-of-war with a transport term

$\left\{ \begin{array}{*{35}{l}} {{u}_{t}}-\Delta _{\infty }^{N}u-\langle \xi ,Du\rangle = f(x,t),\ \ \ \ \ \ \text{in}\ \ {{Q}_{T}}, \\ u = g,\ \ \ \ \ \ \ \ \text{on}\ \ \ \ \ {{\partial }_{p}}{{Q}_{T}}, \\\end{array} \right. $

where $ \Delta _{\infty }^{N}u = \frac{1}{{{\left| Du \right|}^{2}}}\sum\limits_{i,j = 1}^{n}{{{u}_{{{x}_{i}}}}}{{u}_{{{x}_{j}}}}{{u}_{{{x}_{i}}{{x}_{j}}}}$ denotes the normalized infinity Laplacian, $ ξ: Q_T\to R^n$ is a continuous vector field, $ f$ and $ g$ are continuous. When $ ξ$ is a fixed field and the inhomogeneous term $ f$ is constant, the existence is obtained by the approximate procedure. When $ f$ and $ ξ$ are Lipschitz continuous, we also establish the Lipschitz continuity of the viscosity solutions. Furthermore we establish the comparison principle of the generalized equation with the first order term with initial-boundary condition

${u_t}(x,t) -Δ _∞ ^N u (x,t) -H(x,t,Du(x,t)) = f(x,t),$

where $ H(x,t,p):Q_T× R^n\to R$ is continuous, $ H(x,t,0) = 0$ and grows at most linearly at infinity with respect to the variable $ p$. And the existence result is also obtained when $ H(x,t,p) = H(p)$ and $ f$ is constant for the generalized equation.