This paper studies the Euler equations with non-homogeneous Navier slip boundary conditions in a 2D bounded domain. The equations describe the motion of an incompressible, inviscid fluid with flows through the boundary. The boundary conditions are given by $ \mathbf{v}\cdot\mathbf{n}=a $ and $ 2D(\mathbf{v})\mathbf{n}\cdot\mathbf{s}+\alpha\mathbf{v}\cdot\mathbf{s}=b $, where $ \mathbf{n} $ and $ \mathbf{s} $ are the normal and tangent vectors to the boundary. The authors establish the solvability of this problem in the class of solutions with $ L_p $-bounded vorticity, $ p \in (2, \infty] $, by passing to the limit in Navier-Stokes equations with vanishing viscosity. The paper is structured into four main sections. Section 1 introduces the problem and its mathematical formulation. Section 2 presents the main and auxiliary results, including the definition of a weak solution and some well-known results. Section 3 discusses the construction of approximate solutions using Schauder's fixed point argument and estimates independent of the viscosity. Section 4 provides the existence result, including a Gronwall-type inequality for the vorticity and estimates independent of $ \theta $. The authors prove the existence of a weak solution to the Euler equations with non-homogeneous Navier slip boundary conditions, showing that the solution satisfies the boundary condition. The solution is constructed by considering a sequence of approximate solutions with viscosity and regularization parameters, and then passing to the limit as the viscosity and regularization parameters go to zero. The key steps involve using Schauder's fixed point theorem, deriving a priori estimates, and applying compactness arguments to establish the existence of the weak solution. The paper also discusses the implications of the results for the inviscid limit of the Navier-Stokes equations with Navier slip boundary conditions.This paper studies the Euler equations with non-homogeneous Navier slip boundary conditions in a 2D bounded domain. The equations describe the motion of an incompressible, inviscid fluid with flows through the boundary. The boundary conditions are given by $ \mathbf{v}\cdot\mathbf{n}=a $ and $ 2D(\mathbf{v})\mathbf{n}\cdot\mathbf{s}+\alpha\mathbf{v}\cdot\mathbf{s}=b $, where $ \mathbf{n} $ and $ \mathbf{s} $ are the normal and tangent vectors to the boundary. The authors establish the solvability of this problem in the class of solutions with $ L_p $-bounded vorticity, $ p \in (2, \infty] $, by passing to the limit in Navier-Stokes equations with vanishing viscosity. The paper is structured into four main sections. Section 1 introduces the problem and its mathematical formulation. Section 2 presents the main and auxiliary results, including the definition of a weak solution and some well-known results. Section 3 discusses the construction of approximate solutions using Schauder's fixed point argument and estimates independent of the viscosity. Section 4 provides the existence result, including a Gronwall-type inequality for the vorticity and estimates independent of $ \theta $. The authors prove the existence of a weak solution to the Euler equations with non-homogeneous Navier slip boundary conditions, showing that the solution satisfies the boundary condition. The solution is constructed by considering a sequence of approximate solutions with viscosity and regularization parameters, and then passing to the limit as the viscosity and regularization parameters go to zero. The key steps involve using Schauder's fixed point theorem, deriving a priori estimates, and applying compactness arguments to establish the existence of the weak solution. The paper also discusses the implications of the results for the inviscid limit of the Navier-Stokes equations with Navier slip boundary conditions.