Global Martingale Solutions For A Stochastic Cross-Diffusion System With Lévy Noise

In 2018,Dhariwal,Jungel and Zamponi studied the global nonnegative martingale solution to the stochastic cross-diffusion equation du_i-div(sum from j=1 to n A_ij（u）（?）u_j)dt=sum from j=1 to n σ_ij（u）dW_j（t）.On this basis,the paper studied the global martingale solutions to stochastic cross-diffusion equations driven by Lévy noise du_i-div(sum from j=1 to n A_ij（u）（?）u_j)dt=sum from j=1 to n σ_ij（u）dW_j（t）+integral |x|≤1 F（u（t-）,X）（?）（dt,dx）,where （W₁,...,W_n） is an n-dimensional cylindrical Wiener process,n is a real-value Poisson measure with a compensation measure λ（dx）dt,A（u）=A_ij（u）=δ_ij(a_i0+sum from k=1 to n a_iku_k²)+2a_iju_iu_j,i,j=1,...,n.,First,we use the Galerkin approx-imation method to obtain the existence of solutions for finite-dimensional approxima-tion equations.For the finite-dimensional approximation equations,we prove the a priori estimates.Next,we use the Aldous condition to prove the tire tightness of the finite-dimensional approximation solution.The Jakubowski generalization conclusion in Skorokhod’s theorem is used to prove the convergence of the finite-dimensional ap-proximation solution.Finally,we prove the existence of the global martingale solution of the stochastic cross-diffusion equation driven by Lévy process..