Trilinear resonant interactions of semilinear hyperbolic waves
For a single equation \[ Xu_n= (\partial_t- c(t,x)\partial_x)u_n= F(u_n) \] it is well known that there is a weak limit \(\mu_{t,x}(d\lambda)\) of \(u_n\) such that \(f(u_n)\) converges weakly to \(\int f(\lambda) \mu_{t,x}(d\lambda)\) for any continuous function \(f\), and the Young measure \(\mu_{t,x}(d\lambda)\) is determined by the equation \[ X\mu_{t,x}(d\lambda)= -\partial_\lambda(F(\lambda)) \mu_{t,z}(d\lambda)). \] In this paper the authors extend the above result to \(3\times 3\) systems as follows: \[ X_k u_{k,n}= (\partial_t- c_k(t, x)\partial_x) u_{k,n}= F_k(u_{1,n}, u_{2,n}, u_{3,n}),\quad k= 1,2,3, \] where \(c_k\) are real-valued smooth functions and \(c_j\neq c_k\) \((j\neq k)\). Then they prove that if \(u_{k,n}\) \((k= 1,2,3)\) is a bounded sequence of solutions in \(L^\infty(\Omega)\) of the above equations, if \(G\) is a compact Abelian group and \(\rho_n\in \Hom(R; G)\) and if \((G,\rho_n)\) is admissible and complete for the initial data \(u^0_{k,n}(x)= u^0_{k,n}(0, x)\) and the initial phases \(\psi^0_k(x)= \psi_k(0, x)\), then \((g,\rho_n)\) is complete for \(u_{k,n}\) and the phase \(\psi_k\) and moreover the Young measures \(\mu_k t\), \(x(d\lambda)\) of \(u_k\) are the unique solutions of the following equations \[ X_k \mu_k+\partial_\lambda (A_k(t, x,\lambda)\mu_k)= 0,\quad \mu_k|_{t= 0}= \mu_{k,0}, \] with \[ A_1(t, x,\lambda)= \int_{G\times R\times R} F_1(\lambda, \lambda_2,\lambda_3) \mu_{2,t,y,g_2}(d\lambda_2) \mu_{3,-g-g_2}(d\lambda_3) dg_2 \] and with similar definitions for \(A_2\) and \(A_3\), and \(\mu^0_k\) are Young measures associated to the initial data \(u^0_k\).
scientific article; zbMATH DE number 1425161
scientific article