Abstract. In this paper it is proved that the semigroup associated with the onedimensional thermoelastic system with Dirichlet boundary conditions is an exponen-12 2 tially stable C0-semigroup of contraction on the space H0 x L x L . The technique of the proof is completely different from the usual energy method. It is shown that the exponential decay in 3 (s/) recently obtained by Revira is a consequence of our main result. An important application of our main result to the Linear-QuadraticGaussian optimal control problem is also discussed.

Abstract. We are mainly concerned with the Dirichlet initial boundary value problem in one-dimensional nonlinear thermoelasticity. It is proved that if the initial data are close to the equilibrium then the problem admits a unique, global, smooth solution. Moreover, as time tends to infinity, the solution is exponentially stable. As a corollary we also obtain the existence of periodic solutions for small, periodic righthand sides.

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In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010)]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results.

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