In this paper, a one-dimensional Euler-Lagrange equation associated with the total variation energy, and Euler-Lagrange equations generated by approximating total variations with linear growth, are considered. Each of the problems presented can be regarded as a governing equation for steady-states in solid-liquid phase transitions. On the basis of precise structural analysis for the solutions, the continuous dependence between the solution classes of approximating problems and that of the limiting...

In this paper, we consider an initial boundary value problem for the two-dimensional primitive equations of large scale oceanic dynamics. Assuming that the depth of the ocean is a positive constant, we establish rigorous a priori bounds of the solution to problem. With the aid of these a priori bounds, the continuous dependence of the solution on changes in the boundary terms is obtained.

We consider the general parabolic equation :$u_t-\Delta b\left(u\right)+divF\left(u\right)=f$ in $Q=]0,T[\times {ℝ}^N,T\>0$ with ${\displaystyle u_0\in L^{\infty }\left({ℝ}^N\right),}$