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Data Assimilation is important in meteorology and oceanography, because it is a way to improve the models with newly measured data, statically or dynamically. It is a type of inverse problem for which the most popular solution method is least square with regularization and optimal control algorithms. As control theory assumes differentiability, there are mathematical difficulties when viscosity is neglected and the modeling uses a conservation law like the shallow water or Euler equations. In this paper we study the differentiated equations of some systems of conservation laws and show that Calculus of Variation can be applied in a formal and rigorous manner provided that principal values are defined at shocks and equations written in the sense of distribution theory. Numerical illustrations are given for the control of shocks for Burgers’ equation and for the shallow water equations in one space dimension.
We prove the existence and uniqueness of global strong solutions to the Cauchy problem of the compressible Stokes approximation equations for any (specific heat ratio) $\gamma > 1$ in $\Bbb R^3$ when initial data are helically symmetric. Moreover, the large-time behavior of the strong solution and the existence of global weak solutions are obtained simultaneously. The proof is based on a Ladyzhenskaya interpolation type inequality for helically symmetric functions in $\Bbb R^3$ and uniform a priori estimtes. The present paper extends Lions’ and Lu, Kazhikhov and Ukai’s existence theorem in $\Bbb R^2$ to the three-dimensional helically symmetric case.
We show that bounded families of global classical relativistic strings that can be written as graphs are relatively compact in $C^0$ topology, but their accumulation points include many non relativistic strings.
In this paper, we generalize the technique of anti-diffusive flux corrections for high order finite difference WENO schemes solving conservation laws in, to solve Hamilton-Jacobi equations. The objective is to obtain sharp resolution for kinks, which are derivative discontinuities in the viscosity solutions of Hamilton-Jacobi equations. We would like to resolve kinks better while maintaining high order accuracy in smooth regions. Numerical examples for one and two space dimensional problems demonstrate the good quality of these Hamiltonian corrected WENO schemes.