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Stability of fronts for a regularization of the Burgers equation. (English) Zbl 1160.37029

In the previous work [J. Nonl. Sci. 16, No. 6, 615–638 (2006; Zbl 1108.35107)] the following regularization of the Leray-type for the Burgers equation was introduced \[ v_t+uv_x=0, v=u-\alpha^2 u_{xx},\eqno{(1)} \] where \(\alpha>0\) is a constant that has the dimension of the length. It was shown that the system (1) is globally well-posed with initial data \(v(x,0)\) in the Sobolev space \(W^{2,1}(\mathbb{R})\), and also the strong convergence of the solutions \(u^{\alpha}(x,t)\) at \(\alpha\rightarrow 0\) to the entropy solution of the inviscid Burgers equation. In this article the authors prove stability (instability) in some sense for monotone decreasing (monotone increasing) fronts, i.e. travelling waves which are monotonic profiles connecting a left state to a right state. These two types of travelling fronts correspond, respectively, to viscous shocks and rarefaction waves.

MSC:

37K45 Stability problems for infinite-dimensional Hamiltonian and Lagrangian systems

Citations:

Zbl 1108.35107
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References:

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