Nonlinear stability of time-periodic viscous shocks - Mathematics > Analysis of PDEsReport as inadecuate




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Abstract: In order to understand the nonlinear stability of many types of time-periodictravelling waves on unbounded domains, one must overcome two main difficulties:the presence of embedded neutral eigenvalues and the time-dependence of theassociated linear operator. This problem is studied in the context oftime-periodic Lax shocks in systems of viscous conservation laws. Using spatialdynamics and a decomposition into separate Floquet eigenmodes, it is shown thatthe linear evolution for the time-dependent operator can be represented using acontour integral similar to that of the standard time-independent case. Bydecomposing the resulting Green-s distribution, the leading order behaviorassociated with the embedded eigenvalues is extracted. Sharp pointwise boundsare then obtained, which are used to prove that time-periodic Lax shocks arelinearly and nonlinearly stable under the necessary conditions of spectralstability and minimal multiplicity of the translational eigenvalues. The latterconditions hold, for example, for small-oscillation time-periodic waves thatemerge through a supercritical Hopf bifurcation from a family oftime-independent Lax shocks of possibly large amplitude.



Author: Margaret Beck, Bjorn Sandstede, Kevin Zumbrun

Source: https://arxiv.org/







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