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The Degasperis-Procesi (DP) equation is an integrable Camassa-Holm-type model which is an asymptotic approximation for the unidirectional propagation of shallow water waves. This work establishes the orbital stability of localized smooth solitary waves to the DP equation on the real line, extending our previous work on their spectral stability [J. Math. Pures Appl. (9) 142 (2020), pp. 298–314]. The main difficulty stems from the fact that the natural energy space is a subspace of L 3 L^3 , but the translation symmetry for the DP equation gives rise to a conserved quantity equivalent to the L 2 L^2 -norm, resulting in L 3 L^3 higher-order nonlinear terms in the augmented Hamiltonian. But the usual coercivity estimate is in terms of L 2 L^2 norm for DP equation, which cannot be used to control the L 3 L^3 higher order term directly. The remedy is to observe that, given a sufficiently smooth initial condition satisfying some mild constraint, the L ∞ L^\infty orbital norm of the perturbation is bounded above by a function of its L 2 L^2 orbital norm, yielding the higher order control and the orbital stability in the L 2 ∩ L ∞ L^2\cap L^\infty space.
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L-HPG Metabolic labeling of mitochondrial translation in cultured cell lines
This protocol describes the metabolic labeling of mitochondrial translation in cultured cells (such as IMR90) using an amino acid L-Homopropargylglycine (L-HPG).
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- Award ID(s):
- 2339182
- PAR ID:
- 10569641
- Publisher / Repository:
- Springer Nature
- Date Published:
- Edition / Version:
- 1
- Subject(s) / Keyword(s):
- mitochondria, mitochondrial genome, peptide synthesis, amino acids, mitoribosome, gene expression
- Format(s):
- Medium: X
- Institution:
- University of California, Berkeley
- Sponsoring Org:
- National Science Foundation
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- Free Publicly Accessible Full Text
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Accepted Manuscript1.0
- Posted Content:
- https://doi.org/10.17504/protocols.io.3byl4wedrvo5/v1
