Linear damping estimates for periodic roll wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866916983972298752 |
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| author | Rodrigues, L. Miguel Zumbrun, Kevin |
| author_facet | Rodrigues, L. Miguel Zumbrun, Kevin |
| contents | Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions 7 and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov's Lemma for ODE that is to our knowledge new. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12804 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear damping estimates for periodic roll wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws Rodrigues, L. Miguel Zumbrun, Kevin Analysis of PDEs 35B35, 35L67, 15A18, 15A42, 35Q35, 35P15, 15A60, 15A63 Substantially extending previous results of the authors for smooth solutions in the viscous case, we develop linear damping estimates for periodic roll-wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws. Such damping estimates, consisting of $H^s$ energy estimates yielding exponential slaving of high-derivative to low-derivative norms, have served as crucial ingredients in nonlinear stability analyses of traveling waves in hyperbolic or partially parabolic systems, both in obtaining high-frequency resolvent estimates and in closing a nonlinear iteration for which available linearized stability estimates apparently lose regularity. Here, we establish for systems of size $n\leq 6$ a Lyapunov-type theorem stating that such energy estimates are available whenever strict high-frequency spectral stability holds; for dimensions 7 and higher, there may be in general a gap between high-frequency spectral stability and existence of the type of energy estimate that we develop here. A key ingredient is a dimension-dependent linear algebraic lemma reminiscent of Lyapunov's Lemma for ODE that is to our knowledge new. |
| title | Linear damping estimates for periodic roll wave solutions of the inviscid Saint-Venant equations and related systems of hyperbolic balance laws |
| topic | Analysis of PDEs 35B35, 35L67, 15A18, 15A42, 35Q35, 35P15, 15A60, 15A63 |
| url | https://arxiv.org/abs/2504.12804 |