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Seminars

Analysis Seminar

Scalar viscous conservation law with time-shift coupled nonlinearity

2026/03/10 (Tue.) 10:00~13:00
  • Date : 2026/03/10 (Tue.) 10:00~13:00
  • Location : online
  • Speaker : Shih-Hsien Yu (Academia Sinica)
  • Organizer : Tai-Ping Liu (AS)
  • Abstract :
    We investigate the scalar viscous conservation law $\partial_t u + \partial_x {\mathsf F}[u] - \partial_x^2 u = 0$, characterized by a time-delayed nonlinearity ${\mathsf F}[u](x,t) \equiv u(x,t)u(x,t-\tau)/2$ for a constant $\tau > 0$.
    For a sufficiently small time delay $\tau$, we prove the existence of traveling wave solutions corresponding to arbitrary inviscid Burgers' shock profiles $(u_-,u_+)$.
    Furthermore, we establish the nonlinear time asymptotic stability of these traveling waves and determine their pointwise decay rates in the space-time domain.
    By exploiting the specific structure of the Green's function for the Burgers' equation linearized around the stationary profile $\phi(x)=-\tanh(x/2)$, we derive an effective integral representation of the solution. This explicit Green's function enables us to capture the cancellation of effects induced by the time shift.
    Consequently, this integral representation yields a constructive method for obtaining exponentially sharp pointwise estimates.
    The analysis presented in this paper will serve as a foundation for studying nonlinear wave interactions in systems of viscous conservation laws with delayed nonlinearity.

Click here to register: https://forms.gle/EhVyNa94MDEhQkqF9 (Deadline: 4pm, March 9)
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