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Seminars

Analysis Seminar

Local-in-Time Solutions to the Perturbed Riemann Problem for the Multidimensional Compressible Euler Equations with Centered Rarefaction Waves and Other Wave Patterns

2026/04/21 (Tue.) 10:00~13:00
  • Date : 2026/04/21 (Tue.) 10:00~13:00
  • Location : online
  • Speaker : Tao Luo (City University of Hong Kong)
  • Organizer : Tai-Ping Liu (AS)
  • Abstract :
    This talk is based on recent joint work with Prof. Jin Jia at Hunan University.
    For the two-dimensional compressible Euler equations, we study the perturbed Riemann problem with piecewise smooth initial data given by small perturbations of two constant states. We first derive simultaneous energy estimates, without loss of derivatives, for acoustic and vorticity waves within the rarefaction region by introducing acoustic coordinates adapted to the characteristic geometry and by controlling the geometry of characteristic hypersurfaces. Building on these estimates, we construct a family of rarefaction fronts with vorticity that is consistent with the intrinsic singular behavior of centered rarefaction waves near the initial singular set. This shows that small perturbations of suitable rarefaction-wave data give rise to stable multidimensional rarefaction-wave solutions even in the presence of vorticity.
    The main technique is the simultaneous treatment of acoustic and vorticity waves without loss of derivatives. To achieve this, we reformulate the Euler system as a coupled wave-transport system for suitable Riemann invariants and the specific vorticity Ω. The acoustic part is governed by a quasilinear wave operator associated with the acoustical metric, while the vorticity satisfies a transport equation along the material flow. This removes the irrotational assumption used in previous work on rarefaction-wave stability.
    After establishing the existence of multidimensional rarefaction waves connected to prescribed right-hand states, we apply the theory to perturbed Riemann problems. In particular, we prove local structural stability for two important wave patterns:
    shock-rarefaction and rarefaction-vortex sheet-rarefaction.

    For these configurations, we show that the perturbed solution can be constructed as a piecewise smooth flow separated by characteristic or non-characteristic hypersurfaces, with the expected jump and entropy conditions across shock and vortex-sheet interfaces. The rarefaction regions are described by acoustically defined characteristic hypersurfaces, and the interaction geometry is encoded through carefully chosen inner boundaries and compatibility conditions.
    In summary, we extend the theory of multidimensional rarefaction waves from the irrotational setting to flows with vorticity, thereby providing a rigorous framework for analyzing more complex multidimensional Riemann structures involving shocks, rarefaction waves, and vortex sheets.
    If time permits, I will also briefly discuss the multidimensional free expansion of an ideal gas into vacuum through rarefaction waves.

Click here to register: https://forms.gle/EhVyNa94MDEhQkqF9 (Deadline: 4pm, Apr. 20)
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