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Award ID contains: 2306910

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  1. ; ; (, Archive for Rational Mechanics and Analysis)
    Abstract In this paper, we rigorously prove the existence of self-similar converging shock wave solutions for the non-isentropic Euler equations for$$\gamma \in (1,3]$$ γ ∈ ( 1 , 3 ] . These solutions are analytic away from the shock interface before collapse, and the shock wave reaches the origin at the time of collapse. The region behind the shock undergoes a sonic degeneracy, which causes numerous difficulties for smoothness of the flow and the analytic construction of the solution. The proof is based on continuity arguments, nonlinear invariances, and barrier functions. 
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  2. ; ; (, Journal of Mathematical Fluid Mechanics)
    Abstract We examine a micro-scale model of superfluidity derived by Pitaevskii (Sov. Phys. JETP 8:282-287, 1959) which describes the interacting dynamics between superfluid He-4 and its normal fluid phase. This system consists of the nonlinear Schrödinger equation and the incompressible, inhomogeneous Navier-Stokes equations, coupled to each other via a bidirectional nonlinear relaxation mechanism. The coupling permits mass/momentum/energy transfer between the phases, and accounts for the conversion of superfluid into normal fluid. We prove the existence of global weak solutions in$${\mathbb {T}}^3$$ T 3 for a power-type nonlinearity, beginning from small initial data. The main challenge is to control the inter-phase mass transfer in order to ensure the strict positivity of the normal fluid density, while obtaining time-independent a priori estimates. 
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  3. ; ; (, SIAM Journal on Mathematical Analysis)
    Free, publicly-accessible full text available February 28, 2026