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From an open set of initial data, we construct a family of classical solutions to the 1D nonisentropic compressible Euler equations which form $ C^{0,\nu} $ cusps as a first singularity, for any $ \nu \in [\frac{1}{2}, 1) $. For this range of $ \nu $, this is the first result demonstrating the stable formation of such $ C^{0,\nu} $ cusp-type singularities, also known as pre-shocks. The proof uses a new formulation of the differentiated Euler equations along the fast acoustic characteristic, and relies on a novel set of $ L^p $ energy estimates for all $ 1

We consider the 3D isentropic compressible Euler equations with the ideal gas law. We provide a constructive proof of the formation of the first point shock from smooth initial datum of finite energy, with no vacuum regions, with nontrivial vorticity present at the shock, and under no symmetry assumptions. We prove that for an open set of Sobolev‐class initial data that are a small L∞ perturbation of a constant state, there exist smooth solutions to the Euler equations which form a generic stable shock in finite time. The blowup time and location can be explicitly computed, and solutions at the blowup time are smooth except for a single point, where they are of cusp‐type with Hölder C1/3 regularity. Our proof is based on the use of modulated self‐similar variables that are used to enforce a number of constraints on the blowup profile, necessary to establish global existence and asymptotic stability in self‐similar variables. © 2022 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We analyze the shock formation process for the 3D nonisentropic Euler equations with the ideal gas law, in which sound waves interact with entropy waves to produce vorticity. Building on our theory for isentropic flows in [3, 4], we give a constructive proof of shock formation from smooth initial data. Specifically, we prove that there exist smooth solutions to the nonisentropic Euler equations which form a generic stable shock with explicitly computable blowup time, location, and direction. This is achieved by establishing the asymptotic stability of a generic shock profile in modulated self‐similar variables, controlling the interaction of wave families via: (i) pointwise bounds along Lagrangian trajectories, (ii) geometric vorticity structure, and (iii) high‐order energy estimates in Sobolev spaces. © 2022 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

We prove non-uniqueness for a class of weak solutions to the Navier-Stokes equations which have bounded kinetic energy, integrable vorticity, and are smooth outside a fractal set of singular times with Hausdorff dimension strictly less than 1. [ABSTRACT FROM AUTHOR]

We consider the 2D isentropic compressible Euler equations, with pressure law p(ρ) = (1γ)ργ, with γ > 1. We provide an elementary constructive proof of shock formation from smooth initial data of finite energy, with no vacuum regions, and with nontrivial vorticity. We prove that for initial data which has minimum slope −1ε, for ε > 0 taken sufficiently small relative to the O1 amplitude, there exist smooth solutions to the Euler equations which form a shock in time Oε. The blowup time and location can be explicitly computed and solutions at the blowup time are of cusp‐type, with Hölder C13 regularity. Our objective is the construction of solutions with inherent O1 vorticity at the shock. As such, rather than perturbing from an irrotational regime, we instead construct solutions with dynamics dominated by purely azimuthal wave motion. We consider homogenous solutions to the Euler equations and use Riemann‐type variables to obtain a system of forced transport equations. Using a transformation to modulated self‐similar variables and pointwise estimates for the ensuing system of transport equations, we show the global stability, in self‐similar time, of a smooth blowup profile. © 2020 Wiley Periodicals LLC. [ABSTRACT FROM AUTHOR]

The triple‐deck model is a classical high‐order boundary layer model that has been proposed to describe flow regimes where the Prandtl theory is expected to fail. At first sight the model appears to lose two derivatives through the pressure‐displacement relation that links pressure to the tangential slip. In order to overcome this, we split the triple‐deck system into two coupled equations: a Prandtl‐type system on ℍ and a Benjamin‐Ono‐type equation on ℝ. This splitting enables us to extract a crucial leading‐order cancellation at the top of the lower deck. We develop a functional framework to subsequently extend this cancellation into the interior of the lower deck, which enables us to prove the local well‐posedness of the model in tangentially real analytic spaces. © 2019 Wiley Periodicals, Inc. [ABSTRACT FROM AUTHOR]