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Article
Mathematics, Applied
Tristan Buckmaster et al.
Summary: We analyze the formation process of shocks in the 3D nonisentropic Euler equations with the ideal gas law, where sound waves interact with entropy waves to produce vorticity. Building on previous theories, we provide a constructive proof of shock formation from smooth initial data. We prove the existence of smooth solutions to the nonisentropic Euler equations that form a stable shock with computable blowup time, location, and direction by establishing the asymptotic stability of a generic shock profile in modulated self-similar variables and controlling the interaction of wave families through various techniques.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
Article
Mathematics, Applied
Tristan Buckmaster et al.
Summary: In this paper, we study the 3D isentropic compressible Euler equations and provide a proof of the formation of a generic stable shock from smooth initial data. We show that under certain conditions, smooth solutions exist and a shock wave can form in finite time. The blowup time and location can be explicitly computed, and the solutions at the blowup time are smooth except for a single point.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2023)
Article
Mathematics
Nikolaos Athanasiou et al.
Summary: The purpose of this paper is to study the phenomenon of singularity formation in large data problems for C1 solutions to the Cauchy problem of the relativistic Euler equations. By introducing several key artificial quantities and conducting elaborate analysis on the difference of the two Riemann invariants, the decay of mass-energy density lower bound in time is characterized, and concrete progress is made.
TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics
Sanchit Chaturvedi et al.
Summary: We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the first shock forms. The inner part of this expansion has a novel structure based on a fractional spacetime Taylor series for the inviscid solution. We obtain sharp vanishing viscosity rates in a variety of norms, including L-infinity. Comparable prior results break down in the vicinity of shock formation. We partially fill this gap.
Article
Mathematics
Yan Guo et al.
Summary: In this study, Ori and Piran predicted the existence of selfsimilar spacetimes called relativistic Larson-Penston solutions, and provided a mathematical proof for their existence. The proof involved a thorough analysis of nonlinear invariances associated with the dynamical system, the development of a shooting method, and the construction of a maximal analytic extension of the solution. The problem was reformulated in double-null gauge to obtain an asymptotically flat spacetime with an isolated naked singularity.
Article
Mathematics
David Fajman et al.
Summary: This paper studies cosmological solutions to the Einstein-Euler equations. It establishes the future stability of nonlinear perturbations of a class of homogeneous solutions and shows the global regularity and stability of the Milne spacetime under the coupled Einstein-Euler equations.
INTERNATIONAL MATHEMATICS RESEARCH NOTICES
(2023)
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Physics, Mathematical
Tristan Buckmaster et al.
Summary: This paper explores the construction of unstable shocks in 2D isentropic compressible Euler equations in azimuthal symmetry, demonstrating stability under certain conditions using modulation variable techniques in conjunction with a Newton scheme.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Yin Huicheng et al.
Summary: The paper investigates the shock formation and regularities of shock curves for hyperbolic conservation laws. It shows that under certain conditions, a weak entropy solution and shock curve can be locally constructed. When these conditions are violated, the study focuses on the optimal regularities of the shock curve and behavior of the solution near the blowup point.
Article
Mathematics, Applied
Marcelo M. Disconzi et al.
Summary: We prove a series of intimately related results tied to the regularity and geometry of solutions to the 3D compressible Euler equations. Our main result is that the time of existence can be controlled in terms of various norms of the initial data.
SELECTA MATHEMATICA-NEW SERIES
(2022)
Article
Mathematics
Qian Wang
Summary: We prove the local-in-time well-posedness of the solution to the compressible Euler equations in 3-D, and give some restrictions on the solution. By separating different parts of the sound waves and utilizing the nonlinear structure of the system and the geometric structure of acoustic spacetime, the regularity of the solution is significantly improved.
ANNALS OF MATHEMATICS
(2022)
Article
Physics, Mathematical
Xinliang An et al.
Summary: In this paper, a new approach combining algebraic and geometric ideas is used to prove the low regularity ill-posedness for quasilinear hyperbolic systems with non-strict hyperbolicity in three dimensions. These systems are also associated with multiple wave speeds.
JOURNAL OF MATHEMATICAL PHYSICS
(2022)
Article
Mathematics, Applied
Yin Huicheng et al.
Summary: This paper investigates the problem of formation and construction of a multidimensional shock wave for the first-order conservation law \partial tu + \partial xF(u) + \partial yG(u) = 0 with smooth initial data u0(x, y). It establishes a local weak entropy solution u for t \geq T* that is not uniformly Lipschitz continuous on two sides of the shock surface \Sigma under the nondegenerate condition of H(\xi ,\eta ). The constructed shock gradually increases in strength after being zero on the initial blowup curve \Gamma, and detailed descriptions of the singularities of the solution u are provided in the neighborhood of \Gamma.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Charles Collot et al.
Summary: We investigate the properties of a special class of outer Euler flows and solutions of the two-dimensional unsteady Prandtl system. We provide a precise description of singular solutions for a reduced problem and identify a stable blow-up pattern. Additionally, we demonstrate the persistence of analyticity around the axis up to the blow-up time for general solutions that follow the stable blow-up pattern, and establish a universal lower bound for the radius of analyticity.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Tristan Buckmaster et al.
Summary: This study proves the fundamental question in fluid dynamics concerning the formation of discontinuous shock waves from smooth initial data. The research shows that smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called C-1/3 pre-shock, and instantaneously develop into a discontinuous shock. Additionally, two other characteristic surfaces of cusp-type singularities are discovered.
Article
Mathematics
Frank Merle et al.
Summary: This paper and its sequel construct a set of finite energy smooth initial data for which the corresponding solutions to the compressible three-dimensional Navier-Stokes and Euler equations implode at a later time and point, and describe the formation of singularity. The existence of smooth self-similar profiles for the barotropic Euler equations in dimension d >= 2 with decaying density at spatial infinity is studied. The phase portrait of the nonlinear ODE for spherically symmetric self-similar solutions allows the construction of global profiles, but they are generically non-smooth. The existence of non-generic C-infinity self-similar solutions with suitable decay at infinity is proved, leading to the construction of finite energy blow up solutions of the compressible Euler and Navier-Stokes equations in dimensions d = 2, 3.
ANNALS OF MATHEMATICS
(2022)
Article
Mathematics
Frank Merle et al.
Summary: This paper investigates strong singularity formation in compressible fluids, focusing on the compressible three-dimensional Navier-Stokes and Euler equations. By constructing a set of finite energy smooth initial data satisfying suitable barotropic laws, the paper describes in detail the formation of singularity where the corresponding solutions implode at a later time at a point with infinite density. The existence of C-infinity smooth self-similar solutions to the compressible Euler equations, specifically those with quantized values of the speed, plays a crucial role in the proof. The obtained blow up dynamics for the Navier-Stokes problem are all of type II, which are non self-similar.
ANNALS OF MATHEMATICS
(2022)
Article
Mathematics, Applied
Yan Guo et al.
Summary: In this article, we prove the existence of smooth radially symmetric self-similar solutions to the gravitational Euler-Poisson system in the supercritical range of the polytropic indices. These solutions demonstrate gravitational collapse with the density blowing up in finite time. Some of the numerically found solutions by Yahil in 1983 can be seen as polytropic analogues of the Larson-Penston collapsing solutions in the isothermal case. Each solution contains a sonic point, leading to several mathematical difficulties in the existence proof.
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2022)
Article
Mathematics
Charles Collot et al.
Summary: In this paper, we consider Burgers' equation with transverse viscosity and describe a family of solutions which become singular in finite time with unbounded gradients. The leading order solution is given by a backward self-similar solution of Burgers' equation along the x variable, with the scaling parameters evolving according to parabolic equations along the y variable.
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(2022)
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Mathematics, Applied
Tristan Buckmaster et al.
Summary: In this paper, we consider the 2D isentropic compressible Euler equations and provide a constructive proof for the formation of shocks from smooth initial data. By constructing solutions, we demonstrate the existence of shocks with inherent vorticity and analyze their blowup time and location.
COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS
(2022)
Article
Physics, Mathematical
David Fajman et al.
Summary: The study focuses on the global regularity and stability of the relativistic Euler equations with a state equation, for small initial data in expanding spacetime. The result suggests that the Universe's decelerating expansion phase is necessary for structure formation in cosmological evolution.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2021)
Article
Mathematics, Applied
Charles Collot et al.
Summary: This paper studies the inviscid unsteady Prandtl system in two dimensions and provides a sharp expression for the maximal time of existence of regular solutions. It shows that singularities occur only at the boundary or on the set of zero vorticity, corresponding to boundary layer separation. The paper also presents new Lagrangian formulae for backward self-similar profiles and explores a different approach to studying singularities for quasilinear transport equations.
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(2021)
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Mathematics
Nikolaos Athanasiou et al.
Summary: This paper contributes to the study of large data problems for C-1 solutions of the relativistic Euler equations. Necessary and sufficient conditions for singularity formation in finite time are provided, along with insights into the asymptotic behavior of relativistic velocity.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
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Mathematics, Applied
Huali Zhang
Summary: This statement presents the local existence, uniqueness, and stability of local solutions for the Cauchy problem of two-dimensional compressible Euler equations, with specific requirements for the initial data.
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(2021)
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(2020)
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