In this paper, we analyze various types of critical phenomena in one-dimensional gas flows described by Euler equations. We give a geometrical interpretation of thermodynamics with a special emphasis on phase transitions. We use ideas from the geometrical theory of PDEs, in particular, symmetries and differential constraints to find solutions to the Euler system. Solutions obtained are multivalued, have singularities of projection to the plane of independent variables. We analyze the propagation of the shock wave front along with phase transitions.
In this paper, a system of one-dimensional gas dynamics equations is considered. This system is a particular case of Jacobi type systems and has a natural representation in terms of 2-forms on 0-jet space. We use this observation to find a new class of multivalued solutions for an arbitrary thermodynamic state model and discuss singularities of their projections to the space of independent variables for the case of an ideal gas. Caustics and discontinuity lines are found.
In this brief note we show that the author's previous result in \cite{cha} on the nonexistence of self-similar singularities for the 3D incompressible Euler equations implies actually the nonexistence of `locally self-similar' singular solution, which has been sought by many physicists.
We find a sharp condition on the density-dependent coefficient of damping of a one-dimensional repulsive Euler-Poisson system, which makes it possible to suppress the formation of singularities in the solution of the Cauchy problem with arbitrary smooth data. In the context of plasma physics, this means the possibility of suppressing the breakdown of arbitrary oscillations of cold plasma.
We prove that there exists no self-similar finite time blowing up solution to the 3D incompressible Euler equations. By similar method we also show nonexistence of self-similar blowing up solutions to the divergence-free transport equation in $\Bbb R^n$. This result has direct applications to the density dependent Euler equations, the Boussinesq system, and the quasi-geostrophic equations, for which we also show nonexistence of self-similar blowing up solutions.
We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial stationary state. We prove the blowup results using the characteristics of the propagation of the solution in space and find upper and lower bounds for the density of a smooth solution in a given region of space in terms of the initial data. To solve the problems, we introduce a special family of integral functionals and study their temporal dynamics.
In this paper, we prove the existence of smooth initial data for the 2D free boundary incompressible Euler equations (also known for some particular scenarios as the water wave problem), for which the smoothness of the interface breaks down in finite time into a splash singularity or a splat singularity.
We consider solutions to the Euler equations in the whole space from a certain class, which can be characterized, in particular, by finiteness of mass, total energy and momentum. We prove that for a large class of right-hand sides, including the viscous term, such solutions, no matter how smooth initially, develop a singularity within a finite time. We find a sufficient condition for the singularity formation, "the best sufficient condition", in the sense that one can explicitly construct a global in time smooth solution for which this condition is not satisfied "arbitrary little". Also compac…
In this paper we discuss the existence of stationary incompressible fluids with splash singularities. Specifically, we show that there are stationary solutions to the Euler equations with two fluids whose interfaces are arbitrarily close to a splash, and that there are stationary water waves with splash singularities.
We prove that the 3-D free-surface incompressible Euler equations with regular initial geometries and velocity fields have solutions which can form a finite-time "splash" (or "splat") singularity first introduced in [9], wherein the evolving 2-D hypersurface, the moving boundary of the fluid domain, self-intersects at a point (or on surface). Such singularities can occur when the crest of a breaking wave falls unto its trough, or in the study of drop impact upon liquid surfaces. Our approach is founded upon the Lagrangian description of the free-boundary problem, combined with a novel approxim…
By introducing a new averaged quantity with a fast decay weight to perform Sideris's argument (Commun Math Phys, 1985) developed for the Euler Equations, we extend the formation of singularities of classical solution to the 3D Euler Equations established in Sideris (1985) and Makino et al. (Jpn J Appl Math, 1986) for the initial data with compactly supported disturbances to the spherically symmetric solution with general initial data in Sobolev space. Moreover, we also prove the formation of singularities of the spherically symmetric solutions to the 3D Euler-Poisson Equations, but remove the …
An example of a solution branch of the three dimensional Euler equation Cauchy problem is constructed which develops a singular velocity component and a singular vorticity component after finite time for some data which have Hoelder continuous first order spatial derivatives. Such a solution branch can be extended beyond a time section at some positive finite time where a two dimensional set of singularities is located.
We are concerned with the formation of singularities and the existence of global continuous solutions of the Cauchy problem for the one-dimensional non-isentropic Euler equations for compressible fluids. For the isentropic Euler equations, we pinpoint a necessary and sufficient condition for the formation of singularities of solutions with large initial data that allow a far-field vacuum -- there exists a compression in the initial data. For the non-isentropic Euler equations, we identify a sufficient condition for the formation of singularities of solutions with large initial data that allow …
We consider a system of two incompressible fluids separated by a free interface. The first fluid is inviscid, governed by the Euler system, while the second fluid has positive viscosity and is governed by the Navier-Stokes system. We formulate a notion of splash-type self-intersection singularities, and show that this system cannot form such a singularity in finite time. The main obstacle in our analysis is to handle the effect of bulk vorticity induced by the Navier-Stokes flow. To the best of our knowledge this is the first result on preclusion of splash-type singularities in the presence of…
Presented are two results on the formation of finite time singularities of solutions to the compressible Euler equations in two and three space dimensions for isentropic, polytropic, ideal fluid flows. The initial velocity is assumed to be symmetric and the initial sound speed is required to vanish at the origin. They are smooth in Sobolev space $H^3$, but not required to have a compact support. It is shown that the $H^3$ norm of the velocity field and the sound speed will blow up in a finite time.
Classical vorticity solution branches of the three dimensional incompressible Euler equation are constructed where a velocity component can blow up at some point after finite time for regular data in H2. Furthermore, vorticity can blow up after finite time for data in H2, and there are classical solution branches with regular data which develop weak singularities at some point of space time after finite time. The construction of these time-local solution branches is by viscosity limits of viscosity extensions of time-reversed Euler-type equations. The short time (weak) singularities are then i…
In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of self-similar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical pro…
A famous theorem of Bers and Finn states that isolated singularities of solutions to the non-parametric minimal surface equation are removable. We show that this result remains valid, if the area functional is replaced by a general functional of linear growth depending on the modulus of the gradient. We emphasize that Serrin ([1]) in fact proved the removability of singularities on sets of $(n-1)$-dimensional Hausdorff measure zero in an even more general setting. Our main interest is to generalize the comparison principles as outlined, for instance, in Section 10 of [2] without having the par…
Euler-Leray data functions of first and second order are defined by first and second order derivatives of the nonlinear spatial part of the incompressible Euler equation operator in Leray projection form applied to Cauchy data. The Lipschitz continuity of these functions for a certain class of strongly regular Cauchy data is sufficient for the existence of global regular upper bounds of incompressible Navier Stokes equation solutions. Global regular upper bounds of global solution branches of the incompressible Euler equation can be obtained for a class of strongly regular Cauchy data, if the …
In this paper we rule out the possibility of asymptotically self-similar singularities for both of the 3D Euler and the 3D Navier-Stokes equations. The notion means that the local in time classical solutions of the equations develop self-similar profiles as $t$ goes to the possible time of singularity $T$. For the Euler equations we consider the case where the vorticity converges to the corresponding self-similar voriticity profile in the sense of the critical Besov space norm, $\dot{B}^0_{1, \infty}(\Bbb R^3)$. For the Navier-Stokes equations the convergence of the velocity to the self-simila…
We prove local blow-up criterion for smooth axisymmetric solutions to the 3D incompressible Euler equation. If the vorticity satisfies $ \intl_{0}^{t_*} (t_*-t) \| ω(t)\|_{ L^\infty(B(x_{ \ast}, R_0))} dt <+\infty$ for a ball $B(x_{ \ast}, R_0)$ away from the axis of symmetry, then there exists no singularity at $t=t_*$ in the torus $T(x_*, R)$ generated by rotation of the ball $B(x_{ \ast}, R_0)$ around the axis. This implies that possible singularity at $t=t_*$ in the torus $T(x_*, R)$ is excluded if the vorticity satisfies the blow-up rate $ \|ø(t)\|_{L^\infty (T(x_*, R))}= O\left(\frac{1}{…
We study the formation of singularities for the Euler-Alignment system with influence function $ψ=\frac{k_α}{|x|^α}$ in 1D. As in [20] the problem is reduced to the analysis of a nonlocal 1D equation. We show the existence of singularities in finite time for any $α$ in the range $0<α<2$ in both the real line and the periodic case.
We address the question whether a singularity in a three-dimensional incompressible inviscid fluid flow can occur in finite time. Analytical considerations and numerical simulations suggest high-symmetry flows being a promising candidate for a finite-time blowup. Utilizing Lagrangian and geometric non-blowup criteria, we present numerical evidence against the formation of a finite-time singularity for the high-symmetry vortex dodecapole initial condition. We use data obtained from high resolution adaptively refined numerical simulations and inject Lagrangian tracer particles to monitor geometr…
We study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the 3D Euler equations, where the sense of convergence and self-similarity are considered in various sense. We extend much further, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities obtained in \cite{cha1,cha2}. Some implications the notions for the 3D Navier-Stokes equations are also deduced. Generalization of the self-similar transforms is also considered, and by appropriate choice of the transform we obtain new \textit{a priori} es…
This paper contributes to the study of large data problems for $C^1$ solutions of the relativistic Euler equations. In the $(1+1)$-dimensional spacetime setting, if the initial data are away from vacuum, a key difficulty in proving the global well-posedness or finite time blow-up is coming up with a way to obtain sharp enough control on the lower bound of the mass-energy density function $ρ$. First, for $C^1$ solutions of the 1-dimensional classical isentropic compressible Euler equations in the Eulerian setting, we show a novel idea of obtaining a mass density time-dependent lower bound by st…
The purpose of this paper is to study the phenomenon of singularity formation in large data problems for classical solutions to the Cauchy problem of the relativistic Euler equations. The classical theory established by P. D. Lax in 1964 (J. Math. Phys. 5: 611-614) shows that, for 2x2 hyperbolic systems, the break-down of classical solutions occurs in finite time if initial data contain any compression in some truly nonlinear characteristic field under some additional conditions, which include genuine nonlinearity and the strict positivity of the difference between two corresponding eigenvalue…
We show that certain singular structures (Hölderian cusps and mild divergences) are transported by the flow of homeomorphisms generated by an Osgood velocity field. The structure of these singularities is related to the modulus of continuity of the velocity and the results are shown to be sharp in the sense that slightly more singular structures cannot generally be propagated. For the 2D Euler equation, we prove that certain singular structures are preserved by the motion, e.g. a system of $\log\log_+(1/|x|)$ vortices (and those that are slightly less singular) travel with the fluid in a nonli…
Here we consider the 2D free boundary incompressible Euler equation with surface tension. We prove that the surface tension does not prevent a finite time splash or splat singularity, i.e. that the curve touches itself either in a point or along an arc. To do so, the main ingredients of the proof are a transformation to desingularize the curve and a priori energy estimates.
We study the formation of singularity for the Euler-Poisson system equipped with the Boltzmann relation, which describes the dynamics of ions in an electrostatic plasma. In general, it is known that smooth solutions to nonlinear hyperbolic equations fail to exist globally in time. We establish criteria for $C^1$ blow-up of the Euler-Poisson system, both for the isothermal and pressureless cases. In particular, our blow-up condition for the presureless model does not require that the gradient of velocity is negatively large. In fact, our result particularly implies that the smooth solutions can…
We consider self-gravitating fluids in cosmological spacetimes with Gowdy symmetry on the torus $T^3$ and, in this class, we solve the singular initial value problem for the Einstein-Euler system of general relativity, when an initial data set is prescribed on the hypersurface of singularity. We specify initial conditions for the geometric and matter variables and identify the asymptotic behavior of these variables near the cosmological singularity. Our analysis of this class of nonlinear and singular partial differential equations exhibits a condition on the sound speed, which leads us to the…
In this paper we study the singularity formation for the geometric flow of complex curves $$z_t = -z_{xxx} + \frac{3}{2}øz_{x} z_{xx}^2,$$ that was derived [R. E. Goldstein and D. M. Petrich, {\em Phys. Rev. Lett.}, 69 (1992), pp. 555--558] while considering the vortex patch dynamics for the incompressible 2D Euler equation. We prove that arbitrary curve, consisting of two rotating logarithmic spirals, is a finite time singularity developed by a smooth solution of the flow. We provide exact construction of the solution in the terms of appropriate Painlevé II transcendents and furthermore we es…
