Finite-time Singularity Formation Research Articles

Lagrangian dynamics and regularity of the spin Euler equation

We derive the spin Euler equation for ideal flows by applying the spherical Clebsch mapping. This equation is based on the spin vector, a unit vector field encoding vortex lines, instead of the velocity. The spin Euler equation enables a feasible Lagrangian study of fluid dynamics, as the isosurface of a spin-vector component is a vortex surface and material surface in ideal flows. We establish a non-blowup criterion for the spin Euler equation, suggesting that the Laplacian of the spin vector must diverge if the solution forms a singularity at some finite time. The direct numerical simulations (DNS) of three ideal flows – the vortex knot, the vortex link and the modified Taylor–Green flow – are conducted by solving the spin Euler equation. The evolution of the Lagrangian vortex surface illustrates that the regions with large vorticity are rapidly stretched into spiral sheets. The DNS result exhibits a pronounced double-exponential growth of the maximum norm of Laplacian of the spin vector, showing no evidence of the finite-time singularity formation if the double-exponential growth holds at later times. Moreover, the present criterion with Lagrangian nature appears to be more sensitive than the Beale–Kato–Majda criterion in detecting the flows that are incapable of producing finite-time singularities.

Global existence and singularity formation for the generalized Constantin–Lax–Majda equation with dissipation: the real line vs. periodic domains

The question of global existence versus finite-time singularity formation is considered for the generalized Constantin–Lax–Majda equation with dissipation , where , both for the problem on the circle and the real line. In the periodic geometry, two complementary approaches are used to prove global-in-time existence of solutions for and all real values of an advection parameter a when the data is small. We also derive new analytical solutions in both geometries when a = 0, and on the real line when , for various values of σ. These solutions exhibit self-similar finite-time singularity formation, and the similarity exponents and conditions for singularity formation are fully characterized. We revisit an analytical solution on the real line due to Schochet for a = 0 and σ = 2, and reinterpret it terms of self-similar finite-time collapse. The analytical solutions on the real line allow finite-time singularity formation for arbitrarily small data, even for values of σ that are greater than or equal to one, thereby illustrating a critical difference between the problems on the real line and the circle. The analysis is complemented by accurate numerical simulations, which are able to track the formation and motion of singularities in the complex plane. The computations validate and build upon the analytical theory.

Development of singularities in the relativistic Euler equations

The purpose of this paper is to study the phenomenon of singularity formation in large data problems for C 1 C^1 solutions to the Cauchy problem of the relativistic Euler equations. The classical theory established by [P. D. Lax [J. Math. Phys. 5 (1964), pp. 611–613] shows that, for 2 × 2 2\times 2 hyperbolic systems, the break-down of C 1 C^1 solutions occurs in finite time if initial data contain any compression in some truly non-linear characteristic field under some additional conditions, which include genuine non-linearity and the strict positivity of the difference between two corresponding eigenvalues. These harsh structural assumptions mean that it is highly non-trivial to apply this theory to archetypal systems of conservation laws, such as the (1+1)-dimensional relativistic Euler equations. Actually, in the (1+1)-dimensional spacetime setting, if the mass-energy density ρ \rho does not vanish initially at any finite point, the essential difficulty in considering the possible break-down is coming up with a way to obtain sharp enough control on the lower bound of ρ \rho . To this end, based on introducing several key artificial quantities and some elaborate analysis on the difference of the two Riemann invariants, we characterized the decay of mass-energy density lower bound in time, and ultimately made some concrete progress. On the one hand, for the C 1 C^1 solutions with large data and possible far field vacuum to the isentropic flow, we verified the theory obtained by P. D. Lax in 1964. On the other hand, for the C 1 C^1 solutions with large data and strictly positive initial mass-energy density to the non-isentropic flow, we exhibit a numerical value N N , thought of as representing the strength of an initial compression, above which all initial data lead to a finite-time singularity formation. These singularities manifest as a blow-up in the gradient of certain Riemann invariants associated with corresponding systems.

Free surface in two-dimensional potential flow: singularities, invariants and virtual fluid

We study a two-dimensional (2-D) potential flow of an ideal fluid with a free surface with decaying conditions at infinity. By using the conformal variables approach, we study a particular solution of the Euler equations having a pair of square-root branch points in the conformal plane, and find that the analytic continuation of the fluid complex potential and conformal map define a flow in the entire complex plane, excluding a vertical cut between the branch points. The expanded domain is called the ‘virtual’ fluid, and it contains a vortex sheet whose dynamics is equivalent to the equations of motion posed at the free surface. The equations of fluid motion are analytically continued to both sides of the vertical branch cut (the vortex sheet), and additional time invariants associated with the topology of the conformal plane and Kelvin's theorem for a virtual fluid are explored. We called them ‘winding’ and virtual circulation. This result can be generalized to a system of many cuts connecting many branch points, resulting in a pair of invariants for each pair of branch points. We develop an asymptotic theory that shows how a solution originating from a single vertical cut forms a singularity at the free surface in infinite time, the rate of singularity approach is double exponential and supersedes the previous result of the short branch cut theory with finite time singularity formation. The present work offers a new look at fluid dynamics with a free surface by unifying the problem of motion of vortex sheets, and the problem of 2-D water waves. A particularly interesting question that arises in this context is whether instabilities of the virtual vortex sheet are related to breaking of steep ocean waves when gravity effects are included.

Blowup of Solutions to a Damped Euler Equation with Homogeneous Three-Point Boundary Condition

It is known that solutions to the inviscid Proudman-Johnson equation subject to a homogeneous three-point boundary condition can develop singularities in finite time. Given the regularizing effect that damping can have, in this paper we consider the possibility of finite-time singularity formation in solutions of the generalized inviscid Proudman-Johnson equation with damping subject to the same homogeneous three-point boundary condition. In particular, we will derive criteria the initial condition must satisfy in order for solutions to blowup in finite time under the effect of a smooth time-dependent damping term.

On regular solutions and singularity formation for Vlasov/Navier-Stokes equations with degenerate viscosities and vacuum

<p style='text-indent:20px;'>We analyze the Vlasov equation coupled with the compressible Navier–Stokes equations with degenerate viscosities and vacuum. These two equations are coupled through the drag force which depends on the fluid density and the relative velocity between particle and fluid. We first establish the existence and uniqueness of local-in-time regular solutions with arbitrarily large initial data and a vacuum. We then present sufficient conditions on the initial data leading to the finite-time blowup of regular solutions. In particular, our study makes the result on the finite-time singularity formation for Vlasov/Navier–Stokes equations discussed by Choi [J. Math. Pures Appl., 108, (2017), 991–1021] completely rigorous.</p>

The incompressible Euler equations under octahedral symmetry: Singularity formation in a fundamental domain

We consider the 3D incompressible Euler equations in vorticity form in the following fundamental domain for the octahedral symmetry group: {(x1,x2,x3):0<x3<x2<x1}. In this domain, we prove local well-posedness for Cα vorticities not necessarily vanishing on the boundary with any 0<α<1, and establish finite-time singularity formation within the same class for smooth and compactly supported initial data. The solutions can be extended to all of R3 via a sequence of reflections, and therefore we obtain finite-time singularity formation for the 3D Euler equations in R3 with bounded and piecewise smooth vorticities.

Finite-time singularity formation for $C^{1,\alpha}$ solutions to the incompressible Euler equations on $\mathbb{R}^3$

It has been known since work of Lichtenstein and Gunther in the 1920s that the 3D incompressible Euler equation is locally well-posed in the class of velocity fields with Hölder continuous gradient and suitable decay at infinity. It is shown here that these local solutions can develop singularities in finite time, even for some of the simplest three-dimensional flows.

Collapse Versus Blow-Up and Global Existence in the Generalized Constantin–Lax–Majda Equation

The question of finite time singularity formation vs. global existence for solutions to the generalized Constantin-Lax-Majda equation is studied, with particular emphasis on the influence of a parameter $a$ which controls the strength of advection. For solutions on the infinite domain we find a new critical value $a_c=0.6890665337007457\ldots$ below which there is finite time singularity formation % if we write a=a_c=0.6890665337007457\ldots here then \ldots doesn't fit into the line that has a form of self-similar collapse, with the spatial extent of blow-up shrinking to zero. We find a new exact analytical collapsing solution at $a=1/2$ as well as prove the existence of a leading order complex singularity for general values of $a$ in the analytical continuation of the solution from the real spatial coordinate into the complex plane. This singularity controls the leading order behaviour of the collapsing solution. For $a_c<a\leq1$, we find a blow-up solution in which the spatial extent of the blow-up region expands infinitely fast at the singularity time. For $a \gtrsim 1.3$, we find that the solution exists globally with exponential-like growth of the solution amplitude in time. We also consider the case of periodic boundary conditions. We identify collapsing solutions for $a<a_c$ which are similar to the real line case. For $a_c<a\le0.95$, we find new blow-up solutions which are neither expanding nor collapsing. For $ a\ge 1,$ we identify a global existence of solutions.

Finite-time singularity formation for an active scalar equation

We introduce an active scalar equation with a similar structure to the 3D Euler equations. Through studying the behavior of scale-invariant solutions, we show that compactly supported Lipschitz solutions belonging to can become singular in finite time. The interesting feature here is that we can achieve this in the absence of spatial boundaries.

The Singularity Formation on the Coupled Burgers–Constantin–Lax–Majda System with the Nonlocal Term

In this paper, we study the finite-time singularity formation on the coupled Burgers–Constantin–Lax–Majda system with the nonlocal term, which is one nonlinear nonlocal system of combining Burgers equations with Constantin–Lax–Majda equations. We discuss whether the finite-time blow-up singularity mechanism of the system depends upon the domination between the CLM type’s vortex-stretching term and the Burgers type’s convection term in some sense. We give two kinds of different finite-time blow-up results and prove the local smooth solution of the nonlocal system blows up in finite time for two classes of large initial data.

Formation of finite-time singularities for nonlinear hyperbolic systems with small initial disturbances

This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. We propose a universal test function method that works for many nonlinear hyperbolic systems arising from physical applications. We first present a simpler proof of the main result in the work of Sideris [Commun. Math. Phys. 101(4), 475–485 (1985)]: the global classical solution is non-existent for compressible Euler equations even for some small initial data. Then, we apply this approach to nonlinear magnetohydrodynamics in two space dimensions. Finally, we consider second order quasilinear hyperbolic systems with quadratic nonlinearity arising from elastodynamics of isotropic hyperelastic materials by ignoring the cubic and higher order terms. Under some restriction on the coefficients of the nonlinear terms that imply genuine nonlinearity, we show that the classical solutions to these equations can still blow up in finite time even if the initial data are small enough.

Finite-Time Singularity Formation for Strong Solutions to the Boussinesq System

As a follow up to our work [27], we give examples of finite-energy and Lipschitz continuous velocity field and density $$(u_0,\rho _0)$$ which are $$C^\infty $$-smooth away from the origin and belong to a natural local well-posedness class for the Boussinesq equation whose corresponding local solution becomes singular in finite time. That is, while the sup norm of the gradient of the velocity field and the density remain finite on the time interval $$t\in [0,1)$$, both quantities become infinite as $$t\rightarrow 1$$. The key is to use scale-invariant solutions similar to those introduced in [27]. The proof consists of three parts: local well-posedness for the Boussinesq equation in critical spaces, the analysis of certain special infinite-energy solutions belonging to those critical spaces, and finally a cut-off argument to ensure finiteness of energy. All of this is done on spatial domains $$\{(x_1,x_2): x_1 \ge \gamma |x_2|\}$$ for any $$\gamma > 0$$ so that we can get arbitrarily close to the half-space case. We show that the 2D Euler equation is globally well-posed in all of the situations we look at, so that the singularity is not coming from the domain or the lack of smoothness on the data but from the vorticity amplification due to the presence of a density gradient. It is conceivable that our methods can be adapted to produce finite-energy $$C^\infty $$ solutions on $${\mathbb {R}}^2_+$$ which become singular in finite time.

Up-wind difference approximation and singularity formation for a slow erosion model

We consider a model for a granular flow in the slow erosion limit introduced in [31]. We propose an up-wind numerical scheme for this problem and show that the approximate solutions generated by the scheme converge to the unique entropy solution. Numerical examples are also presented showing the reliability of the scheme. We study also the finite time singularity formation for the model with the singularity tracking method, and we characterize the singularities as shocks in the solution.

Integrability-preserving regularizations of Laplacian Growth

The Laplacian Growth (LG) model is known as a universality class of scale-free aggregation models in two dimensions, characterized by classical integrability and featuring finite-time boundary singularity formation. A discrete counterpart, Diffusion-Limited Aggregation (or DLA), has a similar local growth law, but significantly different global behavior. For both LG and DLA, a proper description for the scaling properties of long-time solutions is not available yet. In this note, we outline a possible approach towards finding the correct theory yielding a regularized LG and its relation to DLA.

Poiseuille Flow of Nematic Liquid Crystals via the Full Ericksen–Leslie Model

In this paper, we study the Cauchy problem of the Poiseuille flow of full Ericksen-Leslie model for nematic liquid crystals. The model is a coupled system of a parabolic equation for the velocity and a quasilinear wave equation for the director. For a particular choice of several physical parameter values, we construct solutions with smooth initial data and finite energy that produce, in finite time, cusp singularities - blowups of gradients. The formation of cusp singularity is due to local interactions of wave-like characteristics of solutions, which is different from the mechanism of finite time singularity formations for the parabolic Ericksen-Leslie system. The finite time singularity formation for the physical model might raise some concerns for purposes of applications. This is, however, resolved satisfactorily; more precisely, we are able to establish the global existence of weak solutions that are H\"older continuous and have bounded energy. One major contribution of this paper is our identification of the effect of the flux density of the velocity on the director and the reveal of a singularity cancellation - the flux density remains uniformly bounded while its two components approach infinity at formations of cusp singularities.

Finite-Time Singularity Formation for Strong Solutions to the Axi-symmetric 3D Euler Equations

For all $\epsilon>0$, we prove the existence of finite-energy strong solutions to the axi-symmetric $3D$ Euler equations on the domains $ \{(x,y,z)\in\mathbb{R}^3: (1+\epsilon|z|)^2\leq x^2+y^2\}$ which become singular in finite time. We further show that solutions with 0 swirl are necessarily globally regular. The proof of singularity formation relies on the use of approximate solutions at exactly the critical regularity level which satisfy a $1D$ system which has solutions which blow-up in finite time. The construction bears similarity to our previous result on the Boussinesq system \cite{EJB} though a number of modifications must be made due to anisotropy and since our domains are not scale-invariant. This seems to be the first construction of singularity formation for finite-energy strong solutions to the actual $3D$ Euler system.

On the Effects of Advection and Vortex Stretching

We prove finite-time singularity formation for De Gregorio's model of the three-dimensional vorticity equation in the class of $L^p\cap C^\alpha(\mathbb{R})$ vorticities for some $\alpha>0$ and $p<\infty$. We also prove finite-time singularity formation from smooth initial data for the Okamoto-Sakajo-Wunsch models in a new range of parameter values. As a consequence, we have finite-time singularity for certain infinite-energy solutions of the surface quasi-geostrophic equation which are $C^\alpha$-regular. One of the difficulties in the models we consider is that there are competing \emph{nonlocal} stabilizing effects (advection) and destabilizing effects (vortex stretching) which are of the same size in terms of scaling. Hence, it is difficult to establish the domination of one effect over the other without having strong control of the solution. We conjecture that strong solutions to the De Gregorio model exhibit the following behavior: for each $0<\alpha<1$ there exists an initial $\omega_0\in C^\alpha(\mathbb{R})$ which is compactly supported for which the solution becomes singular in finite-time; on the other hand, solutions to De Gregorio's equation are global whenever $\omega_0\in L^p\cap C^{1}(\mathbb{R})$ for some $p<\infty$. Such a dichotomy seems to be a genuinely non-linear effect which cannot be explained merely by scaling considerations since $C^\alpha$ spaces are scaling subcritical for each $\alpha>0$.

Finite-time singularity formation for C 1 solutions to the compressible Euler equations with time-dependent damping

In this paper, the initial-boundary value problem of the multidimensional compressible Euler equations with time-dependent damping in radial symmetry is considered. It is shown that finite-time singularity will be developed for the solutions of the compressible Euler equations with time-dependent damping coefficients if the initial value of a newly introduced functional, with a time-dependent parameter is sufficiently large. The blowup conditions imply that the initial kinetic energy of the fluid must not be less than a given constant.

Singularities of solutions to compressible Euler equations with damping

In this paper, we study the formation of finite-time singularities of smooth solutions to the compressible Euler equations with damping in arbitrary dimensions for isentropic, polytropic fluid flows. The radial component of initial momentum is assumed to be large enough and the smooth solutions are assumed to be decay at far fields, but it is not required that the initial velocity field has a compact support. It is shown that the smooth solutions to the Cauchy problem will break down in a finite time. Our result partially complements the work for the compactly supported initial velocity field (Sideris et al., 2003).