On the Formation of Singularities in the Critical O(3) Sigma-Model
Igor Rodnianski, Jacob Sterbenz
备注：
51 pages added remarks and references corrected computation of the constant in the Appendix A, leading to the stable blow up with the rate bounded from above by a log-modified self-similar asymptotic and from below by a self-similar rate
We study the phenomena of energy concentration for the critical O(3) sigma model, also known as the wave map flow from R^{2+1} Minkowski space into the sphere S^2. We establish rigorously and constructively existence of a set of smooth initial data resulting in a dynamic finite time formation of singularities. The construction and analysis is done in the context of the k-equivariant symmetry reduction, and we restrict to maps with homotopy class k>3. The concentration mechanism we uncover is essentially due to a resonant self-focusing (shrinking) of a corresponding harmonic map. We show that the phenomenon is generic (e.g. in certain Sobolev spaces) and persists under small perturbations of initial data, while the resulting blowup is bounded by a log-modified self-similar asymptotic.
In this paper we report on numerical studies of the Cauchy problem for equivariant wave maps from 2+1 dimensional Minkowski spacetime into the two-sphere. Our results provide strong evidence for the conjecture that large energy initial data develop singularities in finite time and that singularity formation has the universal form of adiabatic shrinking of the degree-one harmonic map from $\mathbb{R}^2$ into $S^2$.
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.
For the ZND model, we show also singularity formation on the downstream side for arbitrary exponentially-growing weighted norms. For the Majda model on the other hand, we establish for appropriate such weighted norms a set of energy estimates implying not only non-formation of singularities near waves of arbitrary amplitude but also full asymptotic orbital stability for small-amplitude ones.
This paper is concerned with a class of multi-dimensional transport equations with nonlocal velocity. It is shown that the local smooth solution cannot exist globally in time via the De Giorgi iteration technique.
This article concerns the formation of finite-time singularities in solutions to quasilinear hyperbolic systems with small initial data. By constructing a special test function, we first present a simpler proof of the main result in Sideris' "Formation of singularities in three-dimensional compressible fluids": the global classical solution is non-existent for compressible Euler equation even for some small initial data. Then we apply this approach to nonlinear elastodynamics and magnetohydrodynamics, showing that the classical solutions to these equations can still blow up in finite time even…
In this paper, we constuct the multi-point blowup solutions of self-similar type for the inviscid Burgers equation. The shape and blowup dynamics are precisely described. Moreover, the solutions we construct are stable under small perturbations on initial data restricted in a compact set.
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 study the formation of generic singularities of mean curvature flow by combining the different approaches, specifically the methods in studying blowup of nonlinear heat equations, the techniques used by the author and the collaborators for mean curvature flow, and these invented by Colding and Minicozzi. We study the solution in a neighborhood of the blowup point, in some generic regimes, we find the key parameters take favorable signs and have sharp decay rates. We provide the remainder estimates in different norms.
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 …
We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + λc(u)c'(u)( u_x)^2$ with the real parameter $λ$. In previous works, it was reported that there exist finite time blow-up solutions with $λ=1$ and $2$. However, the construction of a blow-up solution depends on the symmetric structure of the equation (e.g., the energy conservation law). In the present paper, we extend the blow-up result with $λ=1$ to the case with $λ\in (0,1]$ by using a new $L^{2/λ}$ estimate. Moreover, some properties for the blow-up solution including the…
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 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 one of the generic regimes of formation of singularities. We obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex. This confirms a conjecture by Ilmanen. And we find that the singularity is isolated from the other ones.
