Self-similar Blow-up Research Articles

On Formation of a Locally Self-Similar Collapse in the Incompressible Euler Equations

The paper addresses the question of the existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the L p -condition for velocity or vorticity and for a range of scaling exponents. In particular, in N dimensions if in self-similar variables $${u \in L^p}$$ and $${u \sim \frac{1}{t^{\alpha/(1+\alpha)}}}$$ , then the blow-up does not occur, provided $${\alpha > N/2}$$ or $${-1 < \alpha \leq N\,/p}$$ . This includes the L 3 case natural for the Navier–Stokes equations. For $${\alpha = N\,/2}$$ we exclude profiles with asymptotic power bounds of the form $${ |y|^{-N-1+\delta} \lesssim |u(y)| \lesssim |y|^{1-\delta}}$$ . Solutions homogeneous near infinity are eliminated, as well, except when homogeneity is scaling invariant.

Self-similar solutions and blow-up phenomena for a two-component shallow water system

In this article, we consider a two-component nonlinear shallow water system, which includes the famous 2-component Camassa-Holm and Degasperis-Procesi equations as special cases. The local well-posedess for this equations is established. Some sufficient conditions for blow-up of the solutions in finite time are given. Moreover, by separation method, the self-similar solutions for the nonlinear shallow water equations are obtained, and which local or global behavior can be determined by the corresponding Emden equation.

Singularity formation in the Smirnov–Chukbar–Zaburdaev equation for the deformation of vortex lattices

An asymptotic equation arising in the theory of vortex lattices (Smirnov and Chukbar 2001 Sov. Phys.—JETP 93 126–35; Zaburdaev et al 2004 Plasma Phys. Rep. 30 214–17) is investigated numerically and mathematically, centring on its singularity formation. A class of self-similar blow-up solutions is identified, all of which have infinite total energy. Detailed numerical results are presented for a number of initial conditions, both for inviscid and dissipative cases. The ill-posed nature of the solutions is revealed by solving a linearized equation. For fully nonlinear problems, we have found finite-time blow-up solutions with finite total energy, in which singularities are hiding in the third-order spatial derivatives or higher. Mathematically, the local existence of solutions is proved with hyper-viscous (double-Laplacian) dissipativity. Finally, remarks are made in connection with the Tkachenko-type lattice.

A study of energy concentration and drain in incompressible fluids

In this paper, we examine two opposite scenarios of energy behaviour for solutions of the Euler equation. We show that if u is a regular solution on a time interval [0, T) and if u ∈ LrL∞ for some , where N is the dimension of the fluid, then the energy at the time T cannot concentrate on a set of Hausdorff dimension smaller than . The same holds for solutions of the three-dimensional Navier-Stokes equation in the range 5/3 < r < 7/4. Oppositely, if the energy vanishes on a subregion of a fluid domain, it must vanish faster than (T − t)1−δ, for any δ > 0. The results are applied to find new exclusions of locally self-similar blow-up in cases not covered previously in the literature.

Numerical Study on the Asymptotic Equation for the Deformation of Vortex Lattice

We study a coarse-grained asymptotic equation which describes deformation of vortex lattices [Smirnov & Chukbar 2001]. It reads φt=φxxφyy–φ2xy, where φ denotes displacement of vortex locations. This equation is valid for a lattice with short-ranged interaction, e.g. geostrophic vortices with a screened potential. New self-similar blow-up solutions with infinite total energy are found. We ask whether or not finite-time blow-up can take place developing from smooth initial data with finite energy. The numerical simulations show finite-time blow-up in such a way that φ∈H3 but φ∉H4.

Numerical investigation of a new class of waves in an open nonlinear heat-conducting medium

AbstractThe paper contributes to the problem of finding all possible structures and waves, which may arise and preserve themselves in the open nonlinear medium, described by the mathematical model of heat structures. A new class of self-similar blow-up solutions of this model is constructed numerically and their stability is investigated. An effective and reliable numerical approach is developed and implemented for solving the nonlinear elliptic self-similar problem and the parabolic problem. This approach is consistent with the peculiarities of the problems — multiple solutions of the elliptic problem and blow-up solutions of the parabolic one.

Stability and clustering of self-similar solutions of aggregation equations

In this paper we consider the linear stability of a family of exact collapsing similarity solutions to the aggregation equation ρt = ∇ · (ρ∇K * ρ) in Rd, d ⩾ 2, where K(r) = rγ/γ with γ > 2. It was previously observed [Y. Huang and A. L. Bertozzi, “Self-similar blowup solutions to an aggregation equation in Rn,” J. SIAM Appl. Math. 70, 2582–2603 (2010)]10.1137/090774495 that radially symmetric solutions are attracted to a self-similar collapsing shell profile in infinite time for γ > 2. In this paper we compute the stability of the similarity solution and show that the collapsing shell solution is stable for 2 < γ < 4. For γ > 4, we show that the shell solution is always unstable and destabilizes into clusters that form a simplex which we observe to be the long time attractor. We then classify the stability of these simplex solutions and prove that two-dimensional (in-)stability implies n-dimensional (in-)stability.

Supercritical Elliptic Equations

Abstract For an elliptic model equation with supercritical power non-linearity we give a complete description of radial solutions and discuss self-similar blow-up solutions.

Anomalous exponents of self-similar blow-up solutions to an aggregation equation in odd dimensions

We calculate the scaling behavior of the second-kind self-similar blow-up solution of an aggregation equation in odd dimensions. This solution describes the radially symmetric finite-time blowup phenomena and has been observed in numerical simulations of the dynamic problem. The nonlocal equation for the self-similar profile is transformed into a system of ODEs that is solved using a shooting method. The anomalous exponents are then retrieved from this transformed system.

Stable Ground States and Self-Similar Blow-Up Solutions for the Gravitational Vlasov--Manev System

In this work, we study the orbital stability of steady states and the existence of self-similar blow-up solutions to the so-called Vlasov--Manev system. This system is a kinetic model which has a similar Vlasov structure to the classical Vlasov--Poisson system but is coupled to a potential in $-1/r- 1/r^2$ (Manev potential) instead of the usual gravitational potential in $-1/r$, and in particular the potential field does not satisfy a Poisson equation but a fractional Laplacian equation. We first prove the orbital stability of the ground state--type solutions which are constructed as minimizers of the Hamiltonian, following the classical strategy: compactness of the minimizing sequences and the rigidity of the flow. However, in driving this analysis, there are two mathematical obstacles: the first one is related to the possible blow-up of solutions to the Vlasov--Manev system, which we overcome by imposing a subcritical condition on the constraints of the variational problem. The second difficulty (and the most important) is related to the nature of the Euler--Lagrange equations (fractional Laplacian equations) to which classical results for the Poisson equation do not extend. We overcome this difficulty by proving the uniqueness of the minimizer under equimeasurability constraints, using only the regularity of the potential and not the fractional Laplacian Euler--Lagrange equations itself. In the second part of this work, we prove the existence of exact self-similar blow-up solutions to the Vlasov--Manev equation, with initial data arbitrarily close to ground states. This construction is based on a suitable variational problem with equimeasurability constraint.

On self-similar blow-up in evolution equations of Monge-Ampere type

We use techniques from reaction-diffusion theory to study the blow-up and existence of solutions of the parabolic Monge–Ampere (M-A) equation with power source, with the following basic 2D model $$u_{t}=-|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where in two-dimensions $|D^{2}u|=u_{xx}u_{yy}-(v_{xy})^{2}$ and p > 1 is a fixed exponent. For a class of ‘dominated concave’ and compactly supported radial initial data $u_{0}(x)\geqslant0$ , the Cauchy problem is shown to be locally well posed and to exhibit finite time blow-up that is described by similarity solutions. For p ∈ (1, 2], similarity solutions, containing domains of concavity and convexity, are shown to be compactly supported and correspond to surfaces with flat sides that persist until the blow-up time. The case p > 2 leads to single-point blow-up. Numerical computations of blow-up solutions without radial symmetry are also presented. The parabolic analogy of the parabolic M-A equation in 3D for which $|D^{2}u|$ is a cubic operator is $$u_{t}=|D^{2}u|+|u|^{p-1}u\quad \mathrm{in}\quad \mathbb{R}^{3}\times \mathbb{R}_{+},$$ and is shown to admit a wider set of (oscillatory) self-similar blow-up patterns. Regional self-similarblow-up in a cubic radial model related to the fourth-order M-A equation $$u_{t}=-|D^{4}u|+u^{3}\quad \mathrm{in}\quad \mathbb{R}^{2}\times \mathbb{R}_{+},$$ where the cubic operator $|D^{4}u|$ is the catalecticant 3 × 3 determinant is also briefly discussed.

Blow-up profiles of solutions for the exponential reaction-diffusion equation

We consider the blow-up of solutions for a semilinear reaction diffusion equation with exponential reaction term. It is know that certain solutions that can be continued beyond the blow-up time possess a nonconstant selfsimilar blow-up profile. Our aim is to find the final time blow-up profile for such solutions. The proof is based on general ideas using semigroup estimates. The same approach works also for the power nonlinearity.

Non-existence of radial backward self-similar blow-up solutions with sign change

We consider a Cauchy problem for a semilinear heat equationwith p &gt; 1. If u(x, t) = (T − t)−1/(p−1)ϕ((T − t)−1/2x) for x ∈ ℝN and t ∈ [0, T),where ϕ ∈ L∞(ℝN) is a solution not identically equal to zero ofthen u is called a backward self-similar solution blowing up at t = T. We show that, for all p &gt; 1, there exists no radial sign-changing solution of (E) which belongs to L∞(ℝN). This implies the non-existence of radial backward self-similar solution with sign change blowing up in finite time.

Erratum: “Self-similar blowup solutions to the 2-component Camassa–Holm equations” [J. Math. Phys. 51, 093524 (2010)

Changes have been made to this article. See the full text for a description of the changes.

Analytical solutions to the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure

We study some particular solutions to the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure, in radial symmetry. With an extension of the previous known blow-up solutions for the Euler-Poisson equations with pressureless Navier-Stokes-Poisson density-dependent viscosity, we constructed the corresponding self-similar blow-up solutions for the Navier-Stokes-Poisson equations with density-dependent viscosity and with pressure. Our solutions can provide concrete examples for testing the validation and stabilities of numerical methods for the systems.

On stable self-similar blowup for equivariant wave maps

We consider corotational wave maps from (3 + 1) Minkowski space into the 3-sphere. This is an energy supercritical model that is known to exhibit finite-time blowup via self-similar solutions. The ground state self-similar solution f0 is known in closed form, and according to numerics, it describes the generic blowup behavior of the system. We prove that the blowup via f0 is stable under the assumption that f0 does not have unstable modes. This condition is equivalent to a spectral assumption for a linear second order ordinary differential operator. In other words, we reduce the problem of stable blowup to a linear ODE spectral problem. Although we are unable at the moment to verify the mode stability of f0 rigorously, it is known that possible unstable eigenvalues are confined to a certain compact region in the complex plane. As a consequence, highly reliable numerical techniques can be applied and all available results strongly suggest the nonexistence of unstable modes, i.e., the assumed mode stability of f0. © 2011 Wiley Periodicals, Inc.

Self-similar blowup solutions to the 2-component Degasperis–Procesi shallow water system

In this article, we study the self-similar solutions of the 2-component Degasperis–Procesi water system: (1) ρ t + k 2 u ρ x + ( k 1 + k 2 ) ρ u x = 0 , u t - u xxt + 4 uu x - 3 u x u xx - uu xxx + k 3 ρ ρ x = 0 . By the separation method, we can obtain a class of self-similar solutions, (2) ρ ( t , x ) = max f ( η ) a ( 4 t ) ( k 1 + k 2 ) / 4 , 0 , u ( t , x ) = a ˙ ( 4 t ) a ( 4 t ) x , a ¨ ( s ) - ξ 4 a ( s ) κ = 0 , a ( 0 ) = a 0 ≠ 0 , a ˙ ( 0 ) = a 1 , f ( η ) = k 3 ξ - η 2 k 3 ξ + α k 3 ξ 2 , where η = x a ( s ) 1 / 4 with s = 4 t ; κ = k 1 2 + k 2 - 1 , α ⩾ 0 , ξ < 0 , a 0 and a 1 are constants, which the local or global behavior can be determined by the corresponding Emden equation a( s), (2) 2. The results are similar to the ones obtained for the 2-component Camassa–Holm equations. Our C 0 solutions (2) can capture the evolution of breaking waves of that system. The constructed solutions could be applied to test the numerical computation for the system. In the last section, with the characteristic line method, blowup phenomenon for k 3 ⩾ 0 is also studied.

Self-similar blowup solutions to the 2-component Camassa–Holm equations

In this article, we study the self-similar solutions of the 2-component Camassa–Holm equations ρt+uρx+ρux=0, mt+2uxm+umx+σρρx=0, with m=u−α2uxx. By the separation method, we can obtain a class of blowup or global solutions for σ=1 or −1. In particular, for the integrable system with σ=1, we have the global solutions, ρ(t,x)=f(η)/a(3t)1/3 for η2&amp;lt;α2/ξ, ρ(t,x)=0 for η2≥α2/ξ, u(t,x)=ȧ(3t)/a(3t)x, ä(s)−ξ/3a(s)1/3=0,a(0)=a0&amp;gt;0,ȧ(0)=a1, f(η)=ξ−1/ξη2+(α/ξ)2, where η=xa(s)1/3 with s=3t; ξ&amp;gt;0 and α≥0 are arbitrary constants. Our analytical solutions could provide concrete examples for testing the validation and stabilities of numerical methods for the systems.

Stable Self-Similar Blow-Up Dynamics for Slightly L 2 Super-Critical Nls Equations

We consider the focusing nonlinear Schrodinger equations i∂ t u + Δu + u|u|p−1 = 0 in dimension 1 ≤ N ≤ 5 and for slightly L 2 super-critical nonlinearities \({{p_{c} < p < (1 +\varepsilon)p_{c}}}\) with \({{p_{c} = 1+\frac{4}{N}}}\) and \({{0 < \varepsilon \ll 1}}\) . We prove the existence and stability in the energy space H 1 of a self-similar finite-time blow-up dynamics and provide a qualitative description of the singularity formation near the blow-up time.

On backward self-similar blow-up solutions to a supercritical semilinear heat equation

We are concerned with a Cauchy problem for the semilinear heat equationthen u is called a backward self-similar solution blowing up at t = T. Let pS and pL be the Sobolev and the Lepin exponents, respectively. It was shown by Mizoguchi (J. Funct. Analysis257 (2009), 2911–2937) that k ≡ (p − 1)−1/(p−1) is a unique regular radial solution of (P) if p &gt; pL. We prove that it remains valid for p = pL. We also show the uniqueness of singular radial solution of (P) for p &gt; pS. These imply that the structure of radial backward self-similar blow-up solutions is quite simple for p ≥ pL.