Explicit Self-similar Solutions Research Articles

A fast regularisation of a Newtonian vortex equation

We consider equations of the form $ u_t = \nabla \cdot ( \gamma(u) \nabla \mathrm{N}(u))$, where $\mathrm{N}$ is the Newtonian potential (inverse of the Laplacian) posed in the whole space $\mathbb R^d$, and $\gamma(u)$ is the mobility. For linear mobility, $\gamma(u)=u$, the equation and some variations have been proposed as a model for superconductivity or superfluidity. In that case the theory leads to uniqueness of bounded weak solutions having the property of compact space support, and in particular there is a special solution in the form of a disk vortex of constant intensity in space $u=c_1t^{-1}$ supported in a ball that spreads in time like $c_2t^{1/d}$, thus showing a discontinuous leading front. In this paper we propose the model with sublinear mobility $\gamma(u)=u^\alpha$, with $0<\alpha<1$, and prove that nonnegative solutions recover positivity everywhere, and moreover display a fat tail at infinity. The model acts in many ways as a regularization of the previous one. In particular, we find that the equivalent of the previous vortex is an explicit self-similar solution decaying in time like $u=O(t^{-1/\alpha})$ with a space tail with size $u=O(|x|^{- d/(1-\alpha)})$. We restrict the analysis to radial solutions and construct solutions by the method of characteristics. We introduce the mass function, which solves an unusual variation of Burger's equation, and plays an important role in the analysis. We show well-posedness in the sense of viscosity solutions. We also construct numerical finite-difference convergent schemes.

Vortex formation for a non-local interaction model with Newtonian repulsion and superlinear mobility

Abstract We consider density solutions for gradient flow equations of the form u t = ∇ · (γ(u)∇ N(u)), where N is the Newtonian repulsive potential in the whole space ℝ d with the nonlinear convex mobility γ(u) = u α , and α &gt; 1. We show that solutions corresponding to compactly supported initial data remain compactly supported for all times leading to moving free boundaries as in the linear mobility case γ(u) = u. For linear mobility it was shown that there is a special solution in the form of a disk vortex of constant intensity in space u = c 1 t −1 supported in a ball that spreads in time like c 2 t 1/d , thus showing a discontinuous leading front or shock. Our present results are in sharp contrast with the case of concave mobilities of the form γ(u) = u α , with 0 &lt; α &lt; 1 studied in [10]. There, we developed a well-posedness theory of viscosity solutions that are positive everywhere and moreover display a fat tail at infinity. Here, we also develop a well-posedness theory of viscosity solutions that in the radial case leads to a very detailed analysis allowing us to show a waiting time phenomena. This is a typical behaviour for nonlinear degenerate diffusion equations such as the porous medium equation. We will also construct explicit self-similar solutions exhibiting similar vortex-like behaviour characterizing the long time asymptotics of general radial solutions under certain assumptions. Convergent numerical schemes based on the viscosity solution theory are proposed analysing their rate of convergence. We complement our analytical results with numerical simulations illustrating the proven results and showcasing some open problems.

The eigenvalue problem of the general Einstein–Weyl metric equation and exact self-similar and multi-traveling waves solutions

The integrability of the general Einstein–Weyl Metric equation is studied . Here, the approach proposed is based on using the extended and generalized unified methods. These methods inspect the integrability conditions, whenever exist. The exact solutions are thus obtained. They provide the existence and uniqueness of solution for initial value problem where initial value is taken from the exact solutions obtained. Similarity variables are introduced, and the extended unified method is applied to find a class of explicit self-similar wave solutions to the GEWM equation. Numerical evaluations of the solutions are done via symbolic computation. These results show periodic solitons and multi-lumps. It is found that the generalized heavenly equation admits a class of infinite solutions, among them the chaotic ones. The generalized unified method is used to find multi-traveling solitary waves solutions. We think that the presented methods generalize the known ones in the literature. This will be illustrated later on.

Dynamical behavior of the strong dispersive nonlinear wave equation

In this paper we consider the dynamical behavior of solutions near explicit self-similar solutions for a strong dispersive nonlinear wave equation. First we construct explicit self-similar solutions, then we investigate dynamical behavior of the solutions near to the self-similar solutions.

Nonlinear stability of explicit self-similar solutions for the timelike extremal hypersurfaces in $${\mathbb {R}}^{1+3}$$

This paper is devoted to the study of the singularity phenomenon of timelike extremal hypersurfaces in Minkowski spacetime $${\mathbb {R}}^{1+3}$$ . We find that there are two explicit lightlike self-similar solutions to a graph representation of timelike extremal hypersurfaces in Minkowski spacetime $${\mathbb {R}}^{1+3}$$ , the geometry of them are two spheres. The linear mode unstable of those lightlike self-similar solutions for the radially symmetric membranes equation is given. After that, we show those self-similar solutions of the radially symmetric membranes equation are nonlinearly stable inside a strictly proper subset of the backward lightcone. This means that the dynamical behavior of those two spheres is as attractors. Meanwhile, we overcome the double roots case (the theorem of Poincare can’t be used) in solving the difference equation by construction of a Newton’s polygon when we carry out the analysis of spectrum for the linear operator.

Explicit self-similar solutions for a class of zero mean curvature equation and minimal surface equation

In this paper, we consider the existence of self-similar solution for a class of zero mean curvature equations including the Born–Infeld equation, the membrane equation and maximal surface equation. By Calabi’s correspondence, this also gives a family of explicit self-similar solutions for the minimal surface equation. Those models arise in string theory and geometric minimal surfaces theory. Moreover, we construct a family of instanton metric obtained from new exact singular solutions for minimal surfaces by noticing the correspondence between minimal surfaces in the three dimensional Euclidean space and gravitational instantons possessing two killing vectors.

Dynamical behavior near explicit self-similar blow–up solutions for the Born–Infeld equation

This paper studies the dynamical behavior near a new family of explicit self-similar solutions for the one-dimensional Born–Infeld equation. This quasilinear scalar field equation arises from nonlinear electromagnetism, as well as branes in string theory and minimal surfaces in Minkowski spacetimes. We show that both this model and the linear wave equation admit the same family of explicit timelike self-similar blow-up solutions; meanwhile, Lyapunov nonlinear stability of those self-similar blow-up solutions is given inside a strictly proper subset of the backward light cone.

Stable blowup for the supercritical Yang–Mills heat flow

In this paper, we consider the heat flow for Yang–Mills connections on $\mathbb{R}^5 \times SO(5)$. In the $SO(5)$-equivariant setting, the Yang–Mills heat equation reduces to a single semilinear reaction-diffusion equation for which an explicit self-similar blowup solution was found by Weinkove [“Singularity formation in the Yang-Mills flow”, Calc. Var. Partial Differential Equations, 19(2):211–220, 2004]. We prove the nonlinear asymptotic stability of this solution under small perturbations. In particular, we show that there exists an open set of initial conditions in a suitable topology such that the corresponding solutions blow up in finite time and converge to a non-trivial self-similar blowup profile on an unbounded domain. Convergence is obtained in suitable Sobolev norms and in $L^{\infty}$.

Asymptotic stability and instability of explicit self-similar waves for a class of nonlinear shallow water equations

This paper considers the dynamical behavior of solutions near explicit self-similar singularity for a class of nonlinear shallow water models including the Dullin–Gottwald–Holm equation, the Korteweg–de Vires equation, the Camassa–Holm equation, the dispersive rod equation and the Benjamin–Bona–Mahony equation. We first show there are explicit self-similar solutions for those five nonlinear shallow water equations, then we find that those explicit self-similar solutions for the Dullin–Gottwald–Holm equation, the Camassa–Holm equation and the dispersive rod equation are asymptotic stable, but for the Korteweg–de Vries equation and the Benjamin–Bona–Mahony equation being unstable.

On the explicit self-similar motion of the relativistic Chaplygin gas

In this letter, the self-similar singularity of the isentropic relativistic Chaplygin gas is considered. It is shown that this model under consideration admits explicit self-similar solutions in all spacial dimension . This reveals the mechanism of the formation of the singularity for this fluid.

Explicitly self-similar solutions for the Euler/Navier–Stokes–Korteweg equations in RN

Abstract In this paper, we present a unified formulae for explicit self-similar solutions of the Euler/Navier–Stokes–Korteweg equations arising in the modeling of capillary fluids. The technique used here is to reduce Euler/Navier–Stokes–Korteweg equations into a series of solvable ordinary differential equations by making use of multi-dimensional self-similar ansatz and variable separation method.

Mode Stability of Self-Similar Wave Maps in Higher Dimensions

We consider co-rotational wave maps from Minkowski space in $d+1$ dimensions to the $d$-sphere. Recently, Bizo\'n and Biernat found explicit self-similar solutions for each dimension $d\geq 4$. We give a rigorous proof for the mode stability of these self-similar wave maps.

On the Stability of Self-Similar Solutions to Nonlinear Wave Equations

We consider an explicit self-similar solution to an energy-supercritical Yang-Mills equation and prove its mode stability. Based on earlier work by one of the authors, we obtain a fully rigorous proof of the nonlinear stability of the self-similar blowup profile. This is a large-data result for a supercritical wave equation. Our method is broadly applicable and provides a general approach to stability problems related to self-similar solutions of nonlinear wave equations.

Lie symmetry analysis, optimal system, exact solutions and conservation laws of a class of high-order nonlinear wave equations

The symmetries, conservation laws and exact solutions to the nonlinear partial differential equations play a significant role in nonlinear science and mathematical physics. Symmetry is derived from physics, and it is a mathematical description for invariance. Symmetry group theory plays an important role in constructing explicit solutions, whether the equations are integrable or not. By using the symmetry method, an original nonlinear system can be reduced to a system with fewer independent variables through any given subgroup. But, since there are almost always an infinite number of such subgroups, it is usually not feasible to list all possible group invariant solutions to the system. It is anticipated to find all those equivalent group invariant solutions, that is to say, to construct the one-dimensional optimal system for the Lie algebra. Construction of explicit forms of conservation laws is meaningful, as they are used for developing the appropriate numerical methods and for making mathematical analyses, in particular, of existence, uniqueness and stability. In addition, the existence of a large number of conservation laws of a partial differential equation (system) is a strong indication of its integrability. The similarity solutions are of importance for investigating the long-time behavior, blow-up profile and asymptotic phenomena of a non-linear system. For instance, in some circumstance, the asymptotic behaviors of finite-mass solutions of non-linear diffusion equation with non-linear source term are described by an explicit self-similar solution, etc. However, how to tackle these matters is a complicated problem that challenges researchers to be solved. In this paper, by using the symmetry method, we obtain the symmetry reduction, optimal systems, and many new exact group invariant solution of a fifth-order nonlinear wave equation. By Lie symmetry analysis method, the point symmetries and an optimal system of the equation are obtained. The exact power series solutions to the equation are provided by the power series method, such solutions can be used for numerical computations in both theory and physical applications conveniently. Finally, a lot of conservation laws of the fifth-order nonlinear wave equation are presented by using the adjoint equation and symmetries of the equation.

Explicit Barenblatt profiles for fractional porous medium equations

Several one-parameter families of explicit self-similar solutions are constructed for the porous medium equations (PMEs) with fractional operators. The corresponding self-similar profiles, also called Barenblatt profiles, have the same forms as those of the classic PMEs. These new exact solutions complement current theoretical analysis of the underlying equations and are expected to provide insights for further quantitative investigations.

Self-similar asymptotics of solutions to heat equation with inverse square potential

We show that the large-time behavior of solutions to the Cauchy problem for the linear heat equation with the inverse square potential is described by explicit self-similar solutions.

Barenblatt profiles for a nonlocal porous medium equation

We study a generalization of the porous medium equation involving nonlocal terms. More precisely, explicit self-similar solutions with compact support generalizing the Barenblatt solutions are constructed. We also present a formal argument to get the L p decay of weak solutions of the corresponding Cauchy problem.

The diffusive limit of Carleman-type models in the range of very fast diffusion equations

We improve the existing results on the limiting behavior of the Cauchy problem for a class of Carleman-like models with power-type interaction rate in the diffusive scaling with data in the spaces L p , 1 ≤ p ≤ ∞. The convergence result, which has been carefully established before for exponents of the interaction rate α ≤ 1, is extended here to the range of exponents 1 < α < 4/3. In addition, we discuss the problem of establishing a good theory in the still remaining range \({\alpha \in [4/3,2)}\) , by introducing a modified kinetic system which admits an explicit self-similar solution. The analysis of this solution clarifies the role of the exponent \({\bar\alpha = 4/3}\) .

Blow-up and Global Smooth Solutions for Incompressible Three-Dimensional Navier–Stokes Equations

We present some explicit self-similar blow-up solutions and some other solutions of the incompressible three-dimensional Navier–Stokes equations. These solutions indicate that in C∞ the solution of Navier–Stokes equations does not always tend to a solution of Euler equations.

Intrinsic Harnack estimates for nonnegative local solutions of degenerate parabolic equations

We establish the intrinsic Harnack inequality for nonnegative solutions of the parabolic p p -Laplacian equation by a proof that uses neither the comparison principle nor explicit self-similar solutions. The significance is that the proof applies to quasilinear p p -Laplacian-type equations, thereby solving a long-standing problem in the theory of degenerate parabolic equations.