Formation Of Singularities In Solutions Research Articles

Formation of singularities in solutions to nonlinear hyperbolic systems with general sources

We consider nonlinear hyperbolic systems with a general source and prove that for appropriately chosen smooth initial data the lifespan of the associated C1-solution u cannot be infinite. We employ ideas of John (1974) and Hörmander (1987) to show that the derivative ux of u becomes unbounded in finite time.

Lifespan estimates of solutions to quasilinear wave equations with damping and negative mass term

<p style='text-indent:20px;'>The main goal of this work is to investigate formation of singularities for solutions to the quasilinear wave equations with damping terms, negative mass terms and divergence form nonlinearities in the critical and sub-critical cases. Upper bound lifespan estimates of solutions are derived by applying the rescaled test function method and iteration technique. The results are the same as corresponding wave equation without damping term and mass term. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems <inline-formula><tex-math id="M2">\begin{document}$ 1.1-1.4 $\end{document}</tex-math></inline-formula> are new. Moreover, the changing trends of semilinear wave equations are illustrated through numerical simulation.</p>

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.

Global Mild Solutions of the Landau and Non‐Cutoff Boltzmann Equations

This paper proves the existence of small‐amplitude global‐in‐time unique mild solutions to both the Landau equation including the Coulomb potential and the Boltzmann equation without angular cutoff. Since the well‐known works [45] and [3, 43] on the construction of classical solutions in smooth Sobolev spaces which in particular are regular in the spatial variables, it still remains an open problem to obtain global solutions in an framework, similar to that in [49], for the Boltzmann equation with the cutoff assumption in general bounded domains. One main difficulty arises from the interaction between the transport operator and the velocity‐diffusion‐type collision operator in the non‐cutoff Boltzmann and Landau equations; another major difficulty is the potential formation of singularities for solutions to the boundary value problem.In the present work we introduce a new function space with low regularity in the spatial variable to treat the problem in cases when the spatial domain is either a torus or a finite channel with boundary. For the latter case, either the inflow boundary condition or the specular reflection boundary condition is considered. An important property of the function space is that the norm, in velocity and time, of the distribution function is in the Wiener algebra A(Ω) in the spatial variables. Besides the construction of global solutions in these function spaces, we additionally study the large‐time behavior of solutions for both hard and soft potentials, and we further justify the property of propagation of regularity of solutions in the spatial variables. © 2019 Wiley Periodicals, Inc.

Improved blowup theorems for the Euler-Poisson equations with attractive forces

Our discussion here mainly focuses on the formation of singularities for solutions to the N-dimensional Euler-Poisson equations with attractive forces, in radial symmetry. Motivated by the integration method of Yuen, we prove two blow-up results under the conditions that the solutions have compact radius R(t) or have no compact support restriction, which generalize the ones Yuen obtained in 2011 [M. W. Yuen, “Blowup for the C1 solution of the Euler-Poisson equations of gaseous stars in RN,” J. Math. Anal. Appl. 383, 627-633 (2011)].

Nonlinear Quasi-Static Surface Plasmons

We derive an asymptotic equation for nonlinear quasi-static surface plasmons propagating on a planar interface between isotropic media. The plasmons are nondispersive with a constant linearized frequency that is independent of their wavenumber. The spatial profile of a weakly nonlinear plasmon satisfies a nonlocal, cubically nonlinear evolution equation that couples its left-moving and right-moving Fourier components. We prove short-time existence of smooth solutions of the asymptotic equation and describe its Hamiltonian structure. We also prove global existence of weak solutions of a unidirectional reduction of the asymptotic equation. Numerical solutions show that nonlinear effects can lead to the strong spatial focusing of plasmons. Solutions of the unidirectional equation appear to remain smooth when they focus, but it is unclear whether focusing can lead to singularity formation in solutions of the bidirectional equation.

Late formation of singularities in solutions to the Navier–Stokes equations

We study how late the first singularity can form in solutions of the Navier–Stokes equations and estimate the size of the potentially dangerous time interval, where it can possibly appear. According to Leray (1934), its size is estimated as when normalized by the local existence time, for a general blowup of the enstrophy at Here is the Reynolds number defined with initial energy E(0) and enstrophy Q(0). Applying dynamic scaling transformations, we give a general estimate parameterized by the behaviour of the scaled enstrophy. In particular, we show that the size is reduced to for a class of type II blow up of the form On the basis on the structure theorem of Leray (1934), we note that the self-similar and asymptotically self-similar blowup are ruled out for any singularities of weak solutions. We also apply the dynamic scaling to weak solutions with more than one singularities to show that the size is estimated as for the type II blowup above.

Formation of singularities in solutions to the compressible radiation hydrodynamics equations with vacuum

We study the Cauchy problem for multi-dimensional compressible radiation hydrodynamics equations with vacuum. First, we present some sufficient conditions on the blow-up of smooth solutions in multi-dimensional space. Then, we obtain the invariance of the support of density for the smooth solutions with compactly supported initial mass density by the property of the system under the vacuum state. Based on the above-mentioned results, we prove that we cannot get a global classical solution, no matter how small the initial data are, as long as the initial mass density is of compact support. Finally, we will see that some of the results that we obtained are still valid for the isentropic flows with degenerate viscosity coefficients as well as for one-dimensional case.

Numerical study of shock formation in the dispersionless Kadomtsev–Petviashvili equation and dispersive regularizations

The formation of singularities in solutions to the dispersionless Kadomtsev–Petviashvili (dKP) equation is studied numerically for different classes of initial data. The asymptotic behavior of the Fourier coefficients is used to quantitatively identify the critical time and location and the type of the singularity. The approach is first tested in detail in 1+1 dimensions for the known case of the Hopf equation, where it is shown that the break-up of the solution can be identified with prescribed accuracy. For dissipative regularizations of this shock formation as the Burgers equation and for dispersive regularizations as the Korteweg–de Vries equation, the Fourier coefficients indicate as expected global regularity of the solutions. The Kadomtsev–Petviashvili (KP) equation can be seen as a dispersive regularization of the dKP equation. The behavior of KP solutions for small dispersion parameter ϵ≪1 near a break-up of corresponding dKP solutions is studied. It is found that the difference between KP and dKP solutions for the same initial data at the critical point scales roughly as ϵ2/7 as for the Korteweg–de Vries equation.

Formation of singularities in solutions to ideal hydrodynamics of freely cooling inelastic gases

We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a finite time in any space dimension n. Furthermore, in the one-dimensional case we introduce a solution depending only on the spatial coordinate outside of a ball containing the origin and prove that this solution under rather general assumptions on initial data cannot be global in time too. Then we construct an exact axially symmetric solution with separable time and space variables having a strong singularity in the density component beginning from the initial moment of time, whereas other components of solution are initially continuous.

微分型シュレディンガー方程式の爆発解とプラズマ中の Alfven 波の崩壊

微分型シュレディンガー方程式の爆発解とプラズマ中の Alfven 波の崩壊

Formation of Singularities in One-Dimensional Hydromagnetic Flow**The project supported by National Natural Science Foundation of China under Grant No. 10001024, the Special Funds for Major State Basic Research Projects No. G2000077306, the Qi Ming Xing programme of Shanghai Government, and the Project sponsored by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry of China

Two results on the formation of singularities in solutions to the system of one-dimensional hydromagnetic dynamics are presented. In particular, it is shown that shocks form from a smooth spatial periodic flow in a finite time if the initial amounts of entropy and the “magnetic field” in each period are smaller than those of sound waves. A quantitative estimate of blow-up time is also given.

On the Formation of Singularities in Solutions of the Critical Nonlinear Schrödinger Equation

For the one-dimensional nonlinear Schrodinger equation with critical power nonlinearity the Cauchy problem with initial data close to a soliton is considered. It is shown that for a certain class of initial perturbations the solution develops a self-similar singularity infinite time T*, the profile being given by the ground state solitary wave and the limiting self-focusing law being of the form¶¶\( \lambda(t) \sim (ln \mid ln(T^* -t)\mid)^{1/2} (T^* - t)^{-1/2} \)

Formation of singularities in solutions of a wave equation

We prove a blow up result for the equation ω tt ( x, t) = a( x) ϕ( ω x ( x, t)) ω xx ( x, t), which can be taken as a model for a transverse motion of a string with nonconstant density.

Global existence and singularity formation in solutions of a modified Fourier law

The aim of this paper is the analysis of formation of singularities from smooth initial data, for rigid conductors. The modified Fourier law used asserts that the governing second-order equation is hyperbolic with first-order dissipation. Global existence for small initial data is also examined.

Formation of singularities in solutions of the nonlinear Schrödinger equation

Formation of singularities in solutions of the nonlinear Schrödinger equation

On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations

(1986). On the structure and formation of singularities in solutions to nonlinear dispersive evolution equations. Communications in Partial Differential Equations: Vol. 11, No. 5, pp. 545-565.

Formation of singularities in three-dimensional compressible fluids

Presented are several results on the formation of singularities in solutions to the three-dimensional Euler equations for a polytropic, ideal fluid under various assumptions on the initial data. In particular, it is shown that a localized fluid which is initially compressed and outgoing, on average, will develop singularities regardless of the size of the initial disturbance.

Formation of singularities in solutions to nonlinear hyperbolic equations

On etudie des systemes de lois de conservation hyperboliques et d'equations d'ondes completement non lineaires

Regularity Properties of Flows Through Porous Media: A Counterexample

Regularity Properties of Flows Through Porous Media: A Counterexample