共查询到20条相似文献,搜索用时 31 毫秒
1.
Methods originally developed to study the finite time blow-up problem of the regular solutions of the three dimensional incompressible Euler equations are used to investigate the regular solutions of the Camassa–Holm equation. We obtain results on the relative behaviors of the momentum density, the deformation tensor and the nonlocal term along the trajectories. In terms of these behaviors, we get new types of asymptotic properties of global solutions, blow-up criterion and blow-up time estimate for local solutions. More precisely, certain ratios of the quantities are shown to be vaguely monotonic along the trajectories of global solutions. Finite time blow-up of the accumulated momentum density is necessary and sufficient for the finite time blow-up of the solution. An upper estimate of the blow-up time and a blow-up criterion are given in terms of the initial short time trajectorial behaviors of the deformation tensor and the nonlocal term. 相似文献
2.
Dongho Chae 《Mathematische Zeitschrift》2007,257(3):563-580
We consider systems of deformed system of equations, which are obtained by some transformations from the system of incompressible
Euler equations. These have similar properties to the original Euler equations including the scaling invariance. For one form
of deformed system we prove that finite time blow-up actually occurs for ‘generic’ initial data, while for the other form
of the deformed system we prove the global in time regularity for smooth initial data. Moreover, using the explicit functional
relations between the solutions of those deformed systems and that of the original Euler system, we derive the condition of
finite time blow-up of the Euler system in terms of solutions of one of its deformed systems. As another application of those
relations we deduce a lower estimate of the possible blow-up time of the 3D Euler equations.
This research was supported partially by the KOSEF Grant no. R01-2005-000-10077-0 相似文献
3.
《数学季刊》2016,(2):125-138
This paper deals with the degenerate and singular parabolic equations coupled via nonlinear nonlocal reactions, subject to zero-Dirichlet boundary conditions. After giving the existence and uniqueness of local classical nonnegative solutions, we show critical blow-up exponents for the solutions of the system. Moreover, uniform blow-up behaviors near the blow-up time are obtained for simultaneous blow-up solutions, divided into four subcases. 相似文献
4.
Dongho Chae 《Journal of Functional Analysis》2010,258(9):2865-2883
In this paper we study blow-up rates and the blow-up profiles of possible asymptotically self-similar singularities of the Euler and the Navier-Stokes equations, where the sense of convergence and self-similarity are considered in various generalized senses. We improve substantially, in particular, the previous nonexistence results of self-similar/asymptotically self-similar singularities. Generalization of the self-similar transforms is also considered, and by appropriate choice of the parameterized transform we obtain new a priori estimates for the Euler and the Navier-Stokes equations depending on a free parameter. 相似文献
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《数学季刊》2016,(2)
This paper deals with the degenerate and singular parabolic equations coupled via nonlinear nonlocal reactions, subject to zero-Dirichlet boundary conditions. After giving the existence and uniqueness of local classical nonnegative solutions, we show critical blowup exponents for the solutions of the system. Moreover, uniform blow-up behaviors near the blow-up time are obtained for simultaneous blow-up solutions, divided into four subcases. 相似文献
7.
In this paper, we investigate some sufficient conditions for the breakdown of local smooth solutions to the three dimensional nonlinear nonlocal dissipative system modeling electro-hydrodynamics. This model is a strongly coupled system by the well-known incompressible Navier–Stokes equations and the classical Poisson–Nernst–Planck equations. We show that the maximum of the vorticity field alone controls the breakdown of smooth solutions, which reveals that the velocity field plays a more dominant role than the density functions of charged particles in the blow-up theory of the system. Moreover, some Prodi–Serrin type blow-up criteria are also established. 相似文献
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Weisheng Niu Xiaotong Sun Xiaojuan Chai 《Journal of Mathematical Analysis and Applications》2010,364(2):508-521
By introducing a stress multiplier we derive a family of Burgers-like equations. We investigate the blow-up phenomena of the equations both on the real line R and on the circle S to get a comparison with the Degasperis-Procesi equation. On the line R, we first establish the local well-posedness and the blow-up scenario. Then we use conservation laws of the equations to get the estimate for the L∞-norm of the strong solutions, by which we prove that the solutions to the equations may blow up in the form of wave breaking for certain initial profiles. Analogous results are provided in the periodic case. Especially, we find differences between the Burgers-like equations and the Degasperis-Procesi equation, see Remark 4.1. 相似文献
11.
主要研究了一类带Robin边界条件的拟线性抛物方程解的整体存在性与爆破问题,利用微分不等式技术,获得了方程的解发生爆破时的爆破时间的下界.然后给出了方程解整体存在的充分条件,最后得到了方程的解发生爆破时发生爆破时间的上界. 相似文献
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Steven Dejak Zhou Gang Israel Michael Sigal Shuangcai Wang 《Advances in Applied Mathematics》2008,40(4):433-481
We study the blow-up of solutions of nonlinear heat equations in dimension 1. We show that for an open set of even initial data which are characterized roughly by having maxima at the origin, the solutions blow up in finite time and at a single point. We find the universal blow-up profile and remainder estimates. Our results extend previous results in several directions and our techniques differ from the techniques previously used for this problem. In particular, they do not rely on maximum principle. 相似文献
14.
In this paper, we prove the blow-up phenomena of smooth solutions to the Cauchy problem for the full compressible magnetohydrodynamic equations and isentropic compressible magnetohydrodynamic equations with constant and degenerate viscosities under some restrictions on the initial data. In particular, our results do not require that the initial data have compact support or contain vacuum in any finite region. 相似文献
15.
主要研究了关于R~2中一类带有幂型非线性的广义Zakharov方程组的Cauchy问题的有限时间爆破解的爆破率的下界估计.在α≤0和p≥3条件下,对于Cauchy问题任意给定的属于能量空间H~1(R~2)×L~2(R~2)×L~2(R~2)的有限时间的爆破解,得到了对于t靠近有限爆破时间T时的爆破率的最优下界估计.此外,给出了Cauchy问题维里等式的一个应用. 相似文献
16.
We study blow-up, global existence and ground state solutions for the N-coupled focusing nonlinear SchrSdinger equations. Firstly, using the Nehari manifold approach and some variational techniques, the existence of ground state solutions to the equations (CNLS) is established. Secondly, under certain conditions, finite time blow-up phenomena of the solutions is derived. Finally, by introducing a refined version of compactness lemma, the L2 concentration for the blow-up solutions is obtained. 相似文献
17.
Gongwei Liu & Shuying Tian 《分析论及其应用》2022,38(4):451-466
We investigate the initial boundary value problem of some semilinear pseudo-parabolic equations with Newtonian nonlocal term. We establish a lower bound for the blow-up time if blow-up does occur. Also both the upper bound for $T$ and blow up rate of the solution are given when $J(u_0)<0$. Moreover, we establish the blow up result for arbitrary initial energy and the upper bound for $T$. As a product, we refine the lifespan when $J(u_0)<0.$ 相似文献
18.
The main propose of this paper is to study the blow-up of solutions of an initial boundary value problem with a nonlocal boundary condition for a system of nonlinear singular viscoelastic equations. where the blow-up of solutions in finite time with nonpositive initial energy combined with a positive initial energy are shown. 相似文献
19.
W. Mydlarczyk W. Okrasiński C.A. Roberts 《Journal of Mathematical Analysis and Applications》2005,301(1):208-218
A system of nonlinear Volterra integral equations with convolution kernels is considered. Estimates are given for the blow-up time when conditions are such that the solution is known to become unbounded in finite time. For two examples that arise in combustion problems, numerical estimates of blow-up time are presented. Additionally, the asymptotic behavior of the blow-up solution in the key limit is established for the power-law and exponential nonlinearity cases. 相似文献
20.
J. Baris 《Journal of Mathematical Analysis and Applications》2008,341(2):1155-1162
We investigate the existence problem for blow-up solutions of cubic differential systems. We find sets of initial values of the blow-up solutions. We also discuss a method of finding upper estimates for the blow-up time of these solutions. Our approach can be applied to systems of partial differential equations. We apply this approach to the Cauchy-Dirichlet problem for systems of semilinear heat equations with cubic nonlinearities. 相似文献