共查询到20条相似文献,搜索用时 303 毫秒
1.
Ryo Takada 《Journal of Evolution Equations》2008,8(4):693-725
We introduce Besov type function spaces, based on the weak L
p
-spaces instead of the standard L
p
-spaces, and prove a local-in-time unique existence and a blow-up criterion of solutions in those spaces for the Euler equations
of perfect incompressible fluid in . For the proof, we establish the Beale-Kato-Majda type logarithmic inequality and commutator type estimates in our weak spaces.
Abbreviate title: Euler equations in Besov spaces of weak type 相似文献
2.
3.
Dongho Chae 《纯数学与应用数学通讯》2002,55(5):654-678
We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ?n, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc. 相似文献
4.
The existence and uniqueness of solutions to the Euler equations for initial vorticity in BΓ∩Lp0∩Lp1 was proved by Misha Vishik, where BΓ is a borderline Besov space parameterized by the function Γ and 1<p0<2<p1. Vishik established short time existence and uniqueness when Γ(n)=O(logn) and global existence and uniqueness when . For initial vorticity in BΓ∩L2, we establish the vanishing viscosity limit in L2(R2) of solutions of the Navier-Stokes equations to a solution of the Euler equations in the plane, convergence being uniform over short time when Γ(n)=O(logn) and uniform over any finite time when Γ(n)=O(logκn), 0?κ<1, and we give a bound on the rate of convergence. This allows us to extend the class of initial vorticities for which both global existence and uniqueness of solutions to the Euler equations can be established to include BΓ∩L2 when Γ(n)=O(logκn) for 0<κ<1. 相似文献
5.
We show the asymptotic behavior of the solution to the Cauchy problem of the two-dimensional damped wave equation. It is shown that the solution of the linear damped wave equation asymptotically decompose into a solution of the heat and wave equations and the difference of those solutions satisfies the Lp−Lq type estimate. This is a two-dimensional generalization of the three-dimensional result due to Nishihara (Math. Z. 244 (2003) 631). To show this, we use the Fourier transform and observe that the evolution operators of the damped wave equation can be approximated by the solutions of the heat and wave equations. By using the Lp−Lq estimate, we also discuss the asymptotic behavior of the semilinear problem of the damped wave equation with the power nonlinearity |u|αu. Our result covers the whole super critical case α>1, where the α=1 is well known as the Fujita exponent when n=2. 相似文献
6.
We are concerned with entropy solutions of the 2×2 relativistic Euler equations for perfect fluids in special relativity. We establish the uniqueness of Riemann solutions in the class of entropy solutions in L∞∩BVloc with arbitrarily large oscillation. Our proof for solutions with large oscillation is based on a detailed analysis of global behavior of shock curves in the phase space and on special features of centered rarefaction waves in the physical plane for this system. The uniqueness result does not require specific reference to any particular method for constructing the entropy solutions. Then the uniqueness of Riemann solutions yields their inviscid large-time stability under arbitrarily largeL1∩L∞∩BVloc perturbation of the Riemann initial data, as long as the corresponding solutions are in L∞ and have local bounded total variation that allows the linear growth in time. We also extend our approach to deal with the uniqueness and stability of Riemann solutions containing vacuum in the class of entropy solutions in L∞ with arbitrarily large oscillation. 相似文献
7.
D. Castorina 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(5):2195-2203
Extending a previous result of Tang [1] we prove the uniqueness of positive radial solutions of Δpu+f(u)=0, subject to Dirichlet boundary conditions on an annulus in Rn with 2<p≤n, under suitable hypotheses on the nonlinearity f. This argument also provides an alternative proof for the uniqueness of positive solutions of the same problem in a finite ball (see [9]), in the complement of a ball or in the whole space Rn (see [10], [3] and [11]). 相似文献
8.
Clayton Bjorland Maria E. Schonbek 《Annales de l'Institut Henri Poincaré (C) Analyse Non Linéaire》2008
We consider the viscous n -dimensional Camassa–Holm equations, with n=2,3,4 in the whole space. We establish existence and regularity of the solutions and study the large time behavior of the solutions in several Sobolev spaces. We first show that if the data is only in L2 then the solution decays without a rate and that this is the best that can be expected for data in L2. For solutions with data in Hm∩L1 we obtain decay at an algebraic rate which is optimal in the sense that it coincides with the rate of the underlying linear part. 相似文献
9.
《Journal of Computational and Applied Mathematics》2002,147(2):485-516
The solution of stochastic evolution equations generally relies on numerical computation. Here, usually the main idea is to discretize the SPDE spatially obtaining a system of SDEs that can be solved by e.g., the Euler scheme. In this paper, we investigate the discretization error of semilinear stochastic evolution equations in Lp-spaces, resp. Banach spaces. The space discretization may be done by Galerkin approximation, for the time discretization we consider the implicit Euler, the explicit Euler scheme and the Crank–Nicholson scheme. In the last section, we give some examples, i.e., we consider an SPDEs driven by nuclear Wiener noise approximated by wavelets and delay equation approximated by finite differences. 相似文献
10.
Zhichun Zhai 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(6):3173-3189
In this paper, we study Keller-Segel systems with fractional diffusion and a nonlocal term. We establish the global existence, uniqueness and stability of solutions for systems with small initial data in critical Besov spaces. Our main tools are the Lp−Lq estimates for in Besov spaces and the perturbation of linearization. 相似文献
11.
In this paper, we first establish a strong convergence criterion of approximate solutions for the 3D steady incompressible Euler equations. For axisymmetric flows, under the assumption that the vorticity is of one sign and uniformly bounded in L1 space, we obtain a sufficient and necessary condition for the strong convergence in of approximate solutions. Furthermore, for one-sign and L1-bounded vorticity, it is shown that if a sequence of approximate solutions concentrates at an isolated point in (r,z)-plane, then the concentration point can appear neither in the region near the axis (including the symmetry axis itself) nor in the region far away from the axis. Finally, we present an example of approximates solutions which converge strongly in by using Hill's spherical vortex. 相似文献
12.
Gary M. Lieberman 《Journal of Functional Analysis》2003,201(2):457-479
In recent years, there has been considerable interest in extending the well-known Calderon-Zygmund estimates for the Laplacian to more general equations, in particular equations with highest order coefficients lying in the Sarason space VMO. In addition, the analogous estimates with Morrey spaces replacing Lebesgue spaces have been considered. These Morrey space estimates have been proved by refining the proofs for the Lp estimates. We shall show that the Morrey space estimates follow from the Lp estimates via an elementary argument which is very similar to that used by Campanato. 相似文献
13.
Guowei LIU 《数学物理学报(B辑英文版)》2017,37(1):79-96
This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal L~p,1≤ p ≤ +∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates. 相似文献
14.
Joachim Toft 《Bulletin des Sciences Mathématiques》2002,126(2):115-142
We study continuity properties for a family {sp}p?1 of increasing Banach algebras under the twisted convolution, which also satisfies that a∈sp, if and only if the Weyl operator aw(x,D) is a Schatten-von Neumann operator of order p on L2. We discuss inclusion relations between the sp-spaces, Besov spaces and Sobolev spaces. We prove also a Young type result on sp for dilated convolution. As an application we prove that f(a)∈s1, when a∈s1 and f is an entire odd function. We finally apply the results on Toeplitz operators and prove that we may extend the definition for such operators. 相似文献
15.
It is well known that the biharmonic equation Δ2u=u|u|p−1 with p∈(1,∞) has positive solutions on Rn if and only if the growth of the nonlinearity is critical or supercritical. We close a gap in the existing literature by proving the existence and uniqueness, up to scaling and symmetry, of oscillatory radial solutions on Rn in the subcritical case. Analyzing the nodal properties of these solutions, we also obtain precise information about sign-changing large radial solutions and radial solutions of the Dirichlet problem on a ball. 相似文献
16.
Nicolas Saintier 《Nonlinear Analysis: Theory, Methods & Applications》2010,72(2):689-703
Let (M,g) be a smooth compact Riemannian manifold. We first give the value of the best first constant for the critical embedding H2(M)?L2?(M) for second-order Sobolev spaces of functions invariant by some subgroup of the isometry group of (M,g). We also prove that we can take ?=0 in the corresponding inequality under some geometric assumptions. As an application we give a sufficient condition for the existence of a smooth positive symmetric solution to a critical equation with a symmetric Paneitz-Branson-type operator. A sufficient condition for the existence of a nodal solution to such an equation is also derived. We eventually prove a multiplicity result for such an equation. 相似文献
17.
In this paper, we are concerned with the global existence and convergence rates of the smooth solutions for the compressible magnetohydrodynamic equations in R3. We prove the global existence of the smooth solutions by the standard energy method under the condition that the initial data are close to the constant equilibrium state in H3-framework. Moreover, if additionally the initial data belong to Lp with , the optimal convergence rates of the solutions in Lq-norm with 2≤q≤6 and its spatial derivatives in L2-norm are obtained. 相似文献
18.
Cornelia Kaiser 《Journal of Mathematical Analysis and Applications》2004,292(2):328-339
We study existence and uniqueness of solutions for linear partial differential equations with delay in Lp-spaces using an approach of Batkai and Piazzera and a recent perturbation result for integrated semigroups. We apply our result to an equation with delay in the highest-order derivatives. 相似文献
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20.
Alex Karasev 《Topology and its Applications》2006,153(17):3241-3254
We extend the definition of quasi-finite complexes from countable complexes to arbitrary ones and provide a characterization of quasi-finite complexes in terms of L-invertible maps and dimensional properties of compactifications. Several results related to the class of quasi-finite complexes are established, such as completion of metrizable spaces, existence of universal spaces and a version of the factorization theorem. Furthermore, we define UV(L)-spaces in the realm of metrizable spaces and show that some properties of UV(n)-spaces and UV(n)-maps remain valid for UV(L)-spaces and UV(L)-maps, respectively. 相似文献