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1.
A NECESSARY AND SUFFICIENT CONDITION OF EXISTENCE OF GLOBAL SOLUTIONS FOR SOME NONLINEAR HYPERBOLIC EQUATIONS 总被引:2,自引:0,他引:2
Zhang Quande 《数学年刊B辑(英文版)》1995,16(4):461-468
ANECESSARYANDSUFFICIENTCONDITIONOFEXISTENCEOFGLOBALSOLUTIONSFORSOMENONLINEARHYPERBOLICEQUATIONS¥ZHANGQUANDE(DepartmentofMathe... 相似文献
2.
Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density 下载免费PDF全文
Jianzhong Zhang & Hongmei Cao 《数学研究通讯:英文版》2023,39(1):79-106
In this paper, we prove the global existence and uniqueness of solutions for the inhomogeneous Navier-Stokes equations with the initial data $(\rho_0,u_0)\in L^∞\times H^s_0$, $s>\frac{1}{2}$ and $||u_0||_{H^s_0}\leq \varepsilon_0$ in bounded domain $\Omega \subset \mathbb{R}^3$, in which the density is assumed to be nonnegative. The regularity of initial data is weaker than the previous $(\rho_0,u_0)\in (W^{1,\gamma}∩L^∞)\times H^1_0$ in [13] and $(\rho_0,u_0)\in L^∞\times H^1_0$ in [7], which constitutes a positive answer to the question raised by Danchin and Mucha in [7]. The methods used in this paper are mainly the classical time weighted energy estimate and Lagrangian approach, and the continuity argument and shift of integrability method are applied to complete our proof. 相似文献
3.
In this paper, we are concerned with interior differentiability of weak solutions u to nonlinear parabolic systems with natural growth and coefficients uniformly monotone in Du. Making use of estimates of Gagliardo–Nirenberg’s type in generalized Sobolev spaces, we show that u belongs to (see Theorem 3). 相似文献
4.
Nikos I. Karachalios Nikos M. Stavrakakis 《NoDEA : Nonlinear Differential Equations and Applications》2002,9(3):347-360
We discuss the asymptotic behaviour of the Schrödinger equation ¶¶$ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 $ iu_{t} + u_{xx} +i\alpha u -k\sigma(|u|^{2})u\, = f, \;\; x \in \mathbb{R}, \;\; t \geq 0,\;\;\;\alpha,\;k>0 ¶¶ with the initial condition u(x,0) = u0 (x) u(x,0) = u_0 (x) . We prove existence of a global attractor in\ H2 (\mathbbR) H^2 (\mathbb{R}) , by using a decomposition of the semigroup in weighted Sobolev spaces to overcome the noncompactness of the classical Sobolev embeddings. 相似文献
5.
Noriko Mizoguchi 《Calculus of Variations and Partial Differential Equations》2013,48(3-4):491-505
This paper is concerned with the Cauchy problem for the Keller–Segel system $$\left\{\begin{array}{l@{\quad}l}u_t = \nabla \cdot (\nabla u - u \nabla v) & \hbox{in } {\bf R}^{2} \times(0,\infty),\\v_t = \Delta v - \lambda v + u & \hbox{ in } {\bf R}^2 \times(0,\infty),\\u(x,0) = u_0 (x) \geq 0, \; v(x,0) = v_0 (x) \geq 0 & \hbox{ in} {\bf R}^2\end{array}\right.$$ with a constant λ ≥ 0, where ${(u_0, v_0) \in (L^1 ({\bf R}^2) \cap L^\infty ({\bf R}^2) ) \times (L^1 ({\bf R}^2) \cap H^1 ({\bf R}^2))}$ . Let $$m (u_0;{\bf R}^2) = \int\limits_{{\bf R}^2} u_0 (x) dx$$ . The same method as in [9] yields the existence of a blowup solution with m (u 0; R 2) > 8π. On the other hand, it was recently shown in [7] that under additional hypotheses ${u_0 \log (1 + |x|^2) \in L^1 ({\bf R}^2)}$ and ${u_0 \log u_0 \in L^1 ({\bf R}^2)}$ , any solution with m(u 0; R 2) < 8π exists globally in time. In[18], the extra assumptions were taken off, but the condition on mass was restricted to m (u 0; R 2) < 4π. In this paper, we prove that any solution with m (u 0; R 2) < 8π exists globally in time under no extra conditions. Furthermore the global existence of solutions is obtained under some condition on u 0 also in the critical case m (u 0; R 2) = 8π. 相似文献
6.
We study the large time behavior of the solution u to an initial and boundary value problem related to the following integro-differential equation $$ u_{tt} = G_0 \Delta u + \int_0^t G'(t-s) \Delta u(x, s)\, ds - a u_t \eqno(0.1) $$ where G 0 , a are real constant coefficients, G 0 > 0, a S 0 and $ G\,' \in L^1({{\shadR}}^ + ) \cap L^2({{\shadR}}^ + ), G\,' \le 0 $ . It is known that, when G ' L 0 and a > 0, the solution u of (0.1) exponentially decays. Here we prove that, for any nonnegative a and for any $ G ' \not \equiv 0 $ , the solution u of the Eq. (0.1) exponentially decays only if the relaxation kernel G ' does. In other words, the introduction of the dissipative term related to G ' does not allow the exponential decay due to the presence of the positive coefficient a . We also prove analogous results for the polynomial decay. 相似文献
7.
Ye Cinan 《数学年刊B辑(英文版)》1986,7(3):384-396
Suppose that there is a variance components model
$$\[\left\{ {\begin{array}{*{20}{c}}
{E\mathop Y\limits_{n \times 1} = \mathop X\limits_{n \times p} \mathop \beta \limits_{p \times 1} }\{DY = \sigma _2^2{V_1} + \sigma _2^2{V_2}}
\end{array}} \right.\]$$
where $\[\beta \]$,$\[\sigma _1^2\]$ and $\[\sigma _2^2\]$ are all unknown, $\[X,V > 0\]$ and $\[{V_2} > 0\]$ are all known, $\[r(X) < n\]$. The author estimates simultaneously $\[(\sigma _1^2,\sigma _2^2)\]$. Estimators are restricted to the class $\[D = \{ d({A_1}{A_2}) = ({Y^''}{A_1}Y,{Y^''}{A_2}Y),{A_1} \ge 0,{A_2} \ge 0\} \]$. Suppose that the loss function is $\[L(d({A_1},{A_2}),(\sigma _1^2,\sigma _2^2)) = \frac{1}{{\sigma _1^4}}({Y^''}{A_1}Y - \sigma _1^2) + \frac{1}{{\sigma _2^4}}{({Y^''}{A_2}Y - \sigma _2^2)^2}\]$.
This paper gives a necessary and sufficient condition for $\[d({A_1},{A_2})\]$ to be an equivariant D-asmissible estimator under the restriction $\[{V_1} = {V_2}\]$, and a sufficient condition and a necessary condition for $\[d({A_1},{A_2})\]$ to equivariant D-asmissible without the restriction. 相似文献
8.
G. Fonseca F. Linares G. Ponce 《Proceedings of the American Mathematical Society》2003,131(6):1847-1855
We discuss results regarding global existence of solutions for the critical generalized Korteweg-de Vries equation,
The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space , i.e. solutions corresponding to data , 3/4$">, with , where is the solitary wave solution of the equation.
The theory established shows the existence of global solutions in Sobolev spaces with order below the one given by the energy space , i.e. solutions corresponding to data , 3/4$">, with , where is the solitary wave solution of the equation.
9.
Yu. Ya. Belov 《Mathematical Notes》1976,20(5):948-953
In a Hilbert space H we consider the approximation by systems $$\frac{{d^2 u_1 }}{{dt^2 }} = A_{11} u_1 + A_{12} u_2 + f_1 ,\varepsilon \frac{{d^2 u_2 }}{{dt^2 }} = A_{21} u_1 + A_{22} u_2 + f_2 ,\varepsilon > 0,$$ of the semievolutionary system obtained from (1) when ∈=0. Under certain conditions on the solutions of the Cauchy problem for system (1) and the existence of a bounded linear operator A 22 ?1 we establish the convergence of the solutions u∈(∈ → 0) to a solution of the corresponding problem for system (1) with ∈=0. We also establish the uniform correctness of the Cauchy problem for the above system. 相似文献
10.
Suppose $\cal{S}^1({\cal T})\subset H^1(\Omega)$ is the $P_1$-finite element
space of $\cal{T}$-piecewise affine functions
based on a regular triangulation $\cal{T}$ of a two-dimensional surface
$\Omega$ into triangles.
The $L^2$ projection $\Pi$ onto $\cal{S}^1(\cal{T})$ is $H^1$ stable
if $\norm{\Pi v}{H^1(\Omega)}\le C\norm{v}{H^1(\Omega)}$ for
all $v$ in the Sobolev space $H^1(\Omega)$ and if the bound $C$
does not depend on the mesh-size in $\cal{T}$ or on the
dimension of $\cal{S}^1(\cal{T})$.
\hskip 1em A red–green–blue refining adaptive algorithm is designed which
refines a coarse mesh $\cal{T}_0$ successively such that each triangle is
divided into one, two, three, or four subtriangles.
This is the newest vertex bisection supplemented with possible red refinements
based on a careful initialization.
The resulting finite element space allows
for an $H^1$ stable $L^2$ projection.
The stability
bound $C$ depends only on the coarse mesh $\cal{T}_0$ through the number of
unknowns, the shapes of the triangles in $\cal{T}_0$, and possible
Dirichlet boundary conditions. Our arguments also
provide a discrete version
$\norm{h_\cal{T}^{-1}\,\Pi v}{L^2(\Omega)}\le C\norm{h_\cal{T}^{-1}\,v}{L^2(\Omega)}$
in $L^2$ norms weighted with the mesh-size $h_\T$. 相似文献
11.
Given initial data(ρ0, u0) satisfying 0 m ρ0≤ M, ρ0- 1 ∈ L2∩˙W1,r(R3) and u0 ∈˙H-2δ∩ H1(R3) for δ∈ ]1/4, 1/2[ and r ∈ ]6, 3/1- 2δ[, we prove that: there exists a small positive constant ε1,which depends on the norm of the initial data, so that the 3-D incompressible inhomogeneous Navier-Stokes system with variable viscosity has a unique global strong solution(ρ, u) whenever‖ u0‖ L2 ‖▽u0 ‖L2 ≤ε1 and ‖μ(ρ0)- 1‖ L∞≤ε0 for some uniform small constant ε0. Furthermore, with smoother initial data and viscosity coefficient, we can prove the propagation of the regularities for such strong solution. 相似文献
12.
Liu Linqi 《数学年刊B辑(英文版)》1988,9(4):379-389
In order better to research the singularities of the solutions $\[u \in H_{loc}^s(\Omega ),\Omega \subset {R^n},s > \frac{n}{2} + 1\]$ , for semilinear hyperbolic equations $\[u = f(u,Du)\]$, in this paper, a kind of weighted Sobolev space $\[({H^s})_{{P_\mu }}^\alpha \],\[\mu = 1,2,{p_1} = {D_i} - \left| {{D_x}} \right|,{P_2} = {D_i} + \left| {{D_x}} \right|\]$, closely related with the solutions of the equations, is presented. It is discussed that their products tacitly keep roughly $\[{H^{3x - n}}\]$ microlocal regularity on the characteristic directions for $\[{P_\mu }\]$ and invariance under nonlinear maps. Then it is obtained that roughly $\[{H^{3x - n}}\]$ propagation of singularities theorem is valid for $\[u = f(u)\]$. 相似文献
13.
Dong Guangchang 《数学年刊B辑(英文版)》1986,7(3):277-302
In this paper, the author proves the existence and uniqueness of nonnegative solution for the first boundary value problem of uniform degenerated parabolic equation
$$\[\left\{ {\begin{array}{*{20}{c}}
{\frac{{\partial u}}{{\partial t}} = \sum {\frac{\partial }{{\partial {x_i}}}\left( {v(u){A_{ij}}(x,t,u)\frac{{\partial u}}{{\partial {x_j}}}} \right) + \sum {{B_i}(x,t,u)} \frac{{\partial u}}{{\partial {x_i}}}} + C(x,t,u)u\begin{array}{*{20}{c}}
{}&{(x,t) \in [0,T]}
\end{array},}\{u{|_{t = 0}} = {u_0}(x),x \in \Omega ,}\{u{|_{x \in \partial \Omega }} = \psi (s,t),0 \le t \le T}
\end{array}} \right.\]$$
$$\[\left( {\frac{1}{\Lambda }{{\left| \alpha \right|}^2} \le \sum {{A_{ij}}{\alpha _i}{\alpha _j}} \le \Lambda {{\left| \alpha \right|}^2},\forall a \in {R^n},0 < \Lambda < \infty ,v(u) > 0\begin{array}{*{20}{c}}
{and}&{v(u) \to 0\begin{array}{*{20}{c}}
{as}&{u \to 0}
\end{array}}
\end{array}} \right)\]$$
under some very weak restrictions, i.e. $\[{A_{ij}}(x,t,r),{B_i}(x,t,r),C(x,t,r),\sum {\frac{{\partial {A_{ij}}}}{{\partial {x_j}}}} ,\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}} \in \overline \Omega } \times [0,T] \times R,\left| {{B_i}} \right| \le \Lambda ,\left| C \right| \le \Lambda ,\],\[\left| {\sum {\frac{{\partial {B_i}}}{{\partial {x_i}}}} } \right| \le \Lambda ,\partial \Omega \in {C^2},v(r) \in C[0,\infty ).v(0) = 0,1 \le \frac{{rv(r)}}{{\int_0^r {v(s)ds} }} \le m,{u_0}(x) \in {C^2}(\overline \Omega ),\psi (s,t) \in {C^\beta }(\partial \Omega \times [0,T]),0 < \beta < 1\],\[{u_0}(s) = \psi (s,0).\]$ 相似文献
14.
The aim of this paper is the study of the convergence of a finite element approximation for a variational inequality related to free boundary problems in non-steady fluid flow through porous media. There have been many results in the stationary case, for example, the steady dam problems, the steady flow well problems, etc. In this paper we shall deal with the axisymmetric non-steady porous flow well problem. It is well know that by means of Torelli's transform this problem, similar to the non-steady rectangular dam problem, can be reduced a variational, inequality, and the existence, uniqueness and regularity of the solution can be obtained ([12, 7]). Now we study the numerical solution of this variational inequality. The main results are as follows: 1. We establish new regularity properties for the solution $W$ of the variation inequality. We prove that $W \in L^\infty(0, T; H^2(D))$, $γ_0W\in L^\infty(0, T; H^2(T_n))$ and $D_1γ_0W\in L^2(0, T; H^1(T_n))$ (see Theorem 2.5). Friedman and Torelli [7] obtained $W\in L^2(0, T; H^2(D))$. Our new regularity properties will be used for error estimation. 2. We prove that the error estimate for the finite element solution of the variational inequality is $$ ( \sum^N_{i=1}\| W^1 - W^1_h \|^2_{H^1(D)}\Delta t)^{1/2} = O(h+\Delta t^{1/2})$$ (see Theorem 3.4). In the stationary case the error estimate is $\|W-W_h\|_{H^1(D)} = O(k)$ ([3,6]). 3. We give a numerical example and compare the result with the corresponding result in the stationary case. The result of this paper are valid for the non-ready rectangular dam problem with stationary or quasi-stationary initial data (see [7], p.534). 相似文献
15.
16.
Let D be a bounded C~3-domain in R~d and(a_(ij))be a bounded symmetric matrixdefined on D.Consider the symmetric form(u,v)=1/2∫_D a_(ij)(x)(u(x))/(x_i) (v(x))/(x_j)dx,u,v∈H~1(D).Under some assumptions it is shown that the diffusion process associated with the regularDirichlet space(,(H~1(D))on L~2(D)can be characterized as a unique solution of acertain stochastic differential equation. 相似文献
17.
{Well-posedness of degenerate differential equations with infinite delay in Holder continuous function spaces 下载免费PDF全文
Using operator-valued $\dot{C}^\alpha$-Fourier multiplier results on vector- valued H\"older continuous function spaces, we give a characterization for the $C^\alpha$-well-posedness of the first order degenerate differential equations with infinite delay $(Mu)"(t) = Au(t) + \int_{-\infty}^t a(t-s)Au(s)ds + f(t)$ ($t\in\R$), where $A, M$ are closed operators on a Banach space $X$ such that $D(A)\cap D(M)\neq \{0\}$, $a\in L^1_{\rm{loc}}(\R_+)\cap L^1(\mathbb{R}_+; t^\alpha dt)$. 相似文献
18.
Tommaso Leonori Francesco Petitta 《Calculus of Variations and Partial Differential Equations》2011,42(1-2):153-187
In this paper we deal with local estimates for parabolic problems in ${\mathbb{R}^N}$ with absorbing first order terms, whose model is $$\left\{\begin{array}{l@{\quad}l}u_t- \Delta u +u |\nabla u|^q = f(t,x) \quad &{\rm in}\, (0,T) \times \mathbb{R}^N\,,\\u(0,x)= u_0 (x) &{\rm in}\, \mathbb{R}^N \,,\quad\end{array}\right.$$ where ${T >0 , \, N\geq 2,\, 1 < q \leq 2,\, f(t,x)\in L^1\left( 0,T; L^1_{\rm loc} \left(\mathbb{R}^N\right)\right)}$ and ${u_0\in L^1_{\rm loc}\left(\mathbb{R}^{N}\right)}$ . 相似文献
19.
该文讨论了如下具有退化粘性的非齐次双曲守恒律方程的Cauchy问题$\left\{\begin{array}{l} u_t+f(u)_x=a^2t^\alpha u_{xx}+g(u),\ \ \ x\in{\bf R},\ \ \ t>0,\\u(x,0)=u_0(x) \in L^\infty({\bf R}).\end{array}\right.\eqno{({\rm I})}$其中$f(u), g(u)$是${\bf R}$上的光滑函数, $a>0, 0<\alpha<1$均为常数.在此条件下, 作者首先给出了Cauchy问题(I)的局部解的存在性, 再利用极值原理获得了解的$L^{\infty}$估计, 从而证明了Cauchy问题(I)整体光滑解的存在性. 相似文献
20.
The purpose of this article is to study the existence and uniqueness of global solution for the nonlinear hyperbolic-parabolic equation of Kirchhoff-Carrier type: $$ u_{tt} + \mu u_t - M\left (\int _{\Omega _t}|\nabla u|^2dx\right )\Delta u = 0\quad \hbox {in}\ \Omega _t\quad \hbox {and}\quad u|_{\Gamma _t} = \dot \gamma $$ where $ \Omega _t = \{x\in {\shadR}^2 | \ x = y\gamma (t), \ y\in \Omega \} $ with boundary o t , w is a positive constant and n ( t ) is a positive function such that lim t M X n ( t ) = + X . The real function M is such that $ M(r) \geq m_0 \gt 0 \forall r\in [0,\infty [ $ . 相似文献