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1.
Fujita-Kato Theorem for the Inhomogeneous Incompressible Navier-Stokes Equations with Nonnegative Density
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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
2.
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
3.
Cesar Enrique Torres Ledesm Ziheng Zhang Amado Mendez 《Journal of Applied Analysis & Computation》2019,9(6):2436-2453
We study the existence of solutions for the following fractional Hamiltonian systems
$$
\left\{
\begin{array}{ll}
- _tD^{\alpha}_{\infty}(_{-\infty}D^{\alpha}_{t}u(t))-\lambda L(t)u(t)+\nabla W(t,u(t))=0,\\[0.1cm]
u\in H^{\alpha}(\mathbb{R},\mathbb{R}^n),
\end{array}
\right.
~~~~~~~~~~~~~~~~~(FHS)_\lambda
$$
where $\alpha\in (1/2,1)$, $t\in \mathbb{R}$, $u\in \mathbb{R}^n$, $\lambda>0$ is a parameter, $L\in C(\mathbb{R},\mathbb{R}^{n^2})$ is a symmetric matrix, $W\in C^1(\mathbb{R} \times \mathbb{R}^n,\mathbb{R})$. Assuming that
$L(t)$ is a positive semi-definite symmetric matrix, that is, $L(t)\equiv 0$ is allowed to occur in some finite interval $T$ of $\mathbb{R}$,
$W(t,u)$ satisfies some superquadratic conditions weaker than Ambrosetti-Rabinowitz condition, we show that (FHS)$_\lambda$ has a solution which vanishes on
$\mathbb{R}\setminus T$ as $\lambda \to \infty$, and converges to some $\tilde{u}\in H^{\alpha}(\R, \R^n)$. Here, $\tilde{u}\in E_{0}^{\alpha}$ is a solution
of the Dirichlet BVP for fractional systems on the finite interval $T$. Our results are new and improve recent results in the literature even in the case $\alpha =1$. 相似文献
4.
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
5.
Mu-shengWei 《计算数学(英文版)》2004,22(3):427-436
It is known that for a given matrix A of rank r, and a set D of positive diagonal matrices, supw∈D‖(W^1/2A) W^1/2‖ = (miniσ (A^(i))^-1, in which (A^(i) is a submatrix of A formed with r = (rank(A)) rows of A, such that (A^(i) has full row rank r. In many practical applications this value is too large to be used. In this paper we consider the case that both A and W(∈D) are fixed with W severely stiff. We show that in this case the weighted pseudoinverse (W^1/2‖A) W^1/2‖ is close to a multilevel constrained weighted pseudoinverse therefore ‖(W^1/2A) W^1/‖2 is uniformly bounded.We also prove that in this case the solution set the stiffly weighted least squares problem is close to that of corresponding multi-level constrained least squares problem. 相似文献
6.
In this note we consider Wente's type inequality on the Lorentz-Sobolev space.If▽f∈L~p1,q1(R~n),G ∈ L~(p2,q2)(R~n) and div G≡0 in the sense of distribution where(1/p1)+(1/P2)=(1/q1)+(1/q2)=1,1P1,P2∞,it is known that G·▽f belongs to the Hardy space H~1 and furthermore‖G·▽f‖H~1≤C‖▽f‖L~(p1,q1)(R~2)‖G‖L~(p2,q2)(R~2).Reader can see[9]Section 4.Here we give a new proof of this result.Our proof depends on an estimate of a maximal operator on the Lorentz space which is of some independent interest.Finally,we use this inequality to get a generalisation of Bethuel's inequality. 相似文献
7.
Any weak solution u to the Navier-Stokes equations is showed to be regular under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small, which is a limiting case of the regularity criteria derived by Kim and Kozono. Our result gives a positive answer to the question proposed by Kim and Kozono. For the incompressible magnetohydrodynamic equations, we also show the regularity of weak solution only under the assumption that ||u|| L 2w (0,T ;L ∞ ( R 3 )) is sufficiently small. 相似文献
8.
H. Beiro da Veiga 《数学年刊B辑(英文版)》1995,16(4):407-412
ANEWREGULARITYCLASSFORTHENAVIER-STOKESEQUATIONSINIR~n¥H.BEIRaODAVEIGA(DepotmentofMathematics,PisaUniversity,Pisa,Italy)Abstra?.. 相似文献
9.
Hartmut Pecher 《NoDEA : Nonlinear Differential Equations and Applications》2008,15(3):279-294
The 1D Cauchy problem for the Dirac-Klein-Gordon system is shown to be locally well-posed for low regularity Dirac data in
and wave data in for under certain assumptions on the parameters r and s, where , generalizing the results for p = 2 by Selberg and Tesfahun. Especially we are able to improve the results from the scaling point of view with respect to
the Dirac part.
相似文献
10.
Let be bounded linear operators. We provide several sufficient conditions for the validity of the inequality . These results can be applied to error and cost estimates for the sparse grid method. 相似文献
11.
We obtain a sharp Remez inequality for the trigonometric polynomial T n of degree n on [0,2π): $$\|T_n \|_{L_\infty([0,2\pi))} \le \biggl(1+2\tan^2 \frac{n\beta}{4m} \biggr) { \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}, $$ where $\frac{2\pi}{m}$ is the minimal period of T n and $|B|=\beta<\frac {2\pi m}{n}$ is a measurable subset of $\mathbb {T}$ . In particular, this gives the asymptotics of the sharp constant in the Remez inequality: for a fixed n, $$\mathcal{C}(n, \beta)=1+ \frac{(n\beta)^2}{8} +O \bigl(\beta^4\bigr), \quad\beta \to0, $$ where $$\mathcal{C}(n,\beta):= \sup_{|B|=\beta}\sup_{T_n} \frac{ \|T_n \|_{L_\infty([0,2\pi ))}}{ \|T_n \|_{L_\infty ([0,2\pi) \setminus B )}}. $$ We also obtain sharp Nikol’skii’s inequalities for the Lorentz spaces and net spaces. Multidimensional variants of Remez and Nikol’skii’s inequalities are investigated. 相似文献
12.
J. M. Aldaz J. Pé rez Lá zaro 《Transactions of the American Mathematical Society》2007,359(5):2443-2461
We prove that if is of bounded variation, then the uncentered maximal function is absolutely continuous, and its derivative satisfies the sharp inequality . This allows us to obtain, under less regularity, versions of classical inequalities involving derivatives.
13.
Asymptotic Behavior in a Quasilinear Fully Parabolic Chemotaxis System with Indirect Signal Production and Logistic Source
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Dan Li & Zhongping Li 《偏微分方程(英文版)》2021,34(2):129-143
In this paper, we study the asymptotic behavior of solutions to a quasilinear
fully parabolic chemotaxis system with indirect signal production and logistic sourceunder homogeneous Neumann boundary conditions in a smooth bounded domain $Ω⊂\mathbb{R}^n$ $(n ≥1)$, where $b ≥0$, $γ ≥1$, $a_i ≥1$, $µ$, $b_i >0$ $(i =1,2)$, $D$, $S∈ C^2([0,∞))$ fulfilling $D(s) ≥ a_0(s+1)^{−α}$, $0 ≤ S(s) ≤ b_0(s+1)^β$ for all $s ≥ 0,$ where $a_0,b_0 > 0$ and $α,β ∈ \mathbb{R}$ are
constants. The purpose of this paper is to prove that if $b ≥ 0$ and $µ > 0$ sufficiently
large, the globally bounded solution $(u,v,w)$ with nonnegative initial data $(u_0,v_0,w_0)$ satisfies $$\Big\| u(·,t)− \Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\|_{L^∞(Ω)}+\Big\| v(·,t)−\frac{b_1b_2}{a_1a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)} +\Big\| w(·,t)−\frac{b_2}{a_2}\Big(\frac{b}{µ}\Big)^{\frac{1}{γ}}\Big\| _{L^∞(Ω)}→0$$ as $t→∞$. 相似文献
14.
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
15.
Natan Kruglyak Eric Setterqvist 《Proceedings of the American Mathematical Society》2008,136(7):2505-2513
It is shown that if we restrict the identity minus Hardy operator on the cone of nonnegative decreasing functions in , then we have the sharp estimate for In other words, for each and each integer . for all .
It is also shown, via a connection between the operator and Laguerre functions, that
16.
Let T be a singular integral operator, and let 0 < α < 1. If t > 0 and the functions f and Tf are both integrable, then there exists a function $g \in B_{Lip_\alpha } (ct)$ such that $\left\| {f - g} \right\|_{L^1 } \leqslant Cdist_{L^1 } (f,B_{Lip_\alpha } (t))$ and $\left\| {Tf - Tg} \right\|_{L^1 } \leqslant C\left\| {f - g} \right\|_{L^1 } + dist_{L^1 } (Tf,B_{Lip_\alpha } (t)).$ . (Here B X (τ) is the ball of radius τ and centered at zero in the space X; the constants C and c do not depend on t and f.) The function g is independent of T and is constructed starting with f by a nearly algorithmic procedure resembling the classical Calderón-Zygmund decomposition. 相似文献
17.
In this paper, we study the Holder regularity of weak solutions to the Dirichlet problem associated with the regional fractional Laplacian (-△)αΩ on a bounded open set Ω ■R(N ≥ 2) with C(1,1) boundary ■Ω. We prove that when f ∈ Lp(Ω), and g ∈ C(Ω), the following problem (-△)αΩu = f in Ω, u = g on ■Ω, admits a unique weak solution u ∈ W(α,2)(Ω) ∩ C(Ω),where p >N/2-2α and 1/2< α < 1. To solve this problem, we consider it into two special cases, i.e.,g ≡ 0 on ■Ω and f ≡ 0 in Ω. Finally, taking into account the preceding two cases, the general conclusion is drawn. 相似文献
18.
Christian Le Merdy 《Journal of Evolution Equations》2012,12(4):789-800
Let (T t ) t?≥ 0 be a bounded analytic semigroup on L p (Ω), with 1?<?p?<?∞. Let ?A denote its infinitesimal generator. It is known that if A and A * both satisfy square function estimates ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{\frac{1}{2}} T_t(x)\vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^p} \lesssim \|x\|_{L^p}}$ and ${\bigl\|\bigl(\int_{0}^{\infty} \vert A^{*\frac{1}{2}} T_t^*(y) \vert^2 {\rm d}t \bigr)^{\frac{1}{2}}\bigr\|_{L^{p^\prime}} \lesssim \|y\|_{L^{p^\prime}}}$ for ${x\in L^p(\Omega)}$ and ${y\in L^{p^\prime}(\Omega)}$ , then A admits a bounded ${H^{\infty}(\Sigma_\theta)}$ functional calculus for any ${\theta>\frac{\pi}{2}}$ . We show that this actually holds true for some ${\theta<\frac{\pi}{2}}$ . 相似文献
19.
Hernán Castro Hui Wang 《Calculus of Variations and Partial Differential Equations》2010,39(3-4):525-531
In this paper, we consider functions ${u\in W^{m,1}(0,1)}$ where m ≥ 2 and u(0) = Du(0) = · · · = D m-1 u(0) = 0. Although it is not true in general that ${\frac{D^ju(x)}{x^{m-j}} \in L^1(0,1)}$ for ${j\in \{0,1,\ldots,m-1\}}$ , we prove that ${\frac{D^ju(x)}{x^{m-j-k}} \in W^{k,1}(0,1)}$ if k ≥ 1 and 1 ≤ j + k ≤ m, with j, k integers. Furthermore, we have the following Hardy type inequality, $$\left\|{D^k\left({\frac{D^ju(x)}{x^{m-j-k}}}\right)}\right\|_{L^1(0,1)} \leq \frac {(k-1)!}{(m-j-1)!} \|{D^mu}\|_{L^1(0,1)},$$ where the constant is optimal. 相似文献
20.
Liu Qiao 《Journal of Applied Analysis & Computation》2014,4(4):355-365
We provide two regularity criteria for the weak solutions of the 3D micropolar fluid equations, the first one in terms of one directional derivative of the velocity, i.e., $\partial_{3}u$, while the second one is is in terms of the behavior of the direction of the velocity $\frac{u}{|u|}$. More precisely, we prove that if \begin{equation*} \partial_{3}u \in L^{\beta}(0,T;L^{\alpha}(\mathbb{R}^{3}))\quad\text{ with }\frac{2}{\beta}+\frac{3}{\alpha}\leq 1+\frac{1}{\alpha}, 2< \alpha \leq\infty, 2\leq\beta< \infty; \end{equation*} or \begin{equation*} \operatorname{div}\left(\frac{u}{|u|}\right)\in L^{\frac{4}{1-2r}}(0,T;\dot{X}_{r}(\mathbb{R}^{3}))\quad \text{ with } 0\leq r< \frac{1}{2}, \end{equation*} then the weak solution $(u(x,t),\omega(x,t))$ is regular on $\mathbb{R}^{3}\times [0,T]$. Here $\dot{X}_{r}(\mathbb{R}^{3})$ is the multiplier space. 相似文献