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1.
We investigate very weak solutions to the instationary Navier–Stokes system being contained in where is a bounded domain and . The chosen space of data is small enough to guarantee uniqueness of solutions and existence in case of small data or short
time intervals. On the other hand, the data space is large enough that every vector field in is a very weak solution for appropriate data. The solutions and the data depend continuously on each other.
相似文献
2.
Anna Maria Candela Giuliana Palmieri 《Calculus of Variations and Partial Differential Equations》2009,34(4):495-530
The aim of this paper is investigating the existence of one or more critical points of a family of functionals which generalizes
the model problem
in the Banach space , being Ω a bounded domain in . In order to use “classical” theorems, a suitable variant of condition (C) is proved and is decomposed according to a “good” sequence of finite dimensional subspaces.
The authors acknowledge the support of M.I.U.R. (research funds ex 40% and 60%). 相似文献
3.
Jérôme Droniou Juan-Luis Vázquez 《Calculus of Variations and Partial Differential Equations》2009,34(4):413-434
We study the existence and uniqueness of solutions of the convective–diffusive elliptic equation
posed in a bounded domain , with pure Neumann boundary conditions
Under the assumption that with p = N if N ≥ 3 (resp. p > 2 if N = 2), we prove that the problem has a solution if ∫Ω
f
dx = 0, and also that the kernel is generated by a function , unique up to a multiplicative constant, which satisfies a.e. on Ω. We also prove that the equation
has a unique solution for all ν > 0 and the map is an isomorphism of the respective spaces. The study is made in parallel with the dual problem, with equation
The dependence on the data is also examined, and we give applications to solutions of nonlinear elliptic PDE with measure
data and to parabolic problems. 相似文献
4.
Nicola Garofalo 《manuscripta mathematica》2008,126(3):353-373
We prove some new a priori estimates for H
2-convex functions which are zero on the boundary of a bounded smooth domain Ω in a Carnot group . Such estimates are global and are geometric in nature as they involve the horizontal mean curvature of ∂Ω. As a consequence of our bounds we show that if has step two, then for any smooth H
2-convex function in vanishing on ∂Ω one has
.
Supported in part by NSF Grant DMS-07010001. 相似文献
5.
Arrigo Cellina Mihai Vornicescu 《Calculus of Variations and Partial Differential Equations》2009,35(2):263-270
In this paper we establish an existence and regularity result for solutions to the problem
for boundary data that are constant on each connected component of the boundary of Ω. The Lagrangean L belongs to a class that contains both extended valued Lagrangeans and Lagrangeans with linear growth. Regularity means that
the solution is Lipschitz continuous and that, in addition, is bounded. 相似文献
6.
We prove regularity results for solutions to a class of quasilinear elliptic equations in divergence form in the Heisenberg
group . The model case is the non-degenerate p-Laplacean operator where , and p is not too far from 2. 相似文献
7.
Pigong Han Zhaoxia Liu 《Calculus of Variations and Partial Differential Equations》2007,30(3):315-352
Let Ω be an open bounded domain in with smooth boundary . We are concerned with the critical Neumann problem
where and Q(x) is a positive continuous function on . Using Moser iteration, we give an asymptotic characterization of solutions for (*) at the origin. Under some conditions
on Q, μ, we, by means of a variational method, prove that there exists such that for every , problem (*) has a positive solution and a pair of sign-changing solutions. 相似文献
8.
Consider the instationary Navier–Stokes system in a smooth bounded domain with vanishing force and initial value . Since the work of Kiselev and Ladyzhenskaya (Am. Math. Soc. Transl. Ser. 2 24:79–106, 1963) there have been found several
conditions on u
0 to prove the existence of a unique strong solution with u(0) = u
0 in some time interval [0, T), 0 < T ≤ ∞, where the exponents 2 < s < ∞, 3 < q < ∞ satisfy . Indeed, such conditions could be weakened step by step, thus enlarging the corresponding solution classes. Our aim is to
prove the following optimal result with the weakest possible initial value condition and the largest possible solution class:
Given u
0, q, s as above and the Stokes operator A
2, we prove that the condition is necessary and sufficient for the existence of such a local strong solution u. The proof rests on arguments from the recently developed theory of very weak solutions. 相似文献
9.
We consider the problem
where Ω is a bounded smooth domain in , 1 < p< + ∞ if N = 2, if N ≥ 3 and ε is a parameter. We show that if the mean curvature of ∂Ω is not constant then, for ε small enough, such a problem
has always a nodal solution u
ε with one positive peak and one negative peak on the boundary. Moreover, and converge to and , respectively, as ε goes to zero. Here, H denotes the mean curvature of ∂Ω.
Moreover, if Ω is a ball and , we prove that for ε small enough the problem has nodal solutions with two positive peaks on the boundary and arbitrarily
many negative peaks on the boundary.
The authors are supported by the M.I.U.R. National Project “Metodi variazionali e topologici nello studio di fenomeni non
lineari”. 相似文献
10.
The Stokes operator in weighted Lq-spaces II: weighted resolvent estimates and maximal Lp-regularity 总被引:1,自引:0,他引:1
Andreas Fr?hlich 《Mathematische Annalen》2007,339(2):287-316
In this paper we establish a general weighted L
q
-theory of the Stokes operator in the whole space, the half space and a bounded domain for general Muckenhoupt weights . We show weighted L
q
-estimates for the Stokes resolvent system in bounded domains for general Muckenhoupt weights. These weighted resolvent estimates
imply not only that the Stokes operator generates a bounded analytic semigroup but even yield the maximal L
p
-regularity of in the respective weighted L
q
-spaces for arbitrary Muckenhoupt weights . This conclusion is archived by combining a recent characterisation of maximal L
p
-regularity by -bounded families due to Weis [Operator-valued Fourier multiplier theorems and maximal L
p
-regularity. Preprint (1999)] with the fact that for L
q
-spaces -boundedness is implied by weighted estimates. 相似文献
11.
Jens Habermann 《Mathematische Zeitschrift》2008,258(2):427-462
For weak solutions of higher order systems of the type , for all , with variable growth exponent p : Ω → (1,∞) we prove that if with , then . We should note that we prove this implication both in the non-degenerate (μ > 0) and in the degenerate case (μ = 0). 相似文献
12.
Thomas Bartsch Shuangjie Peng Zhitao Zhang 《Calculus of Variations and Partial Differential Equations》2007,30(1):113-136
We investigate elliptic equations related to the Caffarelli–Kohn–Nirenberg inequalities: and such that . For various parameters α, β and various domains Ω, we establish some existence and non-existence results of solutions in
rather general, possibly degenerate or singular settings. 相似文献
13.
Let be open, let be the Dirac operator in and let be the Clifford algebra constructed over the quadratic space . If for fixed, denotes the space of r-vectors in , then an -valued smooth function W = W
r
+ W
r+2 in Ω is said to satisfy the Moisil-Théodoresco system if . In terms of differential forms, this means that the corresponding - valued smooth form w = w
r
+ w
r+2 satisfies in Ω the system d
*
w
r
= 0, dw
r
+ d
*
w
r+2 = 0; dw
r+2 = 0.
Based on techniques and results concerning conjugate harmonic functions in the framework of Clifford analysis, a structure
theorem is proved for the solutions of the Moisil-Théodoresco system.
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14.
Juan Dávila Manuel del Pino Monica Musso Juncheng Wei 《Calculus of Variations and Partial Differential Equations》2008,32(4):453-480
We consider the elliptic problem Δu + u
p
= 0, u > 0 in an exterior domain, under zero Dirichlet and vanishing conditions, where is smooth and bounded in , N ≥ 3, and p is supercritical, namely . We prove that this problem has infinitely many solutions with slow decay
at infinity. In addition, a solution with fast decay
O(|x|2-N
) exists if p is close enough from above to the critical exponent. 相似文献
15.
Zhaoli Liu Jiabao Su Zhi-Qiang Wang 《Calculus of Variations and Partial Differential Equations》2009,35(4):463-480
In this paper, we study existence of nontrivial solutions to the elliptic equation
and to the elliptic system
where Ω is a bounded domain in with smooth boundary ∂Ω, , f (x, 0) = 0, with m ≥ 2 and . Nontrivial solutions are obtained in the case in which the nonlinearities have linear growth. That is, for some c > 0, for and , and for and , where I
m
is the m × m identity matrix. In sharp contrast to the existing results in the literature, we do not make any assumptions at infinity
on the asymptotic behaviors of the nonlinearity f and .
Z. Liu was supported by NSFC(10825106, 10831005). J. Su was supported by NSFC(10831005), NSFB(1082004), BJJW-Project(KZ200810028013)
and the Doctoral Programme Foundation of NEM of China (20070028004). 相似文献
16.
Elena I. Kaikina Leonardo Guardado-Zavala Hector F. Ruiz-Paredes Jesus A. Mendez Navarro 《NoDEA : Nonlinear Differential Equations and Applications》2009,16(1):63-77
We study nonlinear nonlocal equations on a half-line in the critical case
where . The linear operator is a pseudodifferential operator defined by the inverse Laplace transform with dissipative symbol , the number . The aim of this paper is to prove the global existence of solutions to the inital-boundary value problem (0.1) and to find
the main term of the large time asymptotic representation of solutions in the critical case.
相似文献
17.
Consider a smooth bounded domain , and the Navier–Stokes system in with initial value and external force f = div F, where , are so-called Serrin exponents. It is an important question what is the optimal (weakest possible) initial value condition
in order to obtain a unique strong solution in some initial interval [0, T), . Up to now several sufficient conditions on u
0 are known which need not be necessary. Our main result, see Theorem 1.1, shows that the condition , A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if , , then any weak solution u in the usual sense does not satisfy Serrin’s condition for each 0 < T ≤ ∞.
相似文献
18.
The solvability in anisotropic spaces
, σ ∈ ℝ+, p, q ∈ (1, ∞), of the heat equation ut − Δu = f in ΩT ≡ (0, T) × Ω is studied under the boundary and initial conditions u = g on ST, u|t=0 = u0 in Ω, where S is the boundary of a bounded domain Ω ⊂ ℝn. The existence of a unique solution
of the above problem is proved under the assumptions that
and under some additional conditions on the data. The existence is proved by the technique of regularizers. For this purpose
the local-in-space solvability near the boundary and near an interior point of Ω is needed. To show the local-in-space existence,
the definition of Besov spaces by the dyadic decomposition of a partition of unity is used. This enables us to get an appropriate
estimate in a new and promising way without applying either the potential technique or the resolvent estimates or the interpolation.
Bibliography: 26 titles.
Published in Zapiski Nauchnykh Seminarov POMI, Vol. 348, 2007, pp. 40–97. 相似文献
19.
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
20.
Peer Christian Kunstmann 《Archiv der Mathematik》2008,91(2):178-186
We consider the Stokes operator A on unbounded domains of uniform C
1,1-type. Recently, it has been shown by Farwig, Kozono and Sohr that – A generates an analytic semigroup in the spaces , 1 < q < ∞, where for q ≥ 2 and for q ∈ (1, 2). Moreover, it was shown that A has maximal L
p
-regularity in these spaces for p ∈ (1,∞). In this paper we show that ɛ + A has a bounded H
∞-calculus in for all q ∈ (1, ∞) and ɛ > 0. This allows to identify domains of fractional powers of the Stokes operator.
Received: 12 October 2007 相似文献