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1
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On dynamics of fluids in meteorology

100%
Poul L.
Open Mathematics
|
2008
|
tom 6
|
nr 3
422-438
EN
We consider the full Navier-Stokes-Fourier system of equations on an unbounded domain with prescribed nonvanishing boundary conditions for the density and temperature at infinity. The topic of this article continues author’s previous works on existence of the Navier-Stokes-Fourier system on nonsmooth domains. The procedure deeply relies on the techniques developed by Feireisl and others in the series of works on compressible, viscous and heat conducting fluids.
2
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On the solvability of Dirichlet problem for the weighted p-Laplacian

100%
Mielczarek D., Rydlewski J., Szlachtowska E.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
|
2014
|
tom 34
|
nr 1
89-103
EN
In this paper we are concerned with the existence and uniqueness of the weak solution for the weighted p-Laplacian. The purpose of this paper is to discuss in some depth the problem of solvability of Dirichlet problem, therefore all proofs are contained in some detail. The main result of the work is the existence and uniqueness of the weak solution for the Dirichlet problem provided that the weights are bounded. Furthermore, under this assumption the solution belongs to the Sobolev space $W₀^{1,p}(Ω)$.
3
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On theorems for weak solutions of nonlinear differential equations with and without delay in Banach spaces

100%
Gomaa A. M.
Commentationes Mathematicae
|
2007
|
tom 47
|
nr 2
EN
In the present work we give an existence theorem for bounded weak solution of the differential equation \[ \dot{x}(t) = A(t)x(t) + f (t, x(t)),\quad t \geq 0 \] where \(\{A(t) : t \in I\mathbb{R}^+ \}\) is a family of linear operators from a Banach space \(E\) into itself, \(B_r = \{x \in E : \|x\| \leq r\}\) and \(f \colon \mathbb{R}^+ \times B_r \to E\) is weakly-weakly continuous. Furthermore, we give existence theorem for the differential equation with delay \[ \dot{x}(t) = \hat{A}(t) x(t) + f^d (t, θ_t x)\quad \text{if}\ t \in [0, T], \] where \(T, d \gt 0\), \(C_{B_r} ([-d, 0])\) is the Banach space of continuous functions from \([-d, 0]\) into \(B_r\), \(f_d\colon [0, T] \times C_{B_r} ([-d, 0]) \to E\) weakly-weakly continuous function, \(\hat{A}(t)\colon [0,T] \to L(E)\) is strongly measurable and Bochner integrable operator on \([0,T]\) and \(θ_t x(s) = x(t + s)\) for all \(s \in [-d, 0]\).
4
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Nontrivial critical points of asymptotically quadratic functions at resonances

100%
Fečkan M.
Annales Polonici Mathematici
|
1997
|
tom 67
|
nr 1
43-57
EN
Asymptotically quadratic functions defined on Hilbert spaces are studied by using some results of the theory of Morse-Conley index. Applications are given to existence of nontrivial weak solutions for asymptotically linear elliptic partial and ordinary differential equations at resonances.
5
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On the local Cauchy problem for nonlinear hyperbolic functional differential equations

100%
Człapiński T.
Annales Polonici Mathematici
|
1997
|
tom 67
|
nr 3
215-232
EN
We consider the local initial value problem for the hyperbolic partial functional differential equation of the first order (1) $Dₓz(x,y) = f(x,y,z(x,y),(Wz)(x,y),D_y z(x,y))$ on E, (2) z(x,y) = ϕ(x,y) on [-τ₀,0]×[-b,b], where E is the Haar pyramid and τ₀ ∈ ℝ₊, b = (b₁,...,bₙ) ∈ ℝⁿ₊. Using the method of bicharacteristics and the method of successive approximations for a certain functional integral system we prove, under suitable assumptions, a theorem on the local existence of weak solutions of the problem (1),(2).
6
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Weak solutions of stochastic differential inclusions and their compactness

88%
Michta M.
Discussiones Mathematicae, Differential Inclusions, Control and Optimization
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2009
|
tom 29
|
nr 1
91-106
EN
In this paper, we consider weak solutions to stochastic inclusions driven by a semimartingale and a martingale problem formulated for such inclusions. Using this we analyze compactness of the set of solutions. The paper extends some earlier results known for stochastic differential inclusions driven by a diffusion process.
7
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The Behaviour of Weak Solutions of Boundary Value Problems for Linear Elliptic Second Order Equations in Unbounded Cone-Like Domains

88%
Wiśniewski D.
Annales Mathematicae Silesianae
|
2016
|
tom 30
|
nr 1
203-217
EN
We investigate the behaviour of weak solutions of boundary value problems (Dirichlet, Neumann, Robin and mixed) for linear elliptic divergence second order equations in domains extending to infinity along a cone. We find an exponent of the solution decreasing rate: we derive the estimate of the weak solution modulus for our problems near the infinity under assumption that leading coefficients of the equations do not satisfy the Dini-continuity condition.
8
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Asymptotic of weak solutions for linear elliptic nonlocal Robin problem without Dini-continuity condition in a plane angle domain

75%
Żyjewski K.
Commentationes Mathematicae
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2017
|
tom 57
|
nr 1
EN
We investigate the behaviour of weak solutions to the nonlocal Robin problem for linear elliptic divergence second order equations in a neighbourhood of the boundary corner point. We find the exponent of the solution decreasing rate under the assumption that the leading coefficients of the equations do not satisfy the Dini-continuity condition.
9
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On weak solutions of functional-differential abstract nonlocal Cauchy problems

63%
Byszewski L.
Annales Polonici Mathematici
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1996-1997
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tom 65
|
nr 2
163-170
EN
The existence, uniqueness and asymptotic stability of weak solutions of functional-differential abstract nonlocal Cauchy problems in a Banach space are studied. Methods of m-accretive operators and the Banach contraction theorem are applied.
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