Displaying similar documents to “On the existence of solutions for the nonstationary Stokes system with slip boundary conditions in general Sobolev-Slobodetskii and Besov spaces”

On the existence of steady-state solutions to the Navier-Stokes system for large fluxes

Antonio Russo, Giulio Starita (2008)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Similarity:

In this paper we deal with the stationary Navier-Stokes problem in a domain Ω with compact Lipschitz boundary Ω and datum a in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of Ω , with possible countable exceptional set, provided a is the sum of the gradient of a harmonic function and a sufficiently small field, with zero total flux for Ω bounded.

Asymptotic behavior of small-data solutions to a Keller-Segel-Navier-Stokes system with indirect signal production

Lu Yang, Xi Liu, Zhibo Hou (2023)

Czechoslovak Mathematical Journal

Similarity:

We consider the Keller-Segel-Navier-Stokes system n t + 𝐮 · n = Δ n - · ( n v ) , x Ω , t > 0 , v t + 𝐮 · v = Δ v - v + w , x Ω , t > 0 , w t + 𝐮 · w = Δ w - w + n , x Ω , t > 0 , 𝐮 t + ( 𝐮 · ) 𝐮 = Δ 𝐮 + P + n φ , · 𝐮 = 0 , x Ω , t > 0 , which is considered in bounded domain Ω N ( N { 2 , 3 } ) with smooth boundary, where φ C 1 + δ ( Ω ¯ ) with δ ( 0 , 1 ) . We show that if the initial data n 0 L N / 2 ( Ω ) , v 0 L N ( Ω ) , w 0 L N ( Ω ) and 𝐮 0 L N ( Ω ) is small enough, an associated initial-boundary value problem possesses a global classical solution which decays to the constant state ( n ¯ 0 , n ¯ 0 , n ¯ 0 , 0 ) exponentially with n ¯ 0 : = ( 1 / | Ω | ) Ω n 0 ( x ) d x .

A blow-up criterion for the strong solutions to the nonhomogeneous Navier-Stokes-Korteweg equations in dimension three

Huanyuan Li (2021)

Applications of Mathematics

Similarity:

This paper proves a Serrin’s type blow-up criterion for the 3D density-dependent Navier-Stokes-Korteweg equations with vacuum. It is shown that if the density ρ and velocity field u satisfy ρ L ( 0 , T ; W 1 , q ) + u L s ( 0 , T ; L ω r ) < for some q > 3 and any ( r , s ) satisfying 2 / s + 3 / r 1 , 3 < r , then the strong solutions to the density-dependent Navier-Stokes-Korteweg equations can exist globally over [ 0 , T ] . Here L ω r denotes the weak L r space.

Profile decomposition for solutions of the Navier-Stokes equations

Isabelle Gallagher (2001)

Bulletin de la Société Mathématique de France

Similarity:

We consider sequences of solutions of the Navier-Stokes equations in  3 , associated with sequences of initial data bounded in  H ˙ 1 / 2 . We prove, in the spirit of the work of H.Bahouri and P.Gérard (in the case of the wave equation), that they can be decomposed into a sum of orthogonal profiles, bounded in  H ˙ 1 / 2 , up to a remainder term small in  L 3 ; the method is based on the proof of a similar result for the heat equation, followed by a perturbation–type argument. If  𝒜 is an “admissible” space (in...

A short note on L q theory for Stokes problem with a pressure-dependent viscosity

Václav Mácha (2016)

Czechoslovak Mathematical Journal

Similarity:

We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on p and on the symmetric part of a gradient of u , namely, it is represented by a stress tensor T ( D u , p ) : = ν ( p , | D | 2 ) D which satisfies r -growth condition with r ( 1 , 2 ] . In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for...

The maximum regularity property of the steady Stokes problem associated with a flow through a profile cascade in L r -framework

Tomáš Neustupa (2023)

Applications of Mathematics

Similarity:

We deal with the steady Stokes problem, associated with a flow of a viscous incompressible fluid through a spatially periodic profile cascade. Using the reduction to domain Ω , which represents one spatial period, the problem is formulated by means of boundary conditions of three types: the conditions of periodicity on curves Γ - and Γ + (lower and upper parts of Ω ), the Dirichlet boundary conditions on Γ in (the inflow) and Γ 0 (boundary of the profile) and an artificial “do nothing”-type boundary...

Real method of interpolation on subcouples of codimension one

S. V. Astashkin, P. Sunehag (2008)

Studia Mathematica

Similarity:

We find necessary and sufficient conditions under which the norms of the interpolation spaces ( N , N ) θ , q and ( X , X ) θ , q are equivalent on N, where N is the kernel of a nonzero functional ψ ∈ (X₀ ∩ X₁)* and N i is the normed space N with the norm inherited from X i (i = 0,1). Our proof is based on reducing the problem to its partial case studied by Ivanov and Kalton, where ψ is bounded on one of the endpoint spaces. As an application we completely resolve the problem of when the range of the operator T θ = S - 2 θ I (S...

Existence of solutions to the nonstationary Stokes system in H - μ 2 , 1 , μ ∈ (0,1), in a domain with a distinguished axis. Part 2. Estimate in the 3d case

W. M. Zajączkowski (2007)

Applicationes Mathematicae

Similarity:

We examine the regularity of solutions to the Stokes system in a neighbourhood of the distinguished axis under the assumptions that the initial velocity v₀ and the external force f belong to some weighted Sobolev spaces. It is assumed that the weight is the (-μ )th power of the distance to the axis. Let f L 2 , - μ , v H - μ ¹ , μ ∈ (0,1). We prove an estimate of the velocity in the H - μ 2 , 1 norm and of the gradient of the pressure in the norm of L 2 , - μ . We apply the Fourier transform with respect to the variable along...

An L q ( L ² ) -theory of the generalized Stokes resolvent system in infinite cylinders

Reinhard Farwig, Myong-Hwan Ri (2007)

Studia Mathematica

Similarity:

Estimates of the generalized Stokes resolvent system, i.e. with prescribed divergence, in an infinite cylinder Ω = Σ × ℝ with Σ n - 1 , a bounded domain of class C 1 , 1 , are obtained in the space L q ( ; L ² ( Σ ) ) , q ∈ (1,∞). As a preparation, spectral decompositions of vector-valued homogeneous Sobolev spaces are studied. The main theorem is proved using the techniques of Schauder decompositions, operator-valued multiplier functions and R-boundedness of operator families.

Interpolation and duality of generalized grand Morrey spaces on quasi-metric measure spaces

Yi Liu, Wen Yuan (2017)

Czechoslovak Mathematical Journal

Similarity:

Let θ ( 0 , 1 ) , λ [ 0 , 1 ) and p , p 0 , p 1 ( 1 , ] be such that ( 1 - θ ) / p 0 + θ / p 1 = 1 / p , and let ϕ , ϕ 0 , ϕ 1 be some admissible functions such that ϕ , ϕ 0 p / p 0 and ϕ 1 p / p 1 are equivalent. We first prove that, via the ± interpolation method, the interpolation L ϕ 0 p 0 ) , λ ( 𝒳 ) , L ϕ 1 p 1 ) , λ ( 𝒳 ) , θ of two generalized grand Morrey spaces on a quasi-metric measure space 𝒳 is the generalized grand Morrey space L ϕ p ) , λ ( 𝒳 ) . Then, by using block functions, we also find a predual space of the generalized grand Morrey space. These results are new even for generalized grand Lebesgue spaces.

Some Hölder-logarithmic estimates on Hardy-Sobolev spaces

Imed Feki, Ameni Massoudi (2024)

Czechoslovak Mathematical Journal

Similarity:

We prove some optimal estimates of Hölder-logarithmic type in the Hardy-Sobolev spaces H k , p ( G ) , where k * , 1 p and G is either the open unit disk 𝔻 or the annular domain G s , 0 < s < 1 of the complex space . More precisely, we study the behavior on the interior of G of any function f belonging to the unit ball of the Hardy-Sobolev spaces H k , p ( G ) from its behavior on any open connected subset I of the boundary G of G with respect to the L 1 -norm. Our results can be viewed as an improvement and generalization of...

𝒞 k -regularity for the ¯ -equation with a support condition

Shaban Khidr, Osama Abdelkader (2017)

Czechoslovak Mathematical Journal

Similarity:

Let D be a 𝒞 d q -convex intersection, d 2 , 0 q n - 1 , in a complex manifold X of complex dimension n , n 2 , and let E be a holomorphic vector bundle of rank N over X . In this paper, 𝒞 k -estimates, k = 2 , 3 , , , for solutions to the ¯ -equation with small loss of smoothness are obtained for E -valued ( 0 , s ) -forms on D when n - q s n . In addition, we solve the ¯ -equation with a support condition in 𝒞 k -spaces. More precisely, we prove that for a ¯ -closed form f in 𝒞 0 , q k ( X D , E ) , 1 q n - 2 , n 3 , with compact support and for ε with 0 < ε < 1 there...

Fourier approximation and embeddings of Sobolev spaces

D. E. Edmunds, V. B. Moscatelli

Similarity:

CONTENTSIntroduction............................................................................................................ 51. Preliminaries............................................................................................................. 82. Embedding into W m , p ( Ω ) into L S ( Ω ) (n>1).......................................... 103. The case n = 1.......................................................................................................... 284. Embedding W m , p ( Ω ) into L φ ( Ω ) ...............................................................

C * -points vs P -points and P -points

Jorge Martinez, Warren Wm. McGovern (2022)

Commentationes Mathematicae Universitatis Carolinae

Similarity:

In a Tychonoff space X , the point p X is called a C * -point if every real-valued continuous function on C { p } can be extended continuously to p . Every point in an extremally disconnected space is a C * -point. A classic example is the space 𝐖 * = ω 1 + 1 consisting of the countable ordinals together with ω 1 . The point ω 1 is known to be a C * -point as well as a P -point. We supply a characterization of C * -points in totally ordered spaces. The remainder of our time is aimed at studying when a point in a product space...