Displaying similar documents to “On local-in-time existence for the Dirichlet problem for equations of compressible viscous fluids”

Long time existence of regular solutions to Navier-Stokes equations in cylindrical domains under boundary slip conditions

W. M. Zajączkowski (2005)

Studia Mathematica

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Long time existence of solutions to the Navier-Stokes equations in cylindrical domains under boundary slip conditions is proved. Moreover, the existence of solutions with no restrictions on the magnitude of the initial velocity and the external force is shown. However, we have to assume that the quantity I = i = 1 2 ( | | x i v ( 0 ) | | L ( Ω ) + | | x i f | | L ( Ω × ( 0 , T ) ) ) is sufficiently small, where x₃ is the coordinate along the axis parallel to the cylinder. The time of existence is inversely proportional to I. Existence of solutions is proved by...

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

Huanyuan Li (2021)

Applications of Mathematics

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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.

Remarks on the a priori bound for the vorticity of the axisymmetric Navier-Stokes equations

Zujin Zhang, Chenxuan Tong (2022)

Applications of Mathematics

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We study the axisymmetric Navier-Stokes equations. In 2010, Loftus-Zhang used a refined test function and re-scaling scheme, and showed that | ω r ( x , t ) | + | ω z ( r , t ) | C r 10 , 0 < r 1 2 . By employing the dimension reduction technique by Lei-Navas-Zhang, and analyzing ω r , ω z and ω θ / r on different hollow cylinders, we are able to improve it and obtain | ω r ( x , t ) | + | ω z ( r , t ) | C | ln r | r 17 / 2 , 0 < r 1 2 .

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

Václav Mácha (2016)

Czechoslovak Mathematical Journal

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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...

Numerical analysis of a Stokes interface problem based on formulation using the characteristic function

Yoshiki Sugitani (2017)

Applications of Mathematics

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Numerical analysis of a model Stokes interface problem with the homogeneous Dirichlet boundary condition is considered. The interface condition is interpreted as an additional singular force field to the Stokes equations using the characteristic function. The finite element method is applied after introducing a regularization of the singular source term. Consequently, the error is divided into the regularization and discretization parts which are studied separately. As a result, error...

Criteria of local in time regularity of the Navier-Stokes equations beyond Serrin's condition

Reinhard Farwig, Hideo Kozono, Hermann Sohr (2008)

Banach Center Publications

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Let u be a weak solution of the Navier-Stokes equations in a smooth bounded domain Ω ⊆ ℝ³ and a time interval [0,T), 0 < T ≤ ∞, with initial value u₀, external force f = div F, and viscosity ν > 0. As is well known, global regularity of u for general u₀ and f is an unsolved problem unless we pose additional assumptions on u₀ or on the solution u itself such as Serrin’s condition | | u | | L s ( 0 , T ; L q ( Ω ) ) < where 2/s + 3/q = 1. In the present paper we prove several local and global regularity properties...

On the existence for the Dirichlet problem for the compressible linearized Navier-Stokes system in the L p -framework

Piotr Boguslaw Mucha, Wojciech Zajączkowski (2002)

Annales Polonici Mathematici

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The existence of solutions to the Dirichlet problem for the compressible linearized Navier-Stokes system is proved in a class such that the velocity vector belongs to W r 2 , 1 with r > 3. The proof is done in two steps. First the existence for local problems with constant coefficients is proved by applying the Fourier transform. Next by applying the regularizer technique the existence in a bounded domain is shown.

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

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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...

Remarks on regularity criteria for the Navier-Stokes equations with axisymmetric data

Zujin Zhang (2016)

Annales Polonici Mathematici

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We consider the axisymmetric Navier-Stokes equations with non-zero swirl component. By invoking the Hardy-Sobolev interpolation inequality, Hardy inequality and the theory of * A β (1 < β < ∞) weights, we establish regularity criteria involving u r , ω z or ω θ in some weighted Lebesgue spaces. This improves many previous results.

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

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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.

Global classical solutions in a self-consistent chemotaxis(-Navier)-Stokes system

Yanjiang Li, Zhongqing Yu, Yumei Huang (2024)

Czechoslovak Mathematical Journal

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The self-consistent chemotaxis-fluid system n t + u · n = Δ n - · ( n c ) + · ( n φ ) , x Ω , t > 0 , c t + u · c = Δ c - n c , x Ω , t > 0 , u t + κ ( u · ) u + P = Δ u - n φ + n c , x Ω , t > 0 , · u = 0 , x Ω , t > 0 , is considered under no-flux boundary conditions for n , c and the Dirichlet boundary condition for u on a bounded smooth domain Ω N ( N = 2 , 3 ) , κ { 0 , 1 } . The existence of global bounded classical solutions is proved under a smallness assumption on c 0 L ( Ω ) . Both the effect of gravity (potential force) on cells and the effect of the chemotactic force on fluid are considered here, and thus the coupling is stronger than the most studied chemotaxis-fluid...

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

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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 .

The internal stabilization by noise of the linearized Navier-Stokes equation

Viorel Barbu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

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One shows that the linearized Navier-Stokes equation in 𝒪 R d , d 2 , around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller V ( t , ξ ) = i = 1 N V i ( t ) ψ i ( ξ ) β ˙ i ( t ) , ξ 𝒪 , where { β i } i = 1 N are independent Brownian motions in a probability space and { ψ i } i = 1 N is a system of functions on 𝒪 with support in an arbitrary open subset 𝒪 0 𝒪 . The stochastic control input { V i } i = 1 N is found in feedback form. One constructs also a tangential boundary noise controller which exponentially stabilizes in probability...

On the existence of solutions for the nonstationary Stokes system with slip boundary conditions in general Sobolev-Slobodetskii and Besov spaces

Wisam Alame (2005)

Banach Center Publications

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We prove the existence of solutions to the evolutionary Stokes system in a bounded domain Ω ⊂ ℝ³. The main result shows that the velocity belongs either to W p 2 s + 2 , s + 1 ( Ω T ) or to B p , q 2 s + 2 , s + 1 ( Ω T ) with p > 3 and s ∈ ℝ₊ ∪ 0. The proof is divided into two steps. First the existence in W p 2 k + 2 , k + 1 for k ∈ ℕ is proved. Next applying interpolation theory the existence in Besov spaces in a half space is shown. Finally the technique of regularizers implies the existence in a bounded domain. The result is generalized to the spaces...

On existence of solutions for the nonstationary Stokes system with boundary slip conditions

Wisam Alame (2005)

Applicationes Mathematicae

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Existence of solutions for equations of the nonstationary Stokes system in a bounded domain Ω ⊂ ℝ³ is proved in a class such that velocity belongs to W p 2 , 1 ( Ω × ( 0 , T ) ) , and pressure belongs to W p 1 , 0 ( Ω × ( 0 , T ) ) for p > 3. The proof is divided into three steps. First, the existence of solutions with vanishing initial data is proved in a half-space by applying the Marcinkiewicz multiplier theorem. Next, we prove the existence of weak solutions in a bounded domain and then we regularize them. Finally, the problem with...

Global regular solutions to the Navier-Stokes equations in a cylinder

Wojciech M. Zajączkowski (2006)

Banach Center Publications

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The existence and uniqueness of solutions to the Navier-Stokes equations in a cylinder Ω and with boundary slip conditions is proved. Assuming that the azimuthal derivative of cylindrical coordinates and azimuthal coordinate of the initial velocity and the external force are sufficiently small we prove long time existence of regular solutions such that the velocity belongs to W 5 / 2 2 , 1 ( Ω × ( 0 , T ) ) and the gradient of the pressure to L 5 / 2 ( Ω × ( 0 , T ) ) . We prove the existence of solutions without any restrictions on the...

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

Reinhard Farwig, Myong-Hwan Ri (2007)

Studia Mathematica

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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.

Global existence of axially symmetric solutions to Navier-Stokes equations with large angular component of velocity

Wojciech M. Zajączkowski (2004)

Colloquium Mathematicae

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Global existence of axially symmetric solutions to the Navier-Stokes equations in a cylinder with the axis of symmetry removed is proved. The solutions satisfy the ideal slip conditions on the boundary. We underline that there is no restriction on the angular component of velocity. We obtain two kinds of existence results. First, under assumptions necessary for the existence of weak solutions, we prove that the velocity belongs to W 4 / 3 2 , 1 ( Ω × ( 0 , T ) ) , so it satisfies the Serrin condition. Next, increasing...

Profile decomposition for solutions of the Navier-Stokes equations

Isabelle Gallagher (2001)

Bulletin de la Société Mathématique de France

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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...

Stability of Constant Solutions to the Navier-Stokes System in ℝ³

Piotr Bogusław Mucha (2001)

Applicationes Mathematicae

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The paper examines the initial value problem for the Navier-Stokes system of viscous incompressible fluids in the three-dimensional space. We prove stability of regular solutions which tend to constant flows sufficiently fast. We show that a perturbation of a regular solution is bounded in W r 2 , 1 ( ³ × [ k , k + 1 ] ) for k ∈ ℕ. The result is obtained under the assumption of smallness of the L₂-norm of the perturbing initial data. We do not assume smallness of the W r 2 - 2 / r ( ³ ) -norm of the perturbing initial data or smallness...

Time regularity of generalized Navier-Stokes equation with p ( x , t ) -power law

Cholmin Sin (2023)

Czechoslovak Mathematical Journal

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We show time regularity of weak solutions for unsteady motion equations of generalized Newtonian fluids described by p ( x , t ) -power law for p ( x , t ) ( 3 n + 2 ) / ( n + 2 ) , n 2 , by using a higher integrability property and fractional difference method. Moreover, as its application we prove that every weak solution to the problem becomes a local in time strong solution and that it is unique.

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

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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...

The Bohr inequality for ordinary Dirichlet series

R. Balasubramanian, B. Calado, H. Queffélec (2006)

Studia Mathematica

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We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if f ( s ) = n = 1 a n - s with | | f | | : = s u p s > 0 | f ( s ) | < , then n = 1 | a | n - 2 | | f | | and even slightly better, and n = 1 | a | n - 1 / 2 C | | f | | , C being an absolute constant.

Existence of three solutions for a class of (p₁,...,pₙ)-biharmonic systems with Navier boundary conditions

Shapour Heidarkhani, Yu Tian, Chun-Lei Tang (2012)

Annales Polonici Mathematici

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We establish the existence of at least three weak solutions for the (p1,…,pₙ)-biharmonic system ⎧ Δ ( | Δ u i | p 2 Δ u i ) = λ F u i ( x , u , , u ) in Ω, ⎨ ⎩ u i = Δ u i = 0 on ∂Ω, for 1 ≤ i ≤ n. The proof is based on a recent three critical points theorem.