On local-in-time existence for the Dirichlet problem for equations of compressible viscous fluids

Piotr Boguslaw Mucha; Wojciech Zajączkowski

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 $I = \sum_{i = 1}^{2} (| | \partial_{x ₃}^{i} {v (0) | |}_{L ₂ (Ω)} + | | \partial_{x ₃}^{i} {f | |}_{L ₂ (Ω \times (0, T))})$ 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 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 $r \in (1, 2]$ . In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for...

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 ${| | u | |}_{L^{s} (0, T; L^{q} (Ω))} < \infty$ 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.

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 $\partial Ω$ and datum $a$ in Lebesgue spaces. We prove existence of a solution for arbitrary values of the fluxes through the connected components of $\partial Ω$ , 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 $\{\begin{matrix} n_{t} + u \cdot \nabla n = Δ n - \nabla \cdot (n \nabla c) + \nabla \cdot (n \nabla φ), & x \in Ω, t > 0, \\ c_{t} + u \cdot \nabla c = Δ c - n c, & x \in Ω, t > 0, \\ u_{t} + κ (u \cdot \nabla) u + \nabla P = Δ u - n \nabla φ + n \nabla c, & x \in Ω, t > 0, \\ \nabla \cdot u = 0, & x \in Ω, t > 0, \end{matrix}$ is considered under no-flux boundary conditions for $n, c$ and the Dirichlet boundary condition for $u$ on a bounded smooth domain $Ω \subset ℝ^{N}$ $(N = 2, 3)$ , $κ \in {0, 1}$ . The existence of global bounded classical solutions is proved under a smallness assumption on $∥ c_{0} ∥_{L^{\infty} (Ω)}$ . 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...

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 $𝒪 \subset R^{d}, d \geq 2$ , around an unstable equilibrium solution is exponentially stabilizable in probability by an internal noise controller $V (t, ξ) = \sum_{i = 1}^{N} V_{i} (t) ψ_{i} (ξ) {\dot{β}}_{i} (t)$ , $ξ \in 𝒪$ , 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 $𝒪_{0} \subset 𝒪$ . 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 $W_{p}^{2, 1} (Ω \times (0, T))$ , and pressure belongs to $W_{p}^{1, 0} (Ω \times (0, T))$ 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 $W_{5 / 2}^{2, 1} (Ω \times (0, T))$ and the gradient of the pressure to $L_{5 / 2} (Ω \times (0, T))$ . 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 $Σ \subset ℝ^{n - 1}$ , a bounded domain of class $C^{1, 1}$ , are obtained in the space $L^{q} (ℝ; L ² (Σ))$ , 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 $W_{4 / 3}^{2, 1} (Ω \times (0, T))$ , 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 $ℝ^{3}$ , associated with sequences of initial data bounded in ${\dot{H}}^{1 / 2}$ . 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 ${\dot{H}}^{1 / 2}$ , 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 $W_{r}^{2, 1} (ℝ ³ \times [k, k + 1])$ 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 $W_{r}^{2 - 2 / r} (ℝ ³)$ -norm of the perturbing initial data or smallness...

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 $f \in L_{2, - μ}$ , $v ₀ \in H_{- μ} ¹$ , 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 $f (s) = \sum_{n = 1}^{\infty} a ₙ n^{- s}$ with ${| | f | |}_{\infty} : = s u p_{ℜ s > 0} | f (s) | < \infty$ , then $\sum_{n = 1}^{\infty} | a ₙ | n^{- 2} {\leq | | f | |}_{\infty}$ and even slightly better, and $\sum_{n = 1}^{\infty} | a ₙ | n^{- 1 / 2} {\leq C | | f | |}_{\infty}$ , 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 ⎧ $Δ (| Δ u_{i} |^{p - 2} Δ u_{i}) = λ F_{u_{i}} (x, u ₁, \dots, u ₙ)$ 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.

A uniqueness theorem for viscous flows on exterior domains with summability assumptions on the gradient of pressure.

Giovanni P. Galdi, Paolo Maremonti (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In questa Nota si fornisce un teorema di unicità per soluzioni regolari delle equazioni di Navier-Stokes in domini esterni. Tale teorema non richiede che le velocità tendano ad un prefissato limite all'infinito, mentre il gradiente di pressione è supposto essere di $q$ -ma potenza sommabile nel cilindro spazio-temporale $(q\in(1,\infty))$ . Questo risultato non può essere ulteriormente generalizzato al caso $q=\infty$ , a causa di noti controesempi.

Long time existence of solutions to 2d Navier-Stokes equations with heat convection

Jolanta Socała, Wojciech M. Zajączkowski (2009)

Applicationes Mathematicae

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Global existence of regular solutions to the Navier-Stokes equations for (v,p) coupled with the heat convection equation for θ is proved in the two-dimensional case in a bounded domain. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First an appropriate estimate is shown and next the existence is proved by the Leray-Schauder fixed point theorem. We prove the existence of solutions such that $v, θ \in W_{s}^{2, 1} (Ω^{T})$ , $\nabla p \in L_{s} (Ω^{T})$ , s>2.

Cauchy problem for the non-newtonian viscous incompressible fluid

Milan Pokorný (1996)

Applications of Mathematics

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We study the Cauchy problem for the non-Newtonian incompressible fluid with the viscous part of the stress tensor $τ^{V} (𝕖) = τ (𝕖) - 2 μ_{1} Δ 𝕖$ , where the nonlinear function $τ (𝕖)$ satisfies $τ_{i j} (𝕖) e_{i j} \geq c {| 𝕖 |}^{p}$ or $τ_{i j} (𝕖) e_{i j} \geq {c (| 𝕖 |}^{2} + {| 𝕖 |}^{p})$ . First, the model for the bipolar fluid is studied and existence, uniqueness and regularity of the weak solution is proved for $p > 1$ for both models. Then, under vanishing higher viscosity $μ_{1}$ , the Cauchy problem for the monopolar fluid is considered. For the first model the existence of the weak solution is proved for $p > \frac{3 n}{n + 2}$ , its uniqueness...

Optimal convergence results for the Brezzi-Pitkäranta approximation of the Stokes problem: Exterior domains

Serguei A. Nazarov, Maria Specovius-Neugebauer (2008)

Banach Center Publications

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This paper deals with a strongly elliptic perturbation for the Stokes equation in exterior three-dimensional domains Ω with smooth boundary. The continuity equation is substituted by the equation -ε²Δp + div u = 0, and a Neumann boundary condition for the pressure is added. Using parameter dependent Sobolev norms, for bounded domains and for sufficiently smooth data we prove $H^{5 / 2 - δ}$ convergence for the velocity part and $H^{3 / 2 - δ}$ convergence for the pressure to the solution of the Stokes problem,...

A direct proof of the Caffarelli-Kohn-Nirenberg theorem

Jörg Wolf (2008)

Banach Center Publications

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In the present paper we give a new proof of the Caffarelli-Kohn-Nirenberg theorem based on a direct approach. Given a pair (u,p) of suitable weak solutions to the Navier-Stokes equations in ℝ³ × ]0,∞[ the velocity field u satisfies the following property of partial regularity: The velocity u is Lipschitz continuous in a neighbourhood of a point (x₀,t₀) ∈ Ω × ]0,∞ [ if $l i m s u p_{R \to 0 ⁺} 1 / R \int_{Q_{R} (x ₀, t ₀)} | c u r l u \times u / | u | | ² d x d t \leq ε_{*}$ for a sufficiently small $ε_{*} > 0$ .

A uniqueness theorem for viscous flows on exterior domains with summability assumptions on the gradient of pressure.

Giovanni P. Galdi, Paolo Maremonti (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

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On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem

Hugo Beirão da Veiga (2009)

Journal of the European Mathematical Society

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We establish regularity results up to the boundary for solutions to generalized Stokes and Navier–Stokes systems of equations in the stationary and evolutive cases. Generalized here means the presence of a shear dependent viscosity. We treat the case $p \geq 2$ . Actually, we are interested in proving regularity results in $L^{q} (Ω)$ spaces for all the second order derivatives of the velocity and all the first order derivatives of the pressure. The main aim of the present paper is to extend our previous...

Existence of three solutions to a double eigenvalue problem for the p-biharmonic equation

Lin Li, Shapour Heidarkhani (2012)

Annales Polonici Mathematici

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Using a three critical points theorem and variational methods, we study the existence of at least three weak solutions of the Navier problem ⎧ ${Δ (| Δ u |}^{p - 2} {Δ u) - d i v (| \nabla u |}^{p - 2} \nabla u) = λ f (x, u) + μ g (x, u)$ in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where $Ω \subset ℝ^{N}$ (N ≥ 1) is a non-empty bounded open set with a sufficiently smooth boundary ∂Ω, λ > 0, μ > 0 and f,g: Ω × ℝ → ℝ are two L¹-Carathéodory functions.

The comparison principle and Dirichlet problem in the class $_{p} (f)$ , p > 0

Pham Hoang Hiep (2006)

Annales Polonici Mathematici

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We establish the comparison principle in the class $_{p} (f)$ . The result obtained is applied to the Dirichlet problem in $_{p} (f)$ .