A direct proof of the Caffarelli-Kohn-Nirenberg theorem

Jörg Wolf

Displaying similar documents to “A direct proof of the Caffarelli-Kohn-Nirenberg theorem”

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 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...

Serrin-type regularity criterion for the Navier-Stokes equations involving one velocity and one vorticity component

Zujin Zhang (2018)

Czechoslovak Mathematical Journal

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We consider the Cauchy problem for the three-dimensional Navier-Stokes equations, and provide an optimal regularity criterion in terms of $u_{3}$ and $ω_{3}$ , which are the third components of the velocity and vorticity, respectively. This gives an affirmative answer to an open problem in the paper by P. Penel, M. Pokorný (2004).

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...

Uniqueness of weak solutions to a Keller-Segel-Navier-Stokes model with a logistic source

Miaochao Chen, Shengqi Lu, Qilin Liu (2022)

Applications of Mathematics

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We prove a uniqueness result of weak solutions to the $n D$ $(n \geq 3)$ Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term.

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.

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...

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 $| ω^{r} (x, t) | + | ω^{z} (r, t) | \leq \frac{C}{r^{10}}, 0 < r \leq \frac{1}{2} .$ 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 $| ω^{r} (x, t) | + | ω^{z} (r, t) | \leq \frac{C | \ln r |}{r^{17 / 2}}, 0 < r \leq \frac{1}{2} .$

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...

On an estimate for the linearized compressible Navier-Stokes equations in the $L_{p}$ -framework

Piotr Bogusław Mucha (2001)

Colloquium Mathematicae

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An $L_{p}$ -estimate with a constant independent of time for solutions of the linearized compressible Navier-Stokes system in the whole space (under the assumption that solutions have compact supports in space) is obtained.

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...

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...

Ill-posedness for the Navier-Stokes and Euler equations in Besov spaces

Yanghai Yu, Fang Liu (2024)

Applications of Mathematics

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We construct a new initial data to prove the ill-posedness of both Navier-Stokes and Euler equations in weaker Besov spaces in the sense that the solution maps to these equations starting from $u_{0}$ are discontinuous at $t = 0$ .

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 ${∥ \nabla ρ ∥}_{L^{\infty} (0, T; W^{1, q})} + {∥ u ∥}_{L^{s} (0, T; L_{ω}^{r})} < \infty$ for some $q > 3$ and any $(r, s)$ satisfying $2 / s + 3 / r \leq 1$ , $3 < r \leq \infty,$ 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.

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...

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 $p (x, t) \geq (3 n + 2) / (n + 2)$ , $n \geq 2,$ 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.

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...

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.

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.

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.

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.

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

Piotr Boguslaw Mucha, Wojciech Zajączkowski (2002)

Annales Polonici Mathematici

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The local existence of solutions for the compressible Navier-Stokes equations with the Dirichlet boundary conditions in the $L_{p}$ -framework is proved. Next an almost-global-in-time existence of small solutions is shown. The considerations are made in Lagrangian coordinates. The result is sharp in the $L_{p}$ -approach, because the velocity belongs to $W_{r}^{2, 1}$ with r > 3.

Stability with respect to domain of the low Mach number limit of compressible heat-conducting viscous fluid

Aneta Wróblewska-Kamińska (2023)

Archivum Mathematicum

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We investigate the asymptotic limit of solutions to the Navier-Stokes-Fourier system with the Mach number proportional to a small parameter $ε \to 0$ , the Froude number proportional to $\sqrt{ε}$ and when the fluid occupies large domain with spatial obstacle of rough surface varying when $ε \to 0$ . The limit velocity field is solenoidal and satisfies the incompressible Oberbeck–Boussinesq approximation. Our studies are based on weak solutions approach and in order to pass to the limit in a convective term we...

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.

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.

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,...

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.