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

Miaochao Chen; Shengqi Lu; Qilin Liu

Displaying similar documents to “Uniqueness of weak solutions to a Keller-Segel-Navier-Stokes model with a logistic source”

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.

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 $ε_{*} > 0$ .

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...

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...

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...

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...

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 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.

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...

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...

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.

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 an existence theorem for the Navier-Stokes equations with free slip boundary condition in exterior domain

Rieko Shimada, Norikazu Yamaguchi (2008)

Banach Center Publications

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This paper deals with a nonstationary problem for the Navier-Stokes equations with a free slip boundary condition in an exterior domain. We obtain a global in time unique solvability theorem and temporal asymptotic behavior of the global strong solution when the initial velocity is sufficiently small in the sense of Lⁿ (n is dimension). The proof is based on the contraction mapping principle with the aid of $L^{p} - L^{q}$ estimates for the Stokes semigroup associated with a linearized problem, which...

Stokes equations in asymptotically flat layers

Helmut Abels (2005)

Banach Center Publications

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We study the generalized Stokes resolvent equations in asymptotically flat layers, which can be considered as compact perturbations of an infinite (flat) layer $Ω ₀ = ℝ^{n - 1} \times (- 1, 1)$ . Besides standard non-slip boundary conditions, we consider a mixture of slip and non-slip boundary conditions on the upper and lower boundary, respectively. We discuss the results on unique solvability of the generalized Stokes resolvent equations as well as the existence of a bounded $H_{\infty}$ -calculus for the associated Stokes operator...

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.

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.

Some linear parabolic system in Besov spaces

Ewa Zadrzyńska, Wojciech M. Zajączkowski (2008)

Banach Center Publications

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We study the solvability in anisotropic Besov spaces $B_{p, q}^{σ / 2, σ} (Ω^{T})$ , σ ∈ ℝ₊, p,q ∈ (1,∞) of an initial-boundary value problem for the linear parabolic system which arises in the study of the compressible Navier-Stokes system with boundary slip conditions. The proof of existence of a unique solution in $B_{p, q}^{σ / 2 + 1, σ + 2} (Ω^{T})$ is divided into three steps: 1° First the existence of solutions to the problem with vanishing initial conditions is proved by applying the Paley-Littlewood decomposition and some ideas of Triebel....

On global regular solutions to the Navier-Stokes equations with heat convection

Piotr Kacprzyk (2013)

Annales Polonici Mathematici

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Global existence of regular solutions to the Navier-Stokes equations for velocity and pressure coupled with the heat convection equation for temperature in a cylindrical pipe is shown. We assume the slip boundary conditions for velocity and the Neumann condition for temperature. First we prove long time existence of regular solutions in [kT,(k+1)T]. Having T sufficiently large and imposing some decay estimates on ${| | f (t) | |}_{L ₂ (Ω)}$ , $| | f_{, x ₃} (t) {| |}_{L ₂ (Ω)}$ we continue the local solutions step by step up to a global one. ...

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 Leray problem for 2D inhomogeneous fluids

Farid Ammar-Khodja, Marcelo M. Santos (2005)

Banach Center Publications

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We formulate the Leray problem for inhomogeneous fluids in two dimensions and outline the proof of the existence of a solution. There are two kinds of results depending on whether the given value for the density is a continuous function or only an $L^{\infty}$ function. In the former case, the given densities are attained in the sense of uniform convergence and in the latter with respect to weak-* convergence.

A remark on the existence of steady Navier-Stokes flows in 2D semi-infinite channel involving the general outflow condition

H. Morimoto, H. Fujita (2001)

Mathematica Bohemica

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We consider the steady Navier-Stokes equations in a 2-dimensional unbounded multiply connected domain $Ω$ under the general outflow condition. Let $T$ be a 2-dimensional straight channel $ℝ \times (- 1, 1)$ . We suppose that $Ω \cap {x_{1} < 0}$ is bounded and that $Ω \cap {x_{1} > - 1} = T \cap {x_{1} > - 1}$ . Let $V$ be a Poiseuille flow in $T$ and $μ$ the flux of $V$ . We look for a solution which tends to $V$ as $x_{1} \to \infty$ . Assuming that the domain and the boundary data are symmetric with respect to the $x_{1}$ -axis, and that the axis intersects every component of the boundary, we have shown...

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 geometric improvement of the velocity-pressure local regularity criterion for a suitable weak solution to the Navier-Stokes equations

Jiří Neustupa (2014)

Mathematica Bohemica

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We deal with a suitable weak solution $(𝐯, p)$ to the Navier-Stokes equations in a domain $Ω \subset ℝ^{3}$ . We refine the criterion for the local regularity of this solution at the point $(𝐟 x_{0}, t_{0})$ , which uses the $L^{3}$ -norm of $𝐯$ and the $L^{3 / 2}$ -norm of $p$ in a shrinking backward parabolic neighbourhood of $(𝐱_{0}, t_{0})$ . The refinement consists in the fact that only the values of $𝐯$ , respectively $p$ , in the exterior of a space-time paraboloid with vertex at $(𝐱_{0}, t_{0})$ , respectively in a ”small” subset of this exterior, are considered. The consequence...

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...

On the Stokes and Navier-Stokes flows in a perturbed half-space

Takayuki Kubo, Yoshihiro Shibata (2005)

Banach Center Publications

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We give the $L_{p} - L_{q}$ estimate for the Stokes semigroup in a perturbed half-space and some global in time existence theorems for small solutions to the Navier-Stokes equation.

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...

Long-time behavior for 2D non-autonomous g-Navier-Stokes equations

Cung The Anh, Dao Trong Quyet (2012)

Annales Polonici Mathematici

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We study the first initial boundary value problem for the 2D non-autonomous g-Navier-Stokes equations in an arbitrary (bounded or unbounded) domain satisfying the Poincaré inequality. The existence of a weak solution to the problem is proved by using the Galerkin method. We then show the existence of a unique minimal finite-dimensional pullback $_{σ}$ -attractor for the process associated to the problem with respect to a large class of non-autonomous forcing terms. Furthermore, when the force...

Long-Time Asymptotics for the Navier-Stokes Equation in a Two-Dimensional Exterior Domain

Thierry Gallay (2012)

Journées Équations aux dérivées partielles

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We study the long-time behavior of infinite-energy solutions to the incompressible Navier-Stokes equations in a two-dimensional exterior domain, with no-slip boundary conditions. The initial data we consider are finite-energy perturbations of a smooth vortex with small circulation at infinity, but are otherwise arbitrarily large. Using a logarithmic energy estimate and some interpolation arguments, we prove that the solution approaches a self-similar Oseen vortex as $t \to \infty$ . This result was...