The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

The search session has expired. Please query the service again.

Displaying similar documents to “Existence of three solutions for a class of (p₁,...,pₙ)-biharmonic systems with Navier boundary conditions”

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

Lin Li, Shapour Heidarkhani (2012)

Annales Polonici Mathematici

Similarity:

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 ( | u | p 2 u ) = λ f ( x , u ) + μ g ( x , u ) in Ω, ⎨ ⎩u = Δu = 0 on ∂Ω, where Ω 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.

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.

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.

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...

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 .

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...

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

Wisam Alame (2005)

Applicationes Mathematicae

Similarity:

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...

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

Similarity:

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...

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

Cholmin Sin (2023)

Czechoslovak Mathematical Journal

Similarity:

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.

Bubbling along boundary geodesics near the second critical exponent

Manuel del Pino, Monica Musso, Frank Pacard (2010)

Journal of the European Mathematical Society

Similarity:

The role of the second critical exponent p = ( n + 1 ) / ( n - 3 ) , the Sobolev critical exponent in one dimension less, is investigated for the classical Lane–Emden–Fowler problem Δ u + u p = 0 , u > 0 under zero Dirichlet boundary conditions, in a domain Ω in n with bounded, smooth boundary. Given Γ , a geodesic of the boundary with negative inner normal curvature we find that for p = ( n + 1 ) / ( n - 3 - ε ) , there exists a solution u ε such that | u ε | 2 converges weakly to a Dirac measure on Γ as ε 0 + , provided that Γ is nondegenerate in the sense of second...

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...

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

Viorel Barbu (2011)

ESAIM: Control, Optimisation and Calculus of Variations

Similarity:

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...

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

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

Similarity:

In this paper we consider a nonlinear elliptic equation with critical growth for the operator Δ 2 in a bounded domain Ω n . We state some existence results when n 8 . Moreover, we consider 5 n 7 , expecially when Ω is a ball in n .

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.

Linking and the Morse complex

Michael Usher (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

Similarity:

For a Morse function f on a compact oriented manifold M , we show that f has more critical points than the number required by the Morse inequalities if and only if there exists a certain class of link in M whose components have nontrivial linking number, such that the minimal value of f on one of the components is larger than its maximal value on the other. Indeed we characterize the precise number of critical points of f in terms of the Betti numbers of M and the behavior of f with respect...

A compactness result for polyharmonic maps in the critical dimension

Shenzhou Zheng (2016)

Czechoslovak Mathematical Journal

Similarity:

For n = 2 m 4 , let Ω n be a bounded smooth domain and 𝒩 L a compact smooth Riemannian manifold without boundary. Suppose that { u k } W m , 2 ( Ω , 𝒩 ) is a sequence of weak solutions in the critical dimension to the perturbed m -polyharmonic maps d d t | t = 0 E m ( Π ( u + t ξ ) ) = 0 with Φ k 0 in ( W m , 2 ( Ω , 𝒩 ) ) * and u k u weakly in W m , 2 ( Ω , 𝒩 ) . Then u is an m -polyharmonic map. In particular, the space of m -polyharmonic maps is sequentially compact for the weak- W m , 2 topology.

Asymptotic properties of ground states of scalar field equations with a vanishing parameter

Vitaly Moroz, Cyrill B. Muratov (2014)

Journal of the European Mathematical Society

Similarity:

We study the leading order behaviour of positive solutions of the equation - Δ u + ϵ u - | u | p - 2 u + | u | q - 2 u = 0 , x N , where N 3 , q > p > 2 and when ϵ > 0 is a small parameter. We give a complete characterization of all possible asymptotic regimes as a function of p , q and N . The behavior of solutions depends sensitively on whether p is less, equal or bigger than the critical Sobolev exponent 2 * = 2 N N - 2 . For p < 2 * the solution asymptotically coincides with the solution of the equation in which the last term is absent. For p > 2 * the solution asymptotically...

On a class of ( p , q ) -Laplacian problems involving the critical Sobolev-Hardy exponents in starshaped domain

M.S. Shahrokhi-Dehkordi (2017)

Communications in Mathematics

Similarity:

Let Ω n be a bounded starshaped domain and consider the ( p , q ) -Laplacian problem - Δ p u - Δ q u = λ ( 𝐱 ) | u | p - 2 u + μ | u | r - 2 u where μ is a positive parameter, 1 < q p < n , r p and p : = n p n - p is the critical Sobolev exponent. In this short note we address the question of non-existence for non-trivial solutions to the ( p , q ) -Laplacian problem. In particular we show the non-existence of non-trivial solutions to the problem by using a method based on Pohozaev identity.

Location of the critical points of certain polynomials

Somjate Chaiya, Aimo Hinkkanen (2013)

Annales Universitatis Mariae Curie-Sklodowska, sectio A – Mathematica

Similarity:

Let 𝔻 denote the unit disk { z : | z | < 1 } in the complex plane . In this paper, we study a family of polynomials P with only one zero lying outside 𝔻 ¯ .  We establish  criteria for P to satisfy implying that each of P and P '   has exactly one critical point outside 𝔻 ¯ .

Fourth-order nonlinear elliptic equations with critical growth

David E. Edmunds, Donato Fortunato, Enrico Jannelli (1989)

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

Similarity:

In this paper we consider a nonlinear elliptic equation with critical growth for the operator Δ 2 in a bounded domain Ω n . We state some existence results when n 8 . Moreover, we consider 5 n 7 , expecially when Ω is a ball in n .

Critical points of the Moser-Trudinger functional on a disk

Andrea Malchiodi, Luca Martinazzi (2014)

Journal of the European Mathematical Society

Similarity:

On the unit disk B 1 2 we study the Moser-Trudinger functional E ( u ) = B 1 e u 2 - 1 d x , u H 0 1 ( B 1 ) and its restrictions E | M Λ , where M Λ : = { u H 0 1 ( B 1 ) : u H 0 1 2 = Λ } for Λ > 0 . We prove that if a sequence u k of positive critical points of E | M Λ k (for some Λ k > 0 ) blows up as k , then Λ k 4 π , and u k 0 weakly in H 0 1 ( B 1 ) and strongly in C loc 1 ( B ¯ 1 { 0 } ) . Using this fact we also prove that when Λ is large enough, then E | M Λ has no positive critical point, complementing previous existence results by Carleson-Chang, M. Struwe and Lamm-Robert-Struwe.