Mostrando recursos 1 - 20 de 878

  1. Constant sign Green's function for simply supported beam equation

    Cabada, Alberto; Saavedra, Lorena
    The aim of this paper consists on the study of the following fourth-order operator: \begin{equation*} T[M]\,u(t)\equiv u^{(4)}(t)+p_1(t)u'''(t)+p_2(t)u''(t)+Mu(t),\ t\in I \equiv [a,b] , \end{equation*} coupled with the two point boundary conditions: \begin{equation*} u(a)=u(b)=u''(a)=u''(b)=0 . \end{equation*} So, we define the following space: \begin{equation*} X=\left\lbrace u\in C^4(I) : u(a)=u(b)=u''(a)=u''(b)=0 \right\rbrace . \end{equation*} Here, $p_1\in C^3(I)$ and $p_2\in C^2(I)$. By assuming that the second order linear differential equation \begin{equation*} L_2\, u(t)\equiv u''(t)+p_1(t)\,u'(t)+p_2(t)\,u(t)=0\,,\quad t\in I, \end{equation*} is disconjugate on $I$, we characterize the parameter's set where the Green's function related to operator $T[M]$ in $X$ is of constant sign on $I \times I$. Such a characterization is equivalent to the strongly inverse positive (negative) character of operator $T[M]$ on $X$ and comes from the first eigenvalues of operator $T[0]$...

  2. The Cauchy problem for the shallow water type equations in low regularity spaces on the circle

    Yan, Wei; Li, Yongsheng; Zhai, Xiaoping; Zhang, Yimin
    In this paper, we investigate the Cauchy problem for the shallow water type equation \begin{align*} u_{t}+\partial_{x}^{3}u + \tfrac{1}{2}\partial_{x}(u^{2})+\partial_{x} (1-\partial_{x}^{2})^{-1}\left[u^{2}+\tfrac{1}{2} u_{x}^{2}\right]=0, \ \ x\in {\mathbf T}={\mathbf R}/2\pi \lambda, \end{align*} with low regularity data and $\lambda\geq1$. By applying the bilinear estimate in $W^{s}$, Himonas and Misiołek (Commun. Partial Diff. Eqns., 23 (1998), 123-139) proved that the problem is locally well-posed in $H^{s}([0, 2\pi))$ with $s\geq {1}/{2}$ for small initial data. In this paper, we show that, when $s < {1}/{2}$, the bilinear estimate in $W^{s}$ is invalid. We also demonstrate that the bilinear estimate in $Z^{s}$ is indeed valid for ${1}/{6} < s < {1}/{2}$. This enables us to prove that the problem is locally well-posed in $H^{s}(\mathbf{T})$ with ${1}/{6}...

  3. Classical solutions of the generalized Camassa-Holm equation

    Holmes, John; Thompson, Ryan C.
    In this paper, well-posedness in $C^1(\mathbb R)$ (a.k.a. classical solutions) for a generalized Camassa-Holm equation (g-$k$$b$CH) having $(k+1)$-degree nonlinearities is shown. This result holds for the Camassa-Holm, the Degasperis-Procesi and the Novikov equations, which improves upon earlier results in Sobolev and Besov spaces.

  4. On the local pressure of the Navier-Stokes equations and related systems

    Wolf, Jörg
    In the study of local regularity of weak solutions to systems related to incompressible viscous fluids local energy estimates serve as important ingredients. However, this requires certain information on the pressure. This fact has been used by V. Scheffer in the notion of a suitable weak solution to the Navier-Stokes equations, and in the proof of the partial regularity due to Caffarelli, Kohn and Nirenberg. In general domains, or in case of complex viscous fluid models a global pressure does not necessarily exist. To overcome this problem, in the present paper we construct a local pressure distribution by showing that every distribution $ \partial _t \boldsymbol u +\boldsymbol F $, which...

  5. A well posedness result for generalized solutions of Hamilton-Jacobi equations

    Zagatti, Sandro
    We study the Dirichlet problem for stationary Hamilton-Jacobi equations $$ \begin{cases} H(x, u(x), \nabla u(x)) = 0 & \ \textrm{in} \ \Omega \\ u(x)=\varphi(x) & \ \textrm{on} \ \partial \Omega. \end{cases} $$ We consider a Caratheodory hamiltonian $H=H(x,u,p)$, with a Sobolev-type (but not continuous) regularity with respect to the space variable $x$, and prove existence and uniqueness of a Lipschitz continuous maximal generalized solution which, in the continuous case, turns out to be the classical viscosity solution. In addition, we prove a continuous dependence property of the solution with respect to the boundary datum $\varphi$, completing in such a way a well posedness theory.

  6. Insensitizing controls for the Boussinesq system with no control on the temperature equation

    Carreño, N.
    In this paper, we prove the existence of controls insensitizing the $L^2$-norm of the solution of the Boussinesq system. The novelty here is that no control is used on the temperature equation. Furthermore, the control acting on the fluid equation can be chosen to have one vanishing component. It is well known that the insensitizing control problem is equivalent to a null controllability result for a cascade system, which is obtained thanks to a suitable Carleman estimate for the adjoint of the linearized system and an inverse mapping theorem. The particular form of the adjoint equation will allow us to obtain the null controllability of the linearized system.

  7. Inhomogeneous Besov spaces associated to operators with off-diagonal semigroup estimates

    Bui, The Anh; Duong, Xuan
    Let $(X, d, \mu)$ be a space of homogeneous type equipped with a distance $d$ and a measure $\mu$. Assume that $L$ is a closed linear operator which generates an analytic semigroup $e^{-tL}, t > 0$. Also assume that $L$ has a bounded $H_\infty$-calculus on $L^2(X)$ and satisfies the $L^p-L^q$ semigroup estimates (which is weaker than the pointwise Gaussian or Poisson heat kernel bounds). The aim of this paper is to establish a theory of inhomogeneous Besov spaces associated to such an operator $L$. We prove the molecular decompositions for the new Besov spaces and obtain the boundedness of the fractional powers $(I+L)^{-\gamma}, \gamma > 0$ on these Besov spaces. Finally, we...

  8. Evolution families and maximal regularity for systems of parabolic equations

    Gallarati, Chiara; Veraar, Mark
    In this paper, we prove maximal $L^p$-regularity for a system of parabolic PDEs, where the elliptic operator $A$ has coefficients which depend on time in a measurable way and are continuous in the space variable. The proof is based on operator-theoretic methods and one of the main ingredients in the proof is the construction of an evolution family on weighted $L^q$-spaces.

  9. Mixed boundary value problems on cylindrical domains

    Auscher, Pascal; Egert, Moritz
    We study second-order divergence-form systems on half-infinite cylindrical domains with a bounded and possibly rough base, subject to homogeneous mixed boundary conditions on the lateral boundary and square integrable Dirichlet, Neumann, or regularity data on the cylinder base. Assuming that the coefficients $A$ are close to coefficients $A_0$ that are independent of the unbounded direction with respect to the modified Carleson norm of Dahlberg, we prove a priori estimates and establish well-posedness if $A_0$ has a special structure. We obtain a complete characterization of weak solutions whose gradient either has an $L^2$-bounded non-tangential maximal function or satisfies a Lusin area bound. To this end, we combine the first-order approach to elliptic systems with the Kato square...

  10. Removable singularities for degenerate elliptic Pucci operators

    Galise, Giulio; Vitolo, Antonio
    In this paper, we introduce some fully nonlinear second order operators defined as weighted partial sums of the eigenvalues of the Hessian matrix, arising in geometrical contexts, with the aim to extend maximum principles and removable singularities results to cases of highly degenerate ellipticity.

  11. A monotonicity formula and Liouville-type theorems for stable solutions of the weighted elliptic system

    Hu, Liang-Gen
    In this paper, we are concerned with the weighted elliptic system \begin{equation*} \begin{cases} -\Delta u=|x|^{\beta} v^{\vartheta},\\ -\Delta v=|x|^{\alpha} |u|^{p-1}u, \end{cases}\quad \mbox{in}\;\ \mathbb{R}^N, \end{equation*}where $N \ge 5$, $\alpha >-4$, $ 0 \le \beta < N-4$, $p>1$ and $\vartheta=1$. We first apply Pohozaev identity to construct a monotonicity formula and reveal their certain equivalence relation. By the use of {\it Pohozaev identity}, {\it monotonicity formula} of solutions together with a {\it blowing down} sequence, we prove Liouville-type theorems for stable solutions (whether positive or sign-changing) of the weighted elliptic system in the higher dimension.

  12. On the interaction problem between a compressible fluid and a Saint-Venant Kirchhoff elastic structure

    Boulakia, M.; Guerrero, S.
    In this paper, we consider an elastic structure immersed in a compressible viscous fluid. The motion of the fluid is described by the compressible Navier-Stokes equations whereas the motion of the structure is given by the nonlinear Saint-Venant Kirchhoff model. For this model, we prove the existence and uniqueness of regular solutions defined locally in time. To do so, we first rewrite the nonlinearity in the elasticity equation in an adequate way. Then, we introduce a linearized problem and prove that this problem admits a unique regular solution. To obtain time regularity on the solution, we use energy estimates on the unknowns and their successive derivatives in time and to obtain spatial regularity, we use elliptic estimates. At...

  13. A Necessary condition for $H^\infty $ well-posedness of $p$-evolution equations

    Ascanelli, Alessia; Boiti, Chiara; Zanghirati, Luisa
    We consider $p$-evolution equations, for $p\geq2$, with complex valued coefficients. We prove that a necessary condition for $H^\infty$ well-posedness of the associated Cauchy problem is that the imaginary part of the coefficient of the subprincipal part (in the sense of Petrowski) satisfies a decay estimate as $|x|\to+\infty$.

  14. A new optimal transport distance on the space of finite Radon measures

    Kondratyev, Stanislav; Monsaingeon, Léonard; Vorotnikov, Dmitry
    We introduce a new optimal transport distance between nonnegative finite Radon measures with possibly different masses. The construction is based on non-conservative continuity equations and a corresponding modified Benamou-Brenier formula. We establish various topological and geometrical properties of the resulting metric space, derive some formal Riemannian structure, and develop differential calculus following F. Otto's approach. Finally, we apply these ideas to identify a model of animal dispersal proposed by MacCall and Cosner as a gradient flow in our formalism and obtain new long-time convergence results.

  15. Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system

    Yamazaki, Kazuo
    We study the three-dimensional stochastic nonhomogeneous magnetohydrodynamics system with random external forces that involve feedback, i.e., multiplicative noise, and are non-Lipschitz. We prove the existence of a global martingale solution via a semi-Galerkin approximation scheme with stochastic calculus and applications of Prokhorov's and Skorokhod's theorems. Furthermore, using de Rham's theorem for processes, we prove the existence of the pressure term.

  16. Uniqueness and nondegeneracy of sign-changing radial solutions to an almost critical elliptic problem

    Ao, Weiwei; Wei, Juncheng; Yao, Wei
    We study uniqueness of sign-changing radial solutions for the following semi-linear elliptic equation \begin{align*} \Delta u-u+|u|^{p-1}u=0\quad{\rm{in}}\ \mathbb{R}^N,\quad u\in H^1(\mathbb{R}^N), \end{align*} where $1 < p < \frac{N+2}{N-2}$, $N\geq3$. It is well-known that this equation has a unique positive radial solution. The existence of sign-changing radial solutions with exactly $k$ nodes is also known. However, the uniqueness of such solutions is open. In this paper, we show that such sign-changing radial solution is unique when $p$ is close to $\frac{N+2}{N-2}$. Moreover, those solutions are non-degenerate, i.e., the kernel of the linearized operator is exactly $N$-dimensional.

  17. Global weak solutions for Boussinesq system with temperature dependent viscosity and bounded temperature

    De Anna, Francesco
    In this paper, we obtain a result about the global existence of weak solutions to the d-dimensional Boussinesq-Navier-Stokes system, with viscosity dependent on temperature. The initial temperature is only supposed to be bounded, while the initial velocity belongs to some critical Besov Space, invariant to the scaling of this system. We suppose the viscosity close enough to a positive constant, and the $L^\infty$-norm of their difference plus the Besov norm of the horizontal component of the initial velocity is supposed to be exponentially small with respect to the vertical component of the initial velocity. In the preliminaries, and in the appendix, we consider some...

  18. Initial boundary value problem of the Hamiltonian fifth-order KdV equation on a bounded domain

    Zhou, Deqin; Zhang, Bing-Yu
    In this paper, we consider the initial boundary value problem (IBVP) of the Hamiltonian fifth-order KdV equation posed on a finite interval $(0,L)$, $$ \begin{cases} \partial_t u-\partial_x^{5}u= c_1 u\partial_x u+ c_2 u^2 \partial _x u + 2b\partial_x u\partial_x^{2}u+bu\partial_x^{3} u, \quad x\in (0,L), \ t>0 \\ u(0,x)=\phi (x) , \ x\in (0,L)\\ u(t,0)=\partial_x u(t,0)=u(t,L)=\partial_x u(t,L)=\partial_x^{2}u(t,L)=0, \quad t>0, \end{cases} $$ and show that, given $0\leq s\leq 5$ and $T>0$, for any $\phi \in H^s (0,L) $ satisfying the natural compatibility conditions, the IBVP admits a unique solution $$ u\in L^{\infty}_{loc} (\mathbb R^+; H^s(0,L))\cap L^2 _{loc}(\mathbb R^+; H^{s+2} (0,L)). $$ Moreover, the corresponding solution map is shown to be locally Lipschtiz continuous from $L^2 (0,L)$ to $L^{\infty}(0,T; L^2 (0,L))\cap L^2 (0,T; H^2 (0,L))$ and from $H^5 (0,L)$ to $L^{\infty}(0,T; H^5...

  19. On the existence of solitary waves for Boussinesq type equations and Cauchy problem for a new conservative model

    Bellec, Stevan; Colin, Mathieu
    In this paper, we present a long time existence theory for a new enhanced Boussinesq-Type system with constant bathymetry written in a conservative form. We also prove the existence of solitary wave for a large class of asymptotic models, including Beji-Nadaoka, Madsen-Sorensen and Nwogu equations. Furthermore, we give a procedure to calculate numerically these particular solutions and we present some effective computations.

  20. On the spectrum of an elastic solid with cusps

    Kozlov, Vladimir; Nazarov, Sergei A.
    The spectral problem of anisotropic elasticity with traction-free boundary conditions is considered in a bounded domain with a spatial cusp having its vertex at the origin and given by triples $(x_1,x_2,x_3)$ such that $x_3^{-2}(x_1,x_2) \in \omega$, where $\omega$ is a two-dimensional Lipschitz domain with a compact closure. We show that there exists a threshold $\lambda_\dagger>0$ expressed explicitly in terms of the elasticity constants and the area of $\omega$ such that the continuous spectrum coincides with the half-line $[\lambda_\dagger,\infty)$, whereas the interval $[0,\lambda_\dagger)$ contains only the discrete spectrum. The asymptotic formula for solutions to this spectral problem near cusp's vertex is also derived. A principle feature of this asymptotic formula is the dependence of...

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