Mostrando recursos 1 - 20 de 887

  1. Convergence of the Allen-Cahn equation with constraint to Brakke's mean curvature flow

    Takasao, Keisuke
    In this paper, we consider the Allen-Cahn equation with constraint. In 1994, Chen and Elliott [9] studied the asymptotic behavior of the solution of the Allen-Cahn equation with constraint. They proved that the zero level set of the solution converges to the classical solution of the mean curvature flow under the suitable conditions on initial data. In 1993, Ilmanen [20] proved the existence of the mean curvature flow via the Allen-Cahn equation without constraint in the sense of Brakke. We proved the same conclusion for the Allen-Cahn equation with constraint.

  2. On local $L_p$-$L_q$ well-posedness of incompressible two-phase flows with phase transitions: Non-equal densities with large initial data

    Shimizu, Senjo; Yagi, Shintaro
    A basic model for incompressible two-phase flows with phase transitions where the interface is nearly flat in the case of non-equal densities is considered. Local well-posedness of the model in $L_p$ in a time setting $L_q$ in a space setting was proved in [30] under a smallness assumption for the initial data. In this paper, we remove the smallness assumption for the initial data.

  3. Local stabilization of compressible Navier-Stokes equations in one dimension around non-zero velocity

    Mitra, Debanjana; Ramaswamy, Mythily; Raymond, Jean-Pierre
    In this paper, we study the local stabilization of one dimensional compressible Navier-Stokes equations around a constant steady solution $(\rho_s, u_s)$, where $\rho_s>0, u_s\neq 0$. In the case of periodic boundary conditions, we determine a distributed control acting only in the velocity equation, able to stabilize the system, locally around $(\rho_s, u_s)$, with an arbitrary exponential decay rate. In the case of Dirichlet boundary conditions, we determine boundary controls for the velocity and for the density at the inflow boundary, able to stabilize the system, locally around $(\rho_s, u_s)$, with an arbitrary exponential decay rate.

  4. Positive solutions of Schrödinger equations and Martin boundaries for skew product elliptic operators

    Murata, Minoru; Tsuchida, Tetsuo
    We consider positive solutions of elliptic partial differential equations on non-compact domains of Riemannian manifolds. We establish general theorems which determine Martin compactifications and Martin kernels for a wide class of elliptic equations in skew product form, by thoroughly exploiting parabolic Martin kernels for associated parabolic equations developed in [35] and [25]. As their applications, we explicitly determine the structure of all positive solutions to a Schrödinger equation and the Martin boundary of the product of Riemannian manifolds. For their sharpness, we show that the Martin compactification of ${\mathbb R}^2$ for some Schrödinger equation is so much distorted near infinity that no product structures remain.

  5. Analyticity of semigroups generated by higher order elliptic operators in spaces of bounded functions on $C^{1}$ domains

    Suzuki, Takuya
    This paper shows the analyticity of semigroups generated by higher order divergence type elliptic operators in $L^{\infty}$ spaces in $C^{1}$ domains which may be unbounded. For this purpose, we establish resolvent estimates in $L^{\infty}$ spaces by a contradiction argument based on a blow-up method. Our results yield the $L^{\infty}$ analyticity of solutions of parabolic equations for $C^{1}$ domains.

  6. Stable and unstable manifolds for quasilinear parabolic problems with fully nonlinear dynamical boundary conditions

    Schnaubelt, Roland
    We develop a wellposedness and regularity theory for a large class of quasilinear parabolic problems with fully nonlinear dynamical boundary conditions. Moreover, we construct and investigate stable and unstable local invariant manifolds near a given equilibrium. In a companion paper, we treat center, center-stable and center-unstable manifolds for such problems and investigate their stability properties. This theory applies e.g. to reaction-diffusion systems with dynamical boundary conditions and to the two-phase Stefan problem with surface tension.

  7. Local Hardy and Rellich inequalities for sums of squares of vector fields

    Ruzhansky, Michael; Suragan, Durvudkhan
    We prove local refined versions of Hardy's and Rellich's inequalities as well as of uncertainty principles for sums of squares of vector fields on bounded sets of smooth manifolds under certain assumptions on the vector fields. We also give some explicit examples, in particular, for sums of squares of vector fields on Euclidean spaces and for sub-Laplacians on stratified Lie groups.

  8. The Cauchy problem on large time for a Boussinesq-Peregrine equation with large topography variations

    Benoit, Mesognon-Gireau
    We prove, in this paper, a long time existence result for a modified Boussinesq-Peregrine equation in dimension $1$, describing the motion of Water Waves in shallow water, in the case of a non flat bottom. More precisely, the dimensionless equations depend strongly on three parameters $\epsilon,\mu,\beta$ measuring the amplitude of the waves, the shallowness and the amplitude of the bathymetric variations, respectively. For the Boussinesq-Peregrine model, one has small amplitude variations ($\epsilon = O(\mu)$). We first give a local existence result for the original Boussinesq Peregrine equation as derived by Boussinesq ([9], [8]) and Peregrine ([22]) in all dimensions. We then introduce a new model which has formally...

  9. Threshold and strong threshold solutions of a semilinear parabolic equation

    Quittner, Pavol
    If $p>1+2/n$, then the equation $u_t-\Delta u = u^p, \quad x\in{\mathbb R}^n,\ t>0,$ possesses both positive global solutions and positive solutions which blow up in finite time. We study the large time behavior of radial positive solutions lying on the borderline between global existence and blow-up.

  10. 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 $T[0]$...

  11. 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}...

  12. 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.

  13. 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...

  14. 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.

  15. 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.

  16. 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...

  17. 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.

  18. 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...

  19. 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.

  20. 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.

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