Showing papers in "Advances in Differential Equations in 2016"

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

Abstract: 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.

Dissipative structures for thermoelastic plate equations in $\mathbb R^n$

Abstract: We consider the Cauchy problem in $\mathbb R^n$ for linear thermoelastic plate equations where heat conduction is modeled by either the Cattaneo law or by the Fourier law -- described by the relaxation parameter $\tau$, where $\tau>0$ corresponds to Cattaneo's law and $ \tau=0 $ corresponds to Fourier's law. Additionally, we take into account possible inertial effects characterized by a parameter $\mu\geq 0$, where $\mu=0$ corresponds to the situation without inertial terms. For the Catteneo system without inertial term, being a coupling of a Schrodinger type equation (the elastic plate equation) with a hyperbolic system for the temperature and the heat flux, we shall show that a regularity-loss phenomenon appears in the asymptotic behavior as time tends to infinity, while this is not given in the standard model where the Cattaneo law is replaced by the standard Fourier law. This kind of effect of changing the qualitative behavior when moving from Fourier to Cattaneo reflects the effect known for bounded domains, where the system with Fourier law is exponentially stable while this property is lost when going to the Cattaneo law. In particular, we shall describe in detail the singular limit as $\tau\to 0$. For the system with inertial term we demonstrate that it is of standard type, not of regularity loss type. The corresponding limit of a vanishing inertial term is also described. All constants appearing in the main results are given explicitly, allowing for quantitative estimates. The optimality of the estimates is also proved.

Strong solutions for the Beris-Edwards model for nematic liquid crystals with homogeneous Dirichlet boundary conditions

Abstract: Existence and uniqueness of local strong solutions for the Beris--Edwards model for nematic liquid crystals, which couples the Navier-Stokes equations with an evolution equation for the $Q$-tensor, is established on a bounded domain $\Omega\subset\mathbb{R}^d$ in the case of homogeneous Dirichlet boundary conditions. The classical Beris--Edwards model is enriched by including a dependence of the fluid viscosity on the $Q$-tensor. The proof is based on a linearization of the system and Banach's fixed-point theorem.

On certain nonlocal Hardy-Sobolev critical elliptic Dirichlet problems

Abstract: This paper deals with the existence, multiplicity and the asymptotic behavior of nontrivial solutions for nonlinear problems driven by the fractional Laplace operator $(-\Delta)^s$ and involving a critical Hardy potential. In particular, we consider $$ \left\{ \begin{array}{ll} (- \Delta)^{s}u - \gamma \displaystyle \frac{u}{|x|^{2s}} = \lambda u + \theta f(x,u) +g(x,u) & \mbox{ in }\Omega,\\ u=0 & \mbox{in} \mathbb{R}^{N} \setminus \Omega, \end{array} \right. $$ where $\Omega\subset \mathbb R^N$ is a bounded domain, $\gamma, \lambda$ and $\theta$ are real parameters, the function $f$ is a subcritical nonlinearity, while $g$ could be either a critical term or a perturbation.

Existence, regularity and representation of solutions of time fractional diffusion equations

Abstract: Using regularized resolvent families, we investigate the solvability of the fractional order inhomogeneous Cauchy problem $$ \mathbb{D}_t^\alpha u(t)=Au(t)+f(t), \;t > 0,\;\;0 0,\; x\in \mathbb R ^N$ where $\pi/2\le \theta < (1-\alpha/2)\pi$ and $\Delta_p$ is a realization of the Laplace operator on $L^p(\mathbb R ^N)$, $1\le p < \infty$.

Persistence of regularity for solutions of the Boussinesq equations in Sobolev spaces

Abstract: We address the global regularity of solutions to the Boussinesq equations with zero diffusivity in two spatial dimensions. Previously, the persistence in the space $H^{1+s}(\mathbb{R}^2)\times H^{s}(\mathbb{R}^2)$ for all $s\ge 0$ has been obtained. In this paper, we address the persistence in general Sobolev spaces, establishing it on a time interval which is almost independent of the size of the initial data. Namely, we prove that if $(u_0,\rho_0)\in W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for $s\in(0,1)$ and $q\in[2,\infty)$, then the solution $(u(t),\rho(t))$ of the Boussinesq system stays in $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for $t\in[0,T^*)$, where $T^*$ depends logarithmically on the size of initial data. If we furthermore assume that $sq>2$, then we get the global persistence in the space $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$. Moreover, we prove the global persistence in the space $W^{1+s,q}(\mathbb{R}^2)\times W^{s,q}(\mathbb{R}^2)$ for the initial data with compact support, as well us for data in $W^{1+s,q}(\mathbb{T}^2)\times W^{s,q}(\mathbb{T}^2)$, without any restriction on $s\in(0,1)$ and $q\in[2,\infty)$.

Profiles for the radial focusing energy-critical wave equation in odd dimensions

Abstract: In this paper, we consider global and non-global radial solutions of the focusing energy--critical wave equation on $\mathbb{R} \times \mathbb{R}^N$ where $N \geq 5$ is odd. We prove that if the solution remains bounded in the energy space as you approach the maximal forward time of existence, then along a sequence of times converging to the maximal forward time of existence, the solution decouples into a sum of dynamically rescaled solitons, a free radiation term, and an error tending to zero in the energy space. If, in addition, we assume a bound on the evolution that rules out the formation of multiple solitons, then this decoupling holds for all times approaching the maximal forward time of existence.

Local well-posedness for the KdV hierarchy at high regularity

Abstract: We prove well-posedness in $L^2$-based Sobolev spaces $H^s$ at high regularity for a class of nonlinear higher-order dispersive equations generalizing the KdV hierarchy both on the line and on the torus.

Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo $p$-Laplacian equation

Abstract: We consider the pseudo-$p$-Laplacian operator: $$ \tilde \Delta_p u = \sum_{i=1}^N \partial_i (|\partial_i u|^{p-2} \partial_i u)=(p-1) \sum_{i=1}^N |\partial_i u|^{p-2} \partial_{ii} u \ \text{ for $p > 2$.} $$ We prove interior regularity results for the viscosity (resp. weak) solutions in the unit ball $B_1$ of $\tilde \Delta_p u =(p-1) f$ for $f\in { \mathcal C} (\overline{B_1})$ (resp. $f\in L^\infty(B_1)$). First, the Holder local regularity for any exponent $\gamma < 1$, recovering in that way a known result about weak solutions. Second, we prove the Lipschitz local regularity.

Resolvent estimates of the Stokes system with Navier boundary conditions in general unbounded domains

Abstract: Consider the Stokes resolvent system in general unbounded domains $\Omega \subset {\mathbb{R}^n}$, $n\geq 2$, with boundary of uniform class $C^{3}$, and Navier slip boundary condition. The main result is the resolvent estimate in function spaces of the type ${\tilde{L}^q}$ defined as $L^q\cap L^2$ when $q\geq 2$, but as $L^q + L^2$ when $1 < q < 2$, adapted to the unboundedness of the domain. As a consequence, we get that the Stokes operator generates an analytic semigroup on a solenoidal subspace ${\tilde{L}^q}_\sigma(\Omega)$ of ${\tilde{L}^q}(\Omega)$.

A level set crystalline mean curvature flow of surfaces

Abstract: We introduce a new notion of viscosity solutions for the level set formulation of the motion by crystalline mean curvature in three dimensions. The solutions satisfy the comparison principle, stability with respect to an approximation by regularized problems, and we also show the uniqueness and existence of a level set flow for bounded crystals.

Spectral properties of the semigroup for the linearized compressible Navier-Stokes equation around a parallel flow in a cylindrical domain

Abstract: This paper is concerned with the stability of a parallel flow of the compressible Navier-Stokes equation in a cylindrical domain. The spectrum of the linearized operator is analyzed for the purpose of the study of the nonlinear stability. It is shown that, if the Reynolds and Mach numbers are sufficiently small, then the linearized semigroup is decomposed into two parts; one behaves like a solution of a one dimensional heat equation as time goes to infinity and the other one decays exponentially. Some estimates related to the spectral projections are established, which will also be useful for the study of the nonlinear problem.

Existence of solutions to parabolic problems with nonstandard growth and irregular obstacles

Abstract: In this paper, we establish the existence theory to nonlinear parabolic problems with nonstandard $p(x,t)$-growth conditions and irregular obstacles.

Global martingale solution to the stochastic nonhomogeneous magnetohydrodynamics system

Abstract: 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.

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

Abstract: 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.

Multi-level Gevrey solutions of singularly perturbed linear partial differential equations

Abstract: We study the asymptotic behavior of the solutions related to a family of singularly perturbed linear partial differential equations in the complex domain. The analytic solutions obtained by means of a Borel-Laplace summation procedure are represented by a formal power series in the perturbation parameter. Indeed, the geometry of the problem gives rise to a decomposition of the formal and analytic solutions so that a multi-level Gevrey order phenomenon appears. This result leans on a Malgrange-Sibuya theorem in several Gevrey levels.

Bresse system with Fourier law on shear force

Abstract: Our goal in this paper is to investigate the effect of Fourier law on a new coupling for the Bresse system. We give an explicit characterization on the decay rate that can be exponential or polynomial type, depending on the relation between the coefficients of the wave speed propagation.

Solutions to nonlinear higher order Schrödinger equations with small initial data on modulation spaces

Abstract: In this paper, we consider the Cauchy problem for the nonlinear higher order Schrodinger equations on modulation spaces $M_{p,q}^s$ and show the existence of a unique global solution by using integrability of time decay factors of time decay estimates. As a result, we are able to deal with wider classes of a nonlinearity and a solution space. Moreover, we study time decay estimates of a semi--group $e^{it\phi(\sqrt{-\Delta})}$ with a polynomial symbol $\phi$. Considering multiplicities of critical points and inflection points of $\phi$ carefully, we have time decay estimates with better time decay rate.

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

Abstract: We consider p-evolution equations, for p ≥ 2, with complex valued coefficients. We prove that a necessary condition for H 1 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| → +∞.

Evolution PDEs and augmented eigenfunctions. Finite interval

Abstract: The so-called unified method expresses the solution of an initial-boundary value problem for an evolution PDE in the finite interval in terms of an integral in the complex Fourier (spectral) plane. Simple initial-boundary value problems, which will be referred to as problems of type I, can be solved via a classical transform pair. For example, the Dirichlet problem of the heat equation can be solved in terms of the transform pair associated with the Fourier sine series. Such transform pairs can be constructed via the spectral analysis of the associated spatial operator. For more complicated initialboundary value problems, which will be referred to as problems of type II, there does not exist a classical transform pair and the solution cannot be expressed in terms of an infinite series. Here we pose and answer two related questions: first, does there exist a (non-classical) transform pair capable of solving a type II problem, and second, can this transform pair be constructed via spectral analysis? The answer to both of these questions is positive and this motivates the introduction of a novel class of spectral entities. We call these spectral entities augmented eigenfunctions, to distinguish them from the generalised eigenfunctions introduced in the sixties by Gel’fand and his co-authors. AMS MSC2010 35P10 (primary), 35C15, 35G16, 47A70 (secondary). 1 ar X iv :1 30 3. 22 05 v2 [ m at h. SP ] 1 5 A ug 2 01 4

On the spectrum of an elastic solid with cusps

Abstract: 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 the leading term on the spectral parameter.

Global weak solutions for boussinesq system with temperature dependent viscosity and bounded temperature

Abstract: 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 $L^p_t L^q_x$ regularity Theorems for the heat kernel, which play an important role in the main proof of this article.

Existence and blow-up rate of large solutions of $p(x)$-Laplacian equations with large perturbation and gradient terms

Abstract: In this paper, we investigate boundary blow-up solutions of the problem \begin{equation*} \quad \left\{ \begin{array}{l} -\Delta _{p(x)}u+f(x,u)=\rho (x,u)+K(x)| abla u|^{m(x)}\mbox{ in }\Omega , \\[2mm] u(x)\rightarrow +\infty \mbox{ as }d(x,\partial \Omega )\rightarrow 0, \end{array} \right \end{equation*} where $\Delta _{p(x)}u=\mathrm{div}\,(| abla u|^{p(x)-2} abla u)$ is called $p(x)$-Laplacian Our results extend the previous work of J Garcia-Melian, A Suarez [23] from the case where $p(\cdot )\equiv 2 $, without gradient term, to the case where $p(\cdot )$ is a function, with gradient term It also extends the previous work of Y Liang, QH Zhang and CS Zhao [38] from the radial case in the problem to the non-radial case The existence of boundary blow-up solutions is established and the singularity of boundary blow-up solution is also studied for several cases including when $\frac{\rho (x,u(x))}{f(x,u(x))}\rightarrow 1$ as $x\rightarrow \partial \Omega $, which means that $\rho (x,u)$ is a large perturbation We provide an exact estimate of the pointwise different behavior of the solutions near the boundary in terms of $d(x,\partial \Omega )$ Hence, the results of this paper are new even in the canonical case $p(\cdot )\equiv 2$ In particular, we do not have the comparison principle, because we don't make the monotone assumption of nonlinear term

Transverse instability for nonlinear Schrödinger equation with a linear potential

Abstract: In this paper, we consider the transverse instability for a nonlinear Schrodinger equation with a linear potential on ${\mathbb {R} \times \mathbb {T}_L}$, where $2\pi L$ is the period of the torus $\mathbb{T}_L$. Rose and Weinstein [18] showed the existence of a stable standing wave for a nonlinear Schrodinger equation with a linear potential. We regard the standing wave of nonlinear Schrodinger equation on ${\mathbb R}$ as a line standing wave of nonlinear Schrodinger equation on ${\mathbb R} \times {\mathbb T}_L$. We show the stability of line standing waves for all $L>0$ by using the argument of the previous paper [26].

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

Abstract: 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 $L^{\infty}(0,T; H^5 (0,L))\cap L^2 (0,T; H^7 (0,L))$, respectively, for any given $T>0$ . This is in sharp contrast to the pure initial value problem (IVP) of the equation posed on the whole line $\mathbb R$, $$ \begin{cases} \partial_t v-\partial_x^{5}v= c_1 v\partial_x v+ c_2 v^2 \partial _x v + 2b\partial_x v\partial_x^{2}u+bv\partial_x^{3} v, \quad x\in \mathbb R, \ t\in \mathbb R \\ v(0,x)=\psi (x) , \ x\in \mathbb R, \end{cases} $$ which is known to be (globally) well-posed in the space $H^s (\mathbb R)$ for $s\geq 2$ and the corresponding solution map is continuous, but fails to be uniformly continuous on any ball in $H^s(\mathbb R)$.

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

Abstract: 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.

The semigroup governing the generalized Cox-Ingersoll-Ross equation

Abstract: The semigroup of a generalized initial value problem including, as a particular case, the Cox-Ingersoll-Ross (CIR) equation for the price of a zero-coupon bond, is studied on spaces of continuous functions on $[0,\infty]$. The main result is the first proof of the strong continuity of the CIR semigroup. We also derive a semi-explicit representation of the semigroup and a Feynman-Kac type formula, in a generalized sense, for the unique solution of the CIR initial value problem as a useful tool for understanding additional properties of the solution itself. The Feynman-Kac type formula is the second main result of this paper.

Sharp well-posedness results for the Schrödinger-Benjamin-Ono system

Abstract: This work is concerned with the Cauchy problem for a coupled Schrodinger-Benjamin-Ono system \begin{equation*} \left \{ \begin{array}{l} i\partial_tu+\partial_x^2u=\alpha uv, \hfill t\!\in\![-T,T], \ x\!\in\!\mathbb R,\\ \partial_tv+ u\mathcal H\partial^2_xv=\beta \partial_x(|u|^2),\\ u(0,x)=\phi, \ v(0,x)=\psi, \ \ \ \ \ \ \ \ \ \ \ \ \ \hfill (\phi,\psi)\!\in\!H^{s}(\mathbb R)\!\times\!H^{s'}\!(\mathbb R). \end{array} \right. \end{equation*} In the ${\it non-resonant}$ case $(| u| e1)$, we prove local well-posedness for a large class of initial data. This improves the results obtained by Bekiranov, Ogawa and Ponce (1998). Moreover, we prove $C^2$-{\it ill-posedness} at ${\it low-regularity}$, and also when the difference of regularity between the initial data is large enough. As far as we know, this last ill-posedness result is the first of this kind for a nonlinear dispersive system. Finally, we also prove that the local well-posedness result obtained by Pecher (2006) in the ${\it resonant}$ case $(| u|=1)$ is sharp except for the end-point.

Dispersive mixed-order systems in L p -Sobolev spaces and application to the thermoelastic plate equation

Abstract: We study dispersive mixed-order systems of pseudodifferential operators in the setting of $L^p$-Sobolev spaces. Under the weak condition of quasi-hyperbolicity, these operators generate a semigroup in the space of tempered distributions. However, if the basic space is a tuple of $L^p$-Sobolev spaces, a strongly continuous semigroup is in many cases only generated if $p=2$ or $n=1$. The results are applied to the linear thermoelastic plate equation with and without inertial term and with Fourier's or Maxwell-Cattaneo's law of heat conduction.

Saddle type solutions for a class of reversible elliptic equations

Abstract: This paper is concerned with the existence of saddle type solutions for a class of semilinear elliptic equations of the type \begin{equation} \Delta u(x)+F_{u}(x,u)=0,\quad x\in\mathbb R^{n},\;\; n\ge 2, \tag*{(PDE)} \end{equation} where $F$ is a periodic and symmetric nonlinearity. Under a non degeneracy condition on the set of minimal periodic solutions, saddle type solutions of $(PDE)$ are found by a renormalized variational procedure.