Mathematics - Analysis of PDEs Publications (50)


Mathematics - Analysis of PDEs Publications

By using, among other things, the Fourier analysis techniques on hyperbolic and symmetric spaces, we establish the Hardy-Sobolev-Maz'ya inequalities for higher order derivatives on half spaces. The proof relies on a Hardy-Littlewood-Sobolev inequality on hyperbolic spaces which is of its independent interest. We also give an alternative proof of Benguria, Frank and Loss' work concerning the sharp constant in the Hardy-Sobolev-Maz'ya inequality in the three dimensional upper half space. Read More

In this paper we consider the initial boundary value problem of the Korteweg-de Vries equation posed on a finite interval \begin{equation} u_t+u_x+u_{xxx}+uu_x=0,\qquad u(x,0)=\phi(x), \qquad 00 \qquad (1) \end{equation} subject to the nonhomogeneous boundary conditions, \begin{equation} B_1u=h_1(t), \qquad B_2 u= h_2 (t), \qquad B_3 u= h_3 (t) \qquad t>0 \qquad (2) \end{equation} where \[ B_i u =\sum _{j=0}^2 \left(a_{ij} \partial ^j_x u(0,t) + b_{ij} \partial ^j_x u(L,t)\right), \qquad i=1,2,3,\] and $a_{ij}, \ b_{ij}$ $ (j,i=0, 1,2,3)$ are real constants. Under some general assumptions imposed on the coefficients $a_{ij}, \ b_{ij}$, $ j,i=0, 1,2,3$, the IBVPs (1)-(2) is shown to be locally well-posed in the space $H^s (0,L)$ for any $s\geq 0$ with $\phi \in H^s (0,L)$ and boundary values $h_j, j=1,2,3$ belonging to some appropriate spaces with optimal regularity. Read More

We establish sharp Hardy-Adams inequalities on hyperbolic space $\mathbb{B}^{4}$ of dimension four. Namely, we will show that for any $\alpha>0$ there exists a constant $C_{\alpha}>0$ such that \[ \int_{\mathbb{B}^{4}}(e^{32\pi^{2} u^{2}}-1-32\pi^{2} u^{2})dV=16\int_{\mathbb{B}^{4}}\frac{e^{32\pi^{2} u^{2}}-1-32\pi^{2} u^{2}}{(1-|x|^{2})^{4}}dx\leq C_{\alpha}. \] for any $u\in C^{\infty}_{0}(\mathbb{B}^{4})$ with \[ \int_{\mathbb{B}^{4}}\left(-\Delta_{\mathbb{H}}-\frac{9}{4}\right)(-\Delta_{\mathbb{H}}+\alpha)u\cdot udV\leq1. Read More

We study discrete spectral quantities associated to Schr\"odinger operators of the form $-\Delta_{\mathbb{R}^d}+V_N$, $d$ odd. The potential $V_N$ models a highly disordered crystal; it varies randomly at scale $N^{-1} \ll 1$. We use perturbation analysis to obtain almost sure convergence of the eigenvalues and scattering resonances of $-\Delta_{\mathbb{R}^d}+V_N$ as $N \rightarrow \infty$. Read More

We consider the heat equation with a logarithmic nonlinearity, on thereal line. For a suitable sign in front of the nonlinearity, weestablish the existence and uniqueness of solutions of the Cauchyproblem, for a well-adapted class of initial data. Explicitcomputations in the case of Gaussian data lead to various scenariiwhich are richer than the mere comparison with the ODE mechanism,involving (like in the ODE case) double exponential growth or decayfor large time. Read More

We consider local energy decay estimates for solutions to scalar wave equations on nontrapping asymptotically flat space-times. Our goals are two-fold. First we consider the stationary case, where we can provide a full spectral characterization of local energy decay bounds; this characterization simplifies in the stationary symmetric case. Read More

In this work, we study a minimal time problem for a Partial Differential Equation of transport type, that arises in crowd models. The control is a Lipschitz vector field localized on a fixed control set $\omega$. We provide a complete answer for the minimal time problem. Read More

In this paper, we are concerned with the existence and asymptotic behavior of least energy solutions for following nonlinear Choquard equation driven by fractional Laplacian $$(-\Delta)^{s} u+\lambda V(x)u=(I_{\alpha}\ast F(u))f(u) \ \ in \ \ R^{N},$$ where $N> 2s$, $ (N-4s)^{+}<\alpha< N$, $\lambda$ is a positive parameter and the nonnegative potential function $V(x)$ is continuous. By variational methods, we prove the existence of least energy solution which localize near the potential well $int (V^{-1}(0))$ as $\lambda$ large enough. Read More

In the present paper, a continuum model is introduced for fluid flow in a deformable porous medium, where the fluid may undergo phase transitions. Typically, such problems arise in modeling liquid-solid phase transformations in groundwater flows. The system of equations is derived here from the conservation principles for mass, momentum, and energy and from the Clausius-Duhem inequality for entropy. Read More

The paper is devoted to studying the global dynamics of the Boltzmann equation on bounded domains. We allow a class of non-negative initial data which have arbitrary large amplitude and even contain vacuum. The result shows that the oscillation of solutions away from global Maxwellians becomes small after some positive time provided that they are initially close to each other in $L^2$. Read More

In this note we prove the instability by blow-up of the ground state solutions for a class of fourth order Schr\" odinger equations. This extends the first rigorous results on blowing-up solutions for the biharmonic NLS due to Boulenger and Lenzmann \cite{BoLe} and confirm numerical conjectures from \cite{BaFi, BaFiMa1, BaFiMa, FiIlPa}. Read More

We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Read More

We study quantitative periodic homogenization of integral functionals in the context of non-linear elasticity. Under suitable assumptions on the energy densities (in particular frame indifference; minimality, non-degeneracy and smoothness at the identity; $p\geq d$-growth from below; and regularity of the microstructure), we show that in a neighborhood of the set of rotations, the multi-cell homogenization formula of non-convex homogenization reduces to a single-cell formula. The latter can be expressed with help of correctors. Read More

In this paper we study the Cauchy problem for the semilinear damped wave equation for the sub-Laplacian on the Heisenberg group. In the case of the positive mass, we show the global in time well-posedness for small data for power like nonlinearities. We also obtain similar well-posedness results for the wave equations for Rockland operators on general graded Lie groups. Read More

We study a semi-linear version of the Skyrme system due to Adkins and Nappi. The objects in this system are maps from $(1+3)$-dimensional Minkowski space into the $3$-sphere and 1-forms on $\mathbb{R}^{1+3}$, coupled via a Lagrangian action. Under a co-rotational symmetry reduction we establish the existence, uniqueness, and unconditional asymptotic stability of a family of stationary solutions $Q_n$, indexed by the topological degree $n \in \mathbb{N} \cup \{0\}$ of the underlying map. Read More

Motion of a rigid body immersed in a semi-infinite expanse of gas in a $d$-dimensional region bounded by an infinite plane wall is studied for free molecular flow on the basis of the free Vlasov equation under the specular boundary condition. We show that the velocity $V(t)$ of the body approaches its terminal velocity $V_{\infty}$ according to a power law $V_{\infty}-V(t)\approx Ct^{-(d-1)}$ by carefully analyzing the pre-collisions due to the presence of the wall. The exponent $d-1$ is smaller than $d+2$ for the case without the wall found in the classical work by Caprino, Marchioro and Pulvirenti~[Comm. Read More

In this paper we consider the following critical nonlocal problem $$ \left\{\begin{array}{ll} M\left(\displaystyle\iint_{\mathbb{R}^{2N}}\frac{|u(x)-u(y)|^2}{|x-y|^{N+2s}}dxdy\right)(-\Delta)^s u = \displaystyle\frac{\lambda}{u^\gamma}+u^{2^*_s-1}&\quad\mbox{in } \Omega,\\ u>0&\quad\mbox{in } \Omega,\\ u=0&\quad\mbox{in } \mathbb{R}^N\setminus\Omega, \end{array}\right. $$ where $\Omega$ is an open bounded subset of $\mathbb R^N$ with continuous boundary, dimension $N>2s$ with parameter $s\in (0,1)$, $2^*_s=2N/(N-2s)$ is the fractional critical Sobolev exponent, $\lambda>0$ is a real parameter, exponent $\gamma\in(0,1)$, $M$ models a Kirchhoff type coefficient, while $(-\Delta)^s$ is the fractional Laplace operator. In particular, we cover the delicate degenerate case, that is when the Kirchhoff function $M$ is zero at zero. Read More

We investigate uniqueness of weak solutions for a system of partial differential equations capturing behavior of magnetoelastic materials. This system couples the Navier-Stokes equations with evolutionary equations for the deformation gradient and for the magnetization obtained from a special case of the micromagnetic energy. It turns out that the conditions on uniqueness coincide with those for the well-known Navier-Stokes equations in bounded domains: weak solutions are unique in two spatial dimensions, and weak solutions satisfying the Prodi-Serrin conditions are unique among all weak solutions in three dimensions. Read More

We consider the operator ${\mathcal A}_h=-\Delta+iV$ in the semi-classical $h\rightarrow 0$, where $V$ is a smooth real potential with no critical points. We obtain both the left margin of the spectrum, as well as resolvent estimates on the left side of this margin. We extend here previous results obtained for the Dirichlet realization of ${\mathcal A}_h$ by removing significant limitations that were formerly imposed on $V$. Read More

We study the plurisubharmonic envelopes of functions in the setting of domains in $\mathbb C^n$. In particular we prove a complex analogue of a result of De Philippis and Figalli concerning the optimal regularity of such envelopes in smooth strictly pseudoconvex domains. Read More

Here, we introduce a numerical approach for a class of Fokker-Planck (FP) equations. These equations are the adjoint of the linearization of Hamilton-Jacobi (HJ) equations. Using this structure, we show how to transfer the properties of schemes for HJ equations to the FP equations. Read More

For the trace of Besov spaces $B^s_{p,q}$ onto a hyperplane, the borderline case with $s=\frac{n}{p}-(n-1)$ and $0Read More

In this paper, we construct the fundamental solution to a degenerate diffusion of Kolmogorov type and develop a time-discrete variational scheme for its adjoint equation. The so-called mean squared derivative cost function plays a crucial role occurring in both the fundamental solution and the variational scheme. The latter is implemented by minimizing a free energy functional with respect to the Kantorovich optimal transport cost functional associated with the mean squared derivative cost. Read More

We consider a general class of nonlinear parabolic systems corresponding to thermodynamically consistent gradient structure models of bulk-interface interaction. The setting includes non-smooth geometries and linear, slow, fast and entropic diffusion coefficients. The main results are global well-posedness and exponential stability of equilibria. Read More

In this paper, we discuss the existence of solutions for the following non-local critical systems: \begin{equation*} \begin{cases} (-\Delta)^{s}u= \mu_{1}|u|^{2^{\ast}-2}u+\frac{\alpha\gamma}{2^{\ast}}|u|^{\alpha-2}u|v|^{\beta} \ \ \ in \ \ R^{N}, (-\Delta)^{s}v= \mu_{2}|v|^{2^{\ast}-2}v+\frac{\beta\gamma}{2^{\ast}}|u|^{\alpha}|v|^{\beta-2}v\ in \ \ R^{N}, u,v\in D_{s}(R^{N}). \end{cases} \end{equation*} By using the Nehari manifold,\ under proper conditions, we establish the existence and nonexistence of positive least energy solution. Read More

A growth fragmentation equation with constant dislocation density measure is considered, in which growth and division rates balance each other. This leads to a simple example of equation where the so called Malthusian hypothesis $(M_+)$ of J. Bertoin and A. Read More

Here, we study radial solutions for first- and second-order stationary Mean-Field Games (MFG) with congestion on $\mathbb{R}^d$. MFGs with congestion model problems where the agents' motion is hampered in high-density regions. The radial case, which is one of the simplest non one-dimensional MFG, is relatively tractable. Read More

The main purpose of the present paper is to establish a link between quadrature surfaces (potential theoretic concept) and sandpile dynamics (Laplacian growth models). For this aim, we introduce a new model of Laplacian growth on the lattice $\mathbb{Z}^d$ $(d\geq 2)$ which continuously deforms occupied regions of the \emph{divisible sandpile} model of Levine and Peres, by redistributing the total mass of the system onto $\frac 1m$-sub-level sets of the odometer which is a function counting total emissions of mass from lattice vertices. In free boundary terminology this goes in parallel with singular perturbation, which is known to converge to a Bernoulli type free boundary. Read More

Every compact Lie group has a unique covering group which is a product of circles and simply connected semisimple Lie groups. We may equip each component of the covering group a canonical metric, with the requirement that the periods of the geodesic flow on each component are rationally related to each other. Then we push down the metric from the covering group to the original compact Lie group, and call such a metric a "rational metric". Read More

The present study is concerned with the following fractional Schr\"{o}dinger-Poisson system with steep potential well: $$ \left\{% \begin{array}{ll} (-\Delta)^s u+ \la V(x)u+K(x)\phi u= f(u), & x\in\R^3, (-\Delta)^t \phi=K(x)u^2, & x\in\R^3, \end{array}% \right. $$ where $s,t\in(0,1)$ with $4s+2t>3$, and $\la>0$ is a parameter. Under certain assumptions on $V(x)$, $K(x)$ and $f(u)$ behaving like $|u|^{q-2}u$ with $2Read More

We extend the resolvent estimate on the sphere to exponents off the line $\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$. Since the condition $\frac{1}{r}-\frac{1}{s}=\frac{2}{n}$ on the exponents is necessary for a uniform bound, one cannot expect estimates off this line to be uniform still. The essential ingredient in our proof is an $(L^{r}, L^{s})$ norm estimate on the operator $H_{k}$ that projects onto the space of spherical harmonics of degree $k$. Read More

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifold of non-negative sectional curvature. Using a concept of "duality" for strictly convex hypersurfaces, we also obtain a new type of inequalities, so-called "pseudo"-Harnack inequalities, for expanding flows in the sphere and in the hyperbolic space. Read More

In this paper, we study traveling wave solutions and peakon weak solutions of the modified Camassa-Holm (mCH) equation with dispersive term $2ku_x$ for $k\in\mathbb{R}$. We study traveling wave solutions through a Hamiltonian system obtained from the mCH equation by using a nonlinear transformation. The typical traveling wave solutions given by this Hamiltonian system are unbounded or multi-valued. Read More

We consider the Kramers--Smoluchowski equation at a low temperature regime and show how semiclassical techniques developed for the study of the Witten Laplacian and Fokker--Planck equation provide quantitative results. This equation comes from molecular dynamics and temperature plays the role of a semiclassical paramater. The presentation is self-contained in the one dimensional case, with pointers to the recent paper \cite{Mi16} for results needed in higher dimensions. Read More

We consider the identification of nonlinear diffusion coefficients of the form $a(t,u)$ or $a(u)$ in quasi-linear parabolic and elliptic equations. Uniqueness for this inverse problem is established under very general assumptions using partial knowledge of the Dirichlet-to-Neumann map. The proof of our main result relies on the construction of a series of appropriate Dirichlet data and test functions with a particular singular behavior at the boundary. Read More

A second-order regularity theory is developed for solutions to a class of quasilinear elliptic equations in divergence form, including the $p$-Laplace equation, with merely square-integrable right-hand side. Our results amount to the existence and square integrability of the weak derivatives of the nonlinear expression of the gradient under the divergence operator. This provides a nonlinear counterpart of the classical $L^2$-coercivity theory for linear problems, which is missing in the existing literature. Read More

We study the two-dimensional stationary Navier-Stokes equations with rotating effect in the whole space. The unique existence and the asymptotics of solutions are obtained without the smallness assumption on the rotation parameter. Read More

We consider a stochastic control problem with the assumption that the system is controlled until the state process breaks the fixed barrier. Assuming some general conditions, it is proved that the resulting Hamilton Jacobi Bellman equations has smooth solution. The aforementioned result is used to solve the optimal dividend and consumption problem. Read More

From the one-dimensional consolidation of fine-grained soils with threshold gradient, it can be derived a special type of Stefan problems where the seepage front, due to the presence of this threshold gradient, exhibits the features of a moving boundary. In this kind of problems, in contrast with the classical Stefan problem, the latent heat is considered to depend inversely with the rate of change of the seepage front. In this paper a one-phase Stefan problem with a latent heat that not only depends on the rate of change of the free boundary but also on its position is studied. Read More

We study the higher gradient integrability of distributional solutions $u$ to the equation $div(\sigma \nabla u) = 0$ in dimension two, in the case when the essential range of $\sigma$ consists of only two elliptic matrices, i.e., $\sigma\in\{\sigma_1, \sigma_2\}$ a. Read More

We study the existence of separable infinite harmonic functions in any cone of R N vanishing on its boundary under the form u(r, $\sigma$) = r --$\beta$ $\omega$($\sigma$). We prove that such solutions exist, the spherical part $\omega$ satisfies a nonlinear eigenvalue problem on a subdomain of the sphere S N --1 and that the exponents $\beta$ = $\beta$ + \textgreater{} 0 and $\beta$ = $\beta$ -- \textless{} 0 are uniquely determined if the domain is smooth. Read More

This paper is concerned with qualitative properties of bounded steady flows of an ideal incompressible fluid with no stagnation point in the two-dimensional plane R^2. We show that any such flow is a shear flow, that is, it is parallel to some constant vector. The proof of this Liouville-type result is firstly based on the study of the geometric properties of the level curves of the stream function and secondly on the derivation of some estimates on the at most logarithmic growth of the argument of the flow. Read More

In this proceeding we expose a particular case of a recent result obtained by the authors regarding the incompressible Navier-Stokes equations in a smooth bounded and simply connected bounded domain, either in 2D or in 3D, with a Navier slip-with-friction boundary condition except on a part of the boundary. This under-determination encodes that one has control over the remaining part of the boundary. We prove that for any initial data, for any positive time, there exists a weak Leray solution which vanishes at this given time. Read More

We study the resolvent of the massive Dirac operator in the Schwarzschild-Anti-de Sitter space-time. After separation of variables, we use standard one dimensional techniques to obtain an explicit formula. We then make use of this formula to extend the resolvent meromorphically accross the real axis. Read More

We prove regularity and well-posedness results for the mixed Dirichlet-Neumann problem for a second order, uniformly strongly elliptic differential operator on a manifold $M$ with boundary $\partial M$ and bounded geometry. Our well-posedness result for the Laplacian $\Delta_g := d^*d \ge 0$ associated to the given metric require the additional assumption that the pair $(M, \partial_D M)$ be of finite width (in the sense that the distance to $\partial_D M$ is bounded uniformly on $M$, where $\partial_D M$ is the Dirichlet part of the boundary). The proof is a continuation of the ideas in our previous paper on the Dirichlet problem on manifolds with boundary and bounded geometry (joint with Bernd Ammann). Read More

We study the existence and stability of stationary solutions of Poisson-Nernst- Planck equations with steric effects (PNP-steric equations) with two counter-charged species. These equations describe steady current through open ionic channels quite well. The current levels in open ionic channels are known to switch between `open' or `closed' states in a spontaneous stochastic process called gating, suggesting that their governing equations should give rise to multiple stationary solutions that enable such multi-stable behavior. Read More

We discuss an initial-boundary value problem for a fractional diffusion equation with Caputo time-fractional derivative where the coefficients are dependent on spatial and time variables and the zero Dirichlet boundary condition is attached. We prove the unique existence of weak and regular solutions. Read More

We study some overdetermined problems for possibly anisotropic degenerate elliptic PDEs, including the well-known Serrin's overdetermined problem, and we prove the corresponding Wulff shape characterizations by using some integral identities and just one pointwise inequality. Our techniques provide a somehow unified approach to this variety of problems. Read More

We prove convergence results for expanding curvature flows in the Euclidean and hyperbolic space. The flow speeds have the form $F^{-p}$, where $p>1$ and $F$ is a positive, strictly monotone, homogeneous of degree $1$ and concave curvature function. In particular this class includes the mean curvature $F=H$. Read More

It is well known in the combustion community that curvature effect in general slows down flame propagation speeds because it smooths out wrinkled flames. However, such a folklore has never been justified rigorously. In this paper, as the first theoretical result in this direction, we prove that the turbulent flame speed (an effective burning velocity) is decreasing with respect to the curvature diffusivity (Markstein number) for shear flows in the well known G-equation model. Read More