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We consider the nonlinear Schr{ö}dinger equation with a potential, also known as Gross-Pitaevskii equation. By introducing a suitable spectral localization, we prove low regularity error estimates for the time discretization corresponding to an adapted Lie-Trotter splitting scheme. |
An approach to some "optimal" (more precisely, non-improvable) regularity of solutions of the thin film equation u_{t} = -\nabla \cdot(|u|^{n} \nabla \D u) in \ren \times \re_+, u(x,0)=u_0(x) in \re^N, where n in (0,2) is a fixed exponent, with smooth compactly supported initial data u_0(x), in dimensions $N \g... |
We study the relaxation of multiple integrals of the calculus of variations, where the integrands are nonconvex with convex effective domain and can take the value \infty. We use local techniques based on measure arguments to prove integral representation in Sobolev spaces of functions which are almost everywhere diff... |
The two scale convergence of the solution to a Robin's type-like problem of a stationary diffusion problem in a periodically perforated domain is investigated. It is shown that the Robin's problem converges to a problem associated to a new operator which is the sum of a standard homogenized operator plus an ex... |
We consider semilinear Robin problems driven by the negative Laplacian plus an indefinite potential and with a superlinear reaction term which need not satisfy the Ambrosetti-Rabinowitz condition. We prove existence and multiplicity theorems (producing also an infinity of smooth solutions) using variational tools, tru... |
We consider a sequence of Leray-Hopf weak solutions of the 2D Navier-Stokes equations on a bounded domain, in the vanishing viscosity limit. We provide sufficient conditions on the associated vorticity measures, away from the boundary, which ensure that as the viscosity vanishes the sequence converges to a weak soluti... |
We prove sharp bounds on the enstrophy growth in viscous scalar conservation laws. The upper bound is, up to a prefactor, the enstrophy created by the steepest viscous shock admissible by the $L^\infty$ and total variation bounds and viscosity. |
We prove the nonexistence of two-dimensional solitary gravity water waves with subcritical wave speeds and an arbitrary distribution of vorticity. This is a longstanding open problem, and even in the irrotational case there are only partial results relying on sign conditions or smallness assumptions. |
We derive von-Kármán plate theory from three dimensional, purely atomistic models with classical particle interaction. This derivation is established as a $\Gamma$-limit when considering the limit where the interatomic distance $\varepsilon$ as well as the thickness of the plate $h$ tend to zero. |
We establish the small data solvability of suitable quasilinear wave and Klein-Gordon equations in high regularity spaces on a geometric class of spacetimes including asymptotically de Sitter spaces. We obtain our results by proving the global invertibility of linear operators with coefficients in high regularity $L^2... |
This is the third part of our series of work devoted to the dynamics of an epidemic model with nonlocal diffusions and free boundary. This part is concerned with the rate of accelerated spreading for three types of kernel functions when spreading happens. |
We consider the decay rate of solutions to nonlinear Klein-Gordon systems with a critical type nonlinearity. We will specify the optimal decay rate for a specific class of Klein-Gordon systems containing the dissipative nonlinearites. |
We discuss the problem of well-posedness of the compressible (barotropic) Euler system in the framework of weak solutions. The principle of maximal dissipation introduced by C.M. Dafermos is adapted and combined with the concept of admissible weak solutions. |
Motivated by applications to cell biology, we study the constrained minimization of the Helfrich energy among closed surfaces confined to a container. We show existence of minimizers in the class of bubble trees of spherical weak branched immersions and derive the Euler--Lagrange equations which involve a measure-valu... |
We are concerned with the radially symmetric stationary wave for the exterior problem of two-dimensional Burgers equation. A sufficient and necessary condition to guarantee the existence of such a stationary wave is given and it is also shown that such a stationary wave satisfies nice decay estimates and is time-asymp... |
The existence of large-data weak entropy solutions to a nonisothermal immiscible compressible two-phase unsaturated flow model in porous media is proved. The model is thermodynamically consistent and includes temperature gradients and cross-diffusion effects. |
Liquid crystal elastomers are special cross-linked polymer materials combining the large elastic deformability of elastomers with the orientational orders of liquid crystals. This model exhibits markedly different phenomena than the liquid crystal model due to the strong coupling between mechanical elastic deformation... |
This paper considers a stochastically perturbed Keller-Segel-Navier-Stokes (KS-SNS) system arising from the biomathematics in two dimensions, where the diffusion of fluid is expressed by a fractional Laplacian with an exponent in $[1/2,1]$. Our main result demonstrates that, under appropriate assumptions, the Cauchy p... |
The width of a convex curve in the plane is the minimal distance between a pair of parallel supporting lines of the curve. In this paper we study the width of nodal lines of eigenfunctions of the Laplacian on the standard flat torus. |
We first construct the global unique solution by assuming that the initial data is small in the H^3 norm but its higher order derivatives could be large. If further the initial data belongs to \Dot{H}^{-s} (0\le s<3/2) or \dot{B}_{2,\infty}^{-s} (0< s\le3/2), we obtain the various decay rates of the solution and... |
An accurate functional inequality for Div-BV positive symmetric tensors $A$ in a bounded domain $U\subset\mathbb{R}^n$ arises whenever the tangential part of the normal trace $\gamma_\nu A\sim A\vec\nu$ is a finite measure over $\partial U$. The proof involves an extension operator to a neighbourhood of $\bar U$. |
The global analytic hypoellipticity is proved for a class of second order partial differential equations with non-negative characteristic form globally defined on the torus. The class considered in this work generalizes at some degree the class of sum of squares considered by Bove-Chinni and also by Cordaro-Himonas. |
In this paper, we consider two-scale limits obtained with increasing homogenization periods, each period being an entire multiple of the previous one. We establish that, up to a measure preserving rearrangement, these two-scale limits form a martingale which is bounded: the rearranged two-scale limits themselves conve... |
We consider the Dirichlet problem for the Schrödinger-Hénon system $$ -\Delta u + \mu_1 u = |x|^{\alpha}\partial_u F(u,v),\quad \qquad <br>-\Delta v + \mu_2 v = |x|^{\alpha}\partial_v F(u,v) $$ in the unit ball $\Omega \subset \mathbb{R}^N, N\geq 2$, where $\alpha>-1$ is a parameter and $F: \mathbb{R}^2 \to \mathbb{... |
We study the limit behaviour of singularly-perturbed elliptic functionals of the form \[ \mathcal F_k(u,v)=\int_A v^2\,f_k(x,\nabla u)dx+\frac{1}{\varepsilon_k}\int_A g_k(x,v,\varepsilon_k\nabla |
In this paper, we establish optimal hypoelliptic estimates for a class of kinetic equations, which are simplified linear models for the spatially inhomogeneous Boltzmann equation without angular cutoff. |
This paper studies the boundary value problem on the steady compressible Navier-Stokes-Fourier system in a channel domain $(0,1)\times\mathbb{T}^2$ with a class of generalized slip boundary conditions that were systematically derived from the Boltzmann equation by Coron \cite{Coron-JSP-1989} and later by Aoki et al \ci... |
This paper is devoted to the derivation of $L^2$ - $L^2$ decay estimates for the solution of the homogeneous linear damped wave equation on the Heisenberg group $\mathbf{H}_n$, for its time derivative and for its horizontal gradient. Moreover, we consider the improvement of these estimates when further $L^1(\mathbf{H}... |
We consider a $2\times 2$ system of parabolic equations with first and zeroth coupling and establish a Carleman estimate by extra data of only one component without data of initial values. Then we apply the Carleman estimate to inverse problems of determining some or all of the coefficients by observations in an arbit... |
In <a href="https://arxiv.org/abs/2205.02920" data-arxiv-id="2205.02920" class="link-https">arXiv:2205.02920</a> a variant of the classical elastic flow for closed curves in $\mathbb{R}^{n}$ was introduced, that is more suitable for numerical purposes. Here we investigate the long-time properties of such evolution dem... |
Husimi distributions of Laplace eigenfunctions are special types of `microlocal lifts' of eigenfunctions to phase space. Their weak * limits are the well-known quantum limits or microlocal defect measures of an orthonormal basis $\{ \phi_j\}$ of eigenfunctions on a Riemannian manifold $(M,g)$ . |
This paper is a continuation of our recent work in [9]. |
We construct the propagator of the massless Dirac operator $W$ on a closed Riemannian 3-manifold as the sum of two invariantly defined oscillatory integrals, global in space and in time, with distinguished complex-valued phase functions. The two oscillatory integrals -- the positive and the negative propagators -- cor... |
In this paper, we study a hydrodynamic system modeling the deformation of vesicle membranes in incompressible viscous fluids. The system consists of the Navier-Stokes equations coupled with a fourth order phase-field equation. |
We establish the concept of $\alpha$-dissipative solutions for the two-component Hunter-Saxton system under the assumption that either $\alpha(x)=1$ or $0\leq \alpha(x)<1$ for all $x\in \mathbb{R}$. Furthermore, we investigate the Lipschitz stability of solutions with respect to time by introducing a suitable param... |
We consider the Cauchy problem for a time fractional semilinear heat equation with initial data belonging to inhomogeneous/homogeneous Besov--Morrey spaces. We present sufficient conditions for the existence of local/global-in-time solutions to the Cauchy problem, which cover all existing results in the literature and... |
We study an asymptotic analysis of a coupled system of kinetic and fluid equations. More precisely, we deal with the nonlinear Vlasov-Fokker-Planck equation coupled with the compressible isentropic Navier-Stokes system through a drag force in a bounded domain with the specular reflection boundary condition for the kin... |
In this work we investigate the following chemo-attraction with consumption model in bounded domains of \, $\mathbb{R}^N$ ($N=1,2,3$): <br>$$ \partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad <br>\partial_t v - \Delta v = - u^s v <br>$$ where $s\ge 1$, endowed with isolated boundary conditions and initial co... |
The paper deals with the equation $-\Delta u+a(x) u =|u|^{p-1}u $, $u \in H^1(\mathbb{R}^N)$, with $N\ge 2$, $p>1,\ p<{N+2\over N-2}$ if $N\ge 3$, $a\in L^{N/2}_{loc}(\mathbb{R}^N)$, $\inf a>0$, $\lim_{|x| \to <br>\infty} a(x)= a_\infty$. Assuming on the potential that <br>$\lim_{|x| \to \infty}[a(x)-a_\infty... |
We consider the simplified Ericksen-Leslie model in three dimensional bounded Lipschitz domains. Applying a semilinear approach, we prove local and global well-posedness (assuming a smallness condition on the initial data) in critical spaces for initial data in $L^3_{\sigma}$ for the fluid and $W^{1,3}$ for the direct... |
We consider the incompressible Euler and Navier-Stokes equations in a three-dimensional moving thin domain. Under the assumption that the moving thin domain degenerates into a two-dimensional moving closed surface as the width of the thin domain goes to zero, we give a heuristic derivation of singular limit equations ... |
We study the propagation properties of abstract linear Schrödinger equations of the form $i\partial_t\psi = H_0\psi+V(t)\psi$, where $H_0$ is a self-adjoint operator and $V(t)$ a time-dependent potential. We present explicit sufficient conditions ensuring that if the initial state $\psi_0$ has spectral support in $(-\... |
We establish $\mathrm{C}^{\infty}$-partial regularity results for relaxed minimizers of strongly quasiconvex functionals \begin{align*} \mathscr{F}[u;\Omega]:=\int_{\Omega}F(\nabla u)\,\mathrm{d} x,\qquad u\colon\Omega\to\mathbb{R}^{N}, \end{align*} subject to a $q$-growth condition $|F(z)|\leq c(1+|z|^{q})$, $z\in\mat... |
In this paper, we prove the global-in-time existence of strong solutions to a class of fractional parabolic reaction-diffusion systems set in a bounded open subset of $\mathbb{R}^N$. The diffusion operators are of the form $u_i \mapsto d_i (-\Delta)_{Sp}^{s_i} u_i$, where $0<s_i<1$. |
The aim of this paper is to put the problem of vibroacoustic imaging into the mathematical framework of inverse problems (more precisely, coefficient identification in PDEs) and regularization. We present a model in frequency domain, prove uniqueness of recovery of the spatially varying nonlinearity parameter from mea... |
Global existence for weak solutions to systems of nematic liquid crystals, with non-constant fluid density has been established by several authors. In this paper, we establish the regularity and uniqueness results for solutions to the density dependent nematic liquid crystals system. |
We investigate a nonlocal equation $\partial_tu=\int_{\mathbb{R}^n}J(x-y)(u(y,t)-u(x,t))dy+a(x,t)u^p$ in $\mathbb{R}^n$, where $a$ is unbounded and $J$ belongs to a weighted space. Crucial weighted $L^p$ and interpolation estimates for the Green operator are established by a new method based on the sharp Young's i... |
We discuss the life span of the Cauchy problem for the one-dimensional Schrödinger equation with a single power nonlinearity $\lambda |u|^{p-1}u$ ($\lambda\in\mathbb{C}$, $2\le p<3$) prescribed an initial data of the form $\varepsilon\varphi$. Here, $\varepsilon$ stands for the size of the data. |
We consider a physical model where the total energy is governed by surface tension and attractive screened Coulomb potential on the 3-dimensional space. We obtain different periodic equilibrium patterns i.e. stationary sets for this energy, under some volume constraints. |
We consider a simple nonlinear hyperbolic system modeling the flow of an inviscid fluid. The model includes as state variable the mass density fraction of the vapor in the fluid and then phase transitions can be taken into consideration; moreover, phase interfaces are contact discontinuities for the system. |
The model problem of a plane angle for a second-order elliptic system with Dirichlet, mixed, and Neumann boundary conditions is analyzed. The existence of solutions of the form $r^\lambda v$ is, for each boundary condition, reduced to solving a matrix equation. |
We prove global well-posedness of the two-dimensional exterior Navier-Stokes equations for bounded initial data with a finite Dirichlet integral, subject to the non-slip boundary condition. As an application, we construct global solutions for asymptotically constant initial data and arbitrary large Reynolds numbers. |
In this article we derive quantitative uniqueness and approximation properties for (perturbations) of Riesz transforms. Seeking to provide robust arguments, we adopt a PDE point of view and realize our operators as harmonic extensions, which makes the problem accessible to PDE tools. |
In this paper, we establish the existence of solutions for time-dependent linearly viscoelastic bodies confined to a specified half-space. The confinement condition under consideration is of Signorini type, and is given over the boundary of the linearly viscoelastic body under consideration. |
We consider an initial-boundary value problem for the incompressible four-component Keller-Segel-Navier-Stokes system with rotational flux $$\left\{\begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c)-nm,\quad x\in \Omega, t>0,\\ c_t+u\cdot\nabla c=\Delta c-c+m,\quad x\in \Omega, t>0,\\ m_t... |
In this paper, the existence of positive strong solutions to a Dirichlet $p$-Laplacian problem with reaction both singular at zero and highly discontinuous is investigated. In particular, it is only required that the set of discontinuity points has Lebesgue measure zero. |
We consider the linear field-road system, a model for fast diffusion channels in population dynamics and ecology. This system takes the form of a system of PDEs set on domains of different dimensions, with exchange boundary conditions. |
We consider a family of non-autonomous reaction-diffusion equations with almost periodic, rapidly oscillating principal part and nonlinear interactions. As the frequency of the oscillations tends to infinity, we prove that the solutions of the non-autonomous equations converge to the solutions of the autonomous averag... |
In the present paper, we study the existence, uniqueness and behaviour in time of the solutions to the Darcy-Bénard problem for an extended-quasi-thermal-incompressible fluid-saturated porous medium uniformly heated from below. Unlike the classical problem, where the compressibility factor of the fluid vanishes, in th... |
The present paper is concerned with the analysis of two strongly coupled systems of degenerate parabolic partial differential equations arising in multiphase thin film flows. In particular, we consider the two-phase thin film Muskat problem and the two-phase thin film approximation of the Stokes flow under the influen... |
It is well-known that the rarefaction wave, one of the basic wave patterns to the hyperbolic conservation laws, is nonlinearly stable to the one-dimensional compressible Navier-Stokes equations (cf. [14,15,12,17]). In the present paper we proved the time-asymptotically nonlinear stability of the planar rarefaction wav... |
In this survey paper, we report on recent works concerning exact observability (and, by duality, exact controllability) properties of subelliptic wave and Schr{ö}dinger-type equations. These results illustrate the slowdown of propagation in directions transverse to the horizontal distribution. |
Black solitons are identical in the nonlinear Schrödinger (NLS) equation with intensity-dependent dispersion and the cubic defocusing NLS equation. We prove that the intensity-dependent dispersion introduces new properties in the stability analysis of the black soliton. |
We analyze a class of partial differential equations that arise as "backwards Kolmogorov operators" in infinite population limits of the Wright-Fisher models in population genetics and in mathematical finance. These are degenerate elliptic operators defined on manifolds with corners. |
If $u$ is a smooth solution of the Navier--Stokes equations on ${\mathbb R}^3$ with first blowup time $T$, we prove lower bounds for $u$ in the Sobolev spaces $\dot H^{3/2}$, $\dot H^{5/2}$, and the Besov space $\dot B^{5/2}_{2,1}$, with optimal rates of blowup: we prove the strong lower bounds $\|u(t)\|_{\dot H^{3/2}}... |
We study the rate of growth of sharp fronts of the Quasi-geostrophic equation and 2D incompressible Euler equations. . |
We consider the initial value problem for the Dirac-Klein-Gordon equations in two space dimensions. Global regularity for $C^\infty$ data was proved by Grünrock and Pecher. |
In this article, we will prove $L^2(\mathbb{R})$-stability of $1$-solitons for the KdV equation by using exponential stability property of the semigroup generated by the linearized operator. The proof follows the lines of recent stability argument of Mizumachi [Asymptotic stability of lattice solitons in the energy sp... |
We consider the focusing inhomogeneous nonlinear Schrödinger equation \[ i\partial_t u + \Delta u + |x|^{-b}|u|^\alpha u = 0\qtq{on}\R\times\R^N, \] with $\alpha=\tfrac{4-2b}{N-2}$, $N=\{3,4,5\}$ and $0<b\leq \min\Big\{\tfrac{6-N}{2},\tfrac{4}{N}$\Big\}. This paper establishes global well-posedness and scattering f... |
We are concerned with a nonstandard phase field model of Cahn-Hilliard type. The model, which was introduced by Podio-Guidugli (Ric. Mat. 2006), describes two-species phase segregation and consists of a system of two highly nonlinearly coupled PDEs. |
Uniform error estimates of a bi-fidelity method for a kinetic-fluid coupled model with random initial inputs in the fine particle regime are proved in this paper. Such a model is a system coupling the incompressible Navier-Stokes equations to the Vlasov-Fokker-Planck equations for a mixture of the flows with distinct ... |
We study stability times for a family of parameter dependent nonlinear Schrödinger equations on the circle, close to the origin. Imposing a suitable Diophantine condition (first introduced by Bourgain), we prove a rather flexible Birkhoff Normal Form theorem, which implies, e.g., exponential and sub-exponential time e... |
In this paper, we study the number of traveling wave families near a shear flow under the influence of Coriolis force, where the traveling speeds lie outside the range of the flow $u$. Under the $\beta$-plane approximation, if the flow $u$ has a critical point at which $u$ attains its minimal (resp. maximal) value, th... |
A class of linear degenerate elliptic equations inspired by nonlinear diffusions of image processing is considered. It is characterized by an interior degeneration of the diffusion coefficient. |
This paper is motivated by a gauged Schrodinger equation in dimension 2 including the so-called Chern-Simons term. The radially symmetric case leads to an elliptic problem with a nonlocal defocusing term, in competition with a local focusing nonlinearity. |
We consider the Derivative NLS equation with general quadratic nonlinearities. In \cite{be2} the first author has proved a sharp small data local well-posedness result in Sobolev spaces with a decay structure at infinity in dimension $n = 2$. |
We establish a compactness result for solutions of a certain class of hypoelliptic equations. This result allows us to show the existence of global weak solutions to the non-homogeneous Landau-Fermi-Dirac equation with Coulomb potential. |
Decay rates for the energy of solutions of the damped wave equation on the torus are studied. In particular, damping invariant in one direction and equal to a sum of squares of nonnegative functions with a particular number of derivatives of regularity is considered. |
We prove global well-posedness of strong solutions and existence of the global attractor for the 2D Boussinesq system in a periodic channel with fractional Laplacian in subcritical case. The analysis reveals a relation between the Laplacian exponent and the regularity of the spaces of velocity and temperature. |
We prove the existence and stability of smooth solutions to the steady Navier-Stokes equations near plane Poiseuille-Couette flow. Consequently, we also provide the zero viscosity limit of the 2D steady Navier-Stokes equations to the steady Euler equations. |
Assume that $M$ is a CR compact manifold without boundary and CR Yamabe invariant $\mathcal{Y}(M)$ is positive. Here, we devote to study a class of sharp Hardy-Littlewood-Sobolev inequality as follows <br>\begin{equation*} <br>\Bigl| \int_M\int_M [G_\xi^\theta(\eta)]^{\frac{Q-\alpha}{Q-2}} f(\xi) g(\eta) dV_\theta(\xi... |
The leading-order asymptotic behavior of the solution of the Cauchy initial-value problem for the Benjamin-Ono equation in $L^2(\mathbb{R})$ is obtained explicitly for generic rational initial data $u_0$. An explicit asymptotic wave profile $u^\mathrm{ZD}(t,x;\epsilon)$ is given, in terms of the branches of the multiv... |
We investigate models for nonlinear ultrasound propagation in soft biological tissue based on the one that serves as the core for the software package k-Wave. The systems are solved for the acoustic particle velocity, mass density, and acoustic pressure and involve a fractional absorption operator. |
We demonstrate that the second eigenfunction of a perturbed fractional Laplace operator on a bounded interval can exhibit two sign changes, in stark contrast with the classical expectation that it should have exactly one zero. Our construction employs the Kato-Rellich regular perturbation theory to analyse an infinite... |
When the velocity field is not a priori known to be globally almost Lipschitz, global uniqueness of solutions to the two-dimensional Euler equations has been established only in some special cases, and the solutions to which these results apply share the property that the diffuse part of the vorticity is constant near ... |
The paper concerns nonlocal time-periodic boundary value problems for first-order Volterra integro-differential hyperbolic systems with boundary inputs. The systems are subjected to integral boundary conditions. |
Let $(u,b)$ be a smooth enough solution of 3-D incompressible MHD system. We prove that if $(u,b)$ blows up at a finite time $T^*$, then for any $p\in]4,\infty[$, there holds $\int_0^{T^*}\bigl(\|u^3(t')\|^p_{\dH^{\frac 12+\frac 2p}}+\|b(t')\|^p_{\dH^{\frac 12+\frac 2p}}\bigr)dt'=\infty$. |
A representation formula for the solution of the $\infty$-Laplace equation is constructed in a punctured square, the prescribed boundary values being $u=0$ on the sides and $u=1$ at the centre. This so-called $\infty$-potential is obtained with a hodograph method. |
We are concerned with the focusing $L^2$-critical nonlinear Schrödinger equations in $\mathbb{R}^d$ for $d=1,2$. The uniqueness is proved for a large energy class of multi-bubble blow-up solutions, which converge to a sum of $K$ pseudo-conformal blow-up solutions particularly with low rate $(T-t)^{0+}$, as $t\to T$, $... |
We consider the inverse source problem arising in thermo- and photo-acoustic tomography. It consists in reconstructing the initial pressure from the boundary measurements of the acoustic wave. |
We prove Li--Yau-type lower bounds for the eigenvalues of the Stokes operator and give applications to the attractors of the Navier--Stokes equations. |
We investigate the convergence as $p\searrow1$ of the minimizers of the $W^{s,p}$-energy for $s\in(0,1)$ and $p\in(1,\infty)$ to those of the $W^{s,1}$-energy, both in the pointwise sense and by means of $\Gamma$-convergence. We also address the convergence of the corresponding Euler-Lagrange equations, and the equiva... |
This paper is devoted to reaction-diffusion equations with bistable nonlinearities depending periodically on time. These equations admit two linearly stable states. |
In this paper, we study the asymptotic behavior of the heat kernel with respect to the Witten Laplacian. We introduce the localization and the scaling technique in semi-classical analysis, and study the semi-classical asymptotic behavior of the family of the heat kernel, indexed by $k$, near the critical point $p$ of ... |
We study the asymptotic behaviour of positive solutions of the Cauchy problem for the fast diffusion equation near the extinction time. We find a continuum of rates of convergence to a self-similar profile. |
We prove the well-posedness for the non-cutoff Boltzmann equation with soft potentials when the initial datum is close to the {\it global Maxwellian} and has only polynomial decay at the large velocities in $L^2$ space. As a result, we get the {\it propagation of the exponential moments} and the {\it sharp rates} of t... |
Inspired by the recently published paper \cite{Hassainia-Hmidi}, the current paper investigates the local well-posedness for the generalized $2d-$Boussinesq system in the setting of regular/singular vortex patch. Under the condition that the initial vorticity $\omega_{0}={\bf 1}_{D_0}$, with $\partial D_0$ is a Jordan... |
We consider minimizers of a Ginzburg-Landau energy with a discontinuous and rapidly oscillating pinning term, subject to a Dirichlet boundary condition of degree $d > 0$. The pinning term models an unbounded number of small impurities in the domain. |
A free boundary problem for the dynamics of a glasslike binary fluid naturally leads to a singular perturbation problem for a strongly degenerate parabolic partial differential equation in 1D. We present a conjecture for an asymptotic formula for the velocity of the free boundary and prove a weak version of the conjec... |
We study a weighted $\frac{N}{2}$ biharmonic equation involving a positive continuous potential in $\overline{B}$. The non-linearity is assumed to have critical exponential growth in view of logarithmic weighted Adams' type inequalities in the unit ball of $\mathbb{R}^{N}$. |
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