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In this paper, we consider an inverse conductivity problem on a bounded domain $\Omega\subset\mathbb{R}^n$, $n\geq2$, also known as Electrical Impedance Tomography (EIT), for the case where unknown impenetrable obstacles are embedded into $\Omega$. We show that a piecewise-constant conductivity function and embedded o... |
In prior work, the author has characterized the real numbers $a,b,c$ and $1\leq p,q,r<\infty $ such that the weighted Sobolev space $W_{\{a,b\}}^{(q,p)}(R^{N}\backslash \{0}):=\{u\in L_{loc}^{1}(R^{N}\backslash \{0}):|x|^{\frac{a}{q}}u\in L^{q}(R^{N}),|x|^{\frac{b}{p}}\nabla u\in (L^{p}(R^{N}))^{N}\}$ is continuousl... |
In this work we prove the existence of a classical positive solution for an elliptic equation with a sublinear term. We use Galerkin approximations to show existence of such solution on bounded domains in RN. |
We investigate a class of n-dimensional free boundary elliptic problems which includes the dam problem, the aluminum problem, and the lubrication problem. We establish that the free boundary in this class is a porous set, which implies its Hausdorff dimension being less than $n$, which in turn leads to its Lebesgue me... |
In this paper we introduce a unified and compact method for proving $L^p$ Hardy and Rellich-type inequalities. The most important feature of our method is that it allows us to extend these inequalities beyond the traditional Euclidean setting. |
In this paper, we are concerned with the recovery of the geometric shapes of inhomogeneous inclusions from the associated far field data in electrostatics and acoustic scattering. We present a local resolution analysis and show that the local shape around a boundary point with a high magnitude of mean curvature can be... |
The aim of this paper is to get the boundedness of certain multi-sublinear operators generated by multilinear fractional integral operators on the product generalized local Morrey spaces under generic size conditions which are satisfied by most of the operators in harmonic analysis. We also prove that the commutators ... |
We propose an approach for the simultaneous reconstruction of the electromagnetic and acoustic material parameters, in the given medium $\Omega$ where to image, using the photo-acoustic pressure, measured on a single point of the boundary of $\Omega$, generated by plasmonic nanoparticles. We prove that the generated p... |
The paper concerns the weak differentiability of weak solutions to two kinds of nonuniform nonlinear degenerate elliptic systems under the $p,q$-growth condition on the Heisenberg Group. We use the iteration to fractional difference quotients on the Heisenberg Group to get the weak differentiability of weak solution $... |
Many technologically useful materials are polycrystals composed of a myriad of small monocrystalline grains separated by grain boundaries. Dynamics of grain boundaries play an essential role in defining the materials properties across multiple scales. |
We study the existence theory of solutions of the kinetic Bohmian equation, a nonlinear Vlasov-type equation proposed for the phase-space formulation of Bohmian mechanics. Our main idea is to interpret the kinetic Bohmian equation as a Hamiltonian system defined on an appropriate Poisson manifold built on a Wasserstei... |
The article deals with electrodynamics in the presence of anisotropic materials having scalar wave impedance. Maxwell's equations written for differential forms over a 3-manifold are analysed. |
We consider the problem of output feedback regulationfor a linear first-order hyperbolic system with collocatedinput and output in presence of a general class of disturbancesand noise. The proposed control law is designed through abackstepping approach incorporating an integral action. |
This paper is concerned with uniqueness in inverse acoustic scattering with phaseless far-field data at a fixed frequency. In our previous work ({\em SIAM J. Appl. |
In this paper, we prove a sharp anisotropic $L^p$ Minkowski inequality involving the total $L^p$ anisotropic mean curvature and the anisotropic $p$-capacity, for any bounded domains with smooth boundary in $\mathbb{R}^n$. As consequences, we obtain an anisotropic Willmore inequality, a sharp anisotropic Minkowski ineq... |
We establish the conditional asymptotic stability in a local energy norm of the unstable soliton for the one-dimensional quadratic Klein-Gordon equation under even perturbations. A key feature of the problem is the positive gap eigenvalue exhibited by the linearized operator around the soliton. |
Let $(M,g)$ be a closed Riemannian manifold of dimension at least $3$. Let $S$ be the union of the focal submanifolds of an isoparametric function on $(M,g)$. |
We analyze gradient flows with jumps generated by a finite set of complete vector fields in involution using some Radon measures $u\in \mathcal{U}_a$ as admissible perturbations. Both the evolution of a bounded gradient flow $\{x^u(t,ł)\in B(x^*,3\g)\subseteq \mbn: \,t\in[0,T],\,ł\in B(x^*,2\g)\}$ and the unique solut... |
We consider the functional given by the product of the first Dirichlet eigenvalue and the torsional rigidity of planar domains normalized by the area. This scale invariant functional was studied by Pólya and Szegő in 1951 who showed that it is bounded above by 1 for all domains. |
In this note we present the current status of the derivation of a viscous Serre-Green-Naghdi system. For this goal, the flow domain is separated into two regions. |
Various thermodynamical phenomena have occurred with change of pressure and temperature, volume. We can choose these parameters but not these constraints, in order to need the thermodynamics with physical properties in the fields of various situations. |
We consider the Burgers equation posed on the outer communication region of a Schwarzschild black hole spacetime. Assuming spherical symmetry for the fluid flow under consideration, we study the propagation and interaction of shock waves under the effect of random forcing. |
We prove short-time existence for the negative $L^2$-gradient flow of the $p$-elastic energy of curves via a minimising movement scheme. In order to account for the degeneracy caused by the energy's invariance under curve reparametrisations, we write the evolving curves as approximate normal graphs over a fixed sm... |
In this paper, we study quasilinear elliptic equations with the nonlinearity modelled after the $p(x)$-Laplacian on nonsmooth domains and obtain sharp Calderón-Zygmund type estimates in the variable exponent setting. In a recent work of \cite{BO}, the estimates obtained were strictly above the natural exponent and hen... |
$F$-concavity is a generalization of power concavity and, actually, the largest available generalization of the notion of concavity. We characterize the $F$-concavities preserved by the Dirichlet heat flow in convex domains on ${\mathbb R}^n$, and complete the study of preservation of concavity properties by the Diric... |
This note is devoted to a simple proof of the generalized Leibniz rule in bounded domains. The operators under consideration are the so-called spectral Laplacian and the restricted Laplacian. |
A quantitative Brenier decomposition shows that the deviation of a map from volume preserving is bounded by the deviation of the derivative from volume preserving. A study of the matrix nearness problem for $SL(n)$ and $Sp(2n)$ relates the estimate to incompressible deformations of elastic materials. |
We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation u_t+(-\partial^2_x)^{\alpha/2} u+uu_x=0 with alpha in (0,1], supplemented with an initial datum approaching the constant states u+/u- (u_-smaller than u_+) as x goes to +/-infty, respectively. It was shown by Karch, M... |
We investigate a time-harmonic wave problem in a waveguide. We work at low frequency so that only one mode can propagate. |
In this study, we consider a general time-space system, whose model operator and observation operator are locally Lipschitz continuous, over a finite time horizon and parameter identification by using Landweber-Kaczmarz regularization. The problem is investigated in two different modeling settings: An All-at-once and ... |
The linearized Korteweg-De Vries equation can be written as a Hamilton-like system. However, the Hamilton energy depends on the time, and is a nonsymmetric operator on $L^2({\bf R})$. |
Denote with $\mu_{1}(\Omega;e^{h\left(|x|\right)})$ the first nontrivial eigenvalue of the Neumann problem \begin{equation*} \left\{\begin{array}{lll} -\text{div}\left(e^{h\left(|x|\right)}\nabla u\right) =\mu e^{h\left(|x|\right)}u & \text{in} & \Omega & & \frac{\partial u}{\partial \nu}=0 & \text{... |
We have proved theorems on compact classes of homeomorphisms with hydrodynamic normalization that are solutions of the Beltrami equation, whose characteristics are compactly supported and satisfy certain constraints of the theoretical-set type. As a consequence, we obtained results on compact classes of solutions of c... |
The objective of this paper is to establish a connection between the problem of optimal regularity among solutions to elliptic PDEs with measurable coefficients and the Liouville property at infinity. Initially, we address the two-dimensional case by proving an Alt-Caffarelli-Friedman type monotonicity formula, enabli... |
The super-critical Brezis-Nirenberg problem in an annulus is considered. The new uniqueness result of positive radial solutions is established for the three-dimensional case. |
The role of predator evasion mediated by chemical signaling is studied in a diffusive prey-predator model when prey-taxis is taken into account (model A) or not (model B) with taxis strength coefficients $\chi$ and $\xi$ respectively. In the kinetic part of the models it is assumed that the rate of prey consumption in... |
We consider, for $h,E>0$, the semiclassical Schrödinger operator $-h^2\Delta + V - E$ in dimension two and higher. The potential $V$, and its radial derivative $\dell_{r}V$ are bounded away from the origin, have long-range decay and $V$ is bounded by $r^{-\delta}$ near the origin while $\dell_{r}V$ is bounded by $r... |
The convergence of multiple Fourier series of functions of bounded partial $% \Lambda$-variation is investigated. The sufficient and necessary conditions on the sequence $\Lambda=\{\lambda_n\}$ are found for the convergence of multiple Fourier series of functions of bounded partial $\Lambda$-variation. |
In this paper, we develop a Galerkin-type approximation, with quantitative error estimates, for weak solutions to the Cauchy problem for kinetic Fokker-Planck equations in the domain $(0, T) \times D \times \mathbb{R}^d$, where $D$ is either $\mathbb{T}^d$ or $\mathbb{R}^d$. Our approach is based on a Hermite expansio... |
We present a solution of the Weiss operator family generalized for the case of $\mathbb{R}^{d}$ and formulate a d-dimensional analogue of the Weiss Theorem. Most importantly, the generalization of the Weiss Theorem allows us to find a sub-set of null class functions for a partial differential equation with the general... |
We describe completely 2-solitary waves related to the ground state of the nonlinear damped Klein-Gordon equation \begin{equation*} \partial_{tt}u+2\alpha\partial_{t}u-\Delta u+u-|u|^{p-1}u=0 \end{equation*} on $\bf R^N$, for $1\leq N\leq 5$ and energy subcritical exponents $p>2$. The description is twofold. |
We examine the Hénon equation $ -\Delta u =|x|^\alpha u^p$ in $ \Omega \subset \mathbb{R}^N$ with $u=0$ on $ \partial \Omega$ where $ 0 < \alpha$. We show there exists a sequence $ \{p_k\}_k \subset [ \frac{N+2}{N-2}, p_{\alpha}(N)]$ with $p_1 < p_2 <p_3 < ...$, $ p_k \nearrow p_{\alpha}(N)$ such that for ... |
We consider the regular Lagrangian flow X associated to a bounded divergence-free vector field b with bounded variation. We prove a Lusin-Lipschitz regularity result for X and we show that the Lipschitz constant grows at most linearly in time. |
Let $\Omega_0$ denote the unit ball of $\mathbb{R}^N$ ($N\ge 2$) centered at the origin. We suppose that $\Omega_0$ contains a core, given by a smaller concentric ball $D_0$, made of a (possibly) different material. |
The Merriman-Bence-Osher (MBO) scheme, also known as thresholding or diffusion generated motion, is an efficient numerical algorithm for computing mean curvature flow (MCF). It is fairly well understood in the case of hypersurfaces. |
The focusing cubic wave equation in three spatial dimensions has the explicit solution $\sqrt{2}/t$. We study the stability of the blowup described by this solution as $t \to 0$ without symmetry restrictions on the data. |
The Cauchy problem of the compressible Oldroyd-B model without damping mechanism in R^n$ with $n\ge2$ is considered. The lack of dissipation in density and stress tensor in the model is compensated by exploiting an intrinsic structure and introducing new quantities between density, velocity and stress tensor. |
We show existence and uniqueness for a linearized water wave problem in a two dimensional domain $G$ with corner, formed by two semi-axis $\Gamma_1$ and $\Gamma_2$ which intersect under an angle $\alpha\in (0,\pi ]$. The existence and uniqueness of the solution is proved by considering an auxiliary mixed problem with ... |
We study the adjointness problem for the closed extensions of a general b-elliptic operator A in x^{-\nu}Diff^m_b(M;E), \nu>0, initially defined as an unbounded operator A:C_c^\infty(M;E)\subset x^\mu L^2_b(M;E)\to x^\mu L^2_b(M;E), \mu \in \R. The case where A is a symmetric semibounded operator is of particular i... |
In this article we investigate the dynamics of the initial-boundary value problem for the nonlinear marine riser equation in the presence of time dependent boundary conditions at the top end and a time dependent coefficient of the nonlinear drag force. We introduce sufficient conditions on these functions to maintain ... |
In this paper we obtain interior regularity estimates for viscosity solutions of nonlocal Dirichlet problems that degenerate when the gradient of the solution vanishes. Interior Hölder estimates are obtained when the order of the fractional diffusion is less or equal than one, and Lipschitz estimates when it is bigger... |
In this paper, we prove a uniqueness result in the inverse problem of determining several non-constant coefficients of one-dimensional reaction-diffusion equations. Such reaction-diffusion equations include the classical model of Kolmogorov, Petrovsky and Piskunov as well as more sophisticated models from biology. |
Let $(M,\bar g)$ be a compact Riemannian manifold with minimal boundary such that the second fundamental form is nowhere vanishing on $\partial M$. We show that for a generic Riemannian metric $\bar g$, the squared norm of the second fundamental form is a Morse function, i.e. all its critical points are non-degenerate... |
We consider the angle $\theta$ of inclination (with respect to the horizontal) of the profile of a steady 2D inviscid symmetric periodic or solitary water wave subject to gravity. Although $\theta$ may surpass 30$^\circ$ for some irrotational waves close to the extreme wave, Amick [Ami87] proved that for any irrotatio... |
In a gas transport system, the customer behavior is uncertain. Motivated by this situation, we consider a boundary stabilization problem for the flow through a gas pipeline, where the outflow at one end of the pipe that is governed by the customer's behavior is uncertain. |
The thesis deals with the Aw-Rascle-Zhang model for traffic. We have applied the model to describe the influence of a large and slow vehicle (a bus or a truck) on the traffic. |
We consider the problem of finding and describing minimisers of the Rayleigh quotient \[ \Lambda_\infty \, :=\, \inf_{u\in \mathcal{W}^{2,\infty}(\Omega)\setminus\{0\} }\frac{\|\Delta u\|_{L^\infty(\Omega)}}{\|u\|_{L^\infty(\Omega)}}, \] where $\Omega \subseteq \mathbb{R}^n$ is a bounded $C^{1,1}$ domain and $\mathcal{... |
The paper addresses the question of existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the $L^p$-condition for velocity or vorticity and for a range of scaling exponents. |
In this paper, we study the semilinear elliptic equation of the form \begin{eqnarray*} -\Delta u+a(x)|u|^{p-2}u-b(x)|u|^{q-2}u=0 \end{eqnarray*} on lattice graphs $\mathbb{Z}^{N}$, where $N\geq 2$ and $2\leq p<q<+\infty$. By the Brézis-Lieb lemma and concentration compactness principle, we prove the existence of... |
In this paper we examine the inverse problem of determining the twisting function for the dynamicSchrodinger operator in a quantum waveguide from some suitable localized, either interior or boundary. |
In the present article, we study the numerical approximation of a system of Hamilton-Jacobi and transport equations arising in geometrical optics. We consider a semi-Lagrangian scheme. |
This paper shows global uniqueness in an inverse problem for a fractional magnetic Schrödinger equation (FMSE): an unknown electromagnetic field in a bounded domain is uniquely determined up to a natural gauge by infinitely many measurements of solutions taken in arbitrary open subsets of the exterior. The proof is ba... |
We construct unique regular solutions to the minimal nonlinear system of the 1d thermoelasticity. The obtained solution has a positive temperature. |
We establish a nonlinear Calderón-Zygmund $L^2$-theory to the Dirichlet problem $$-|Du|^{\gamma}\Delta^N_p u=f\in L^2(\Omega)\quad {\rm in}\quad \Omega; \quad u=0 \ \mbox{on $\partial\Omega$} $$ for $n\ge2$, $ p>1$ and a large range of $\gamma>-1$, in particular, for all $p>1$ and all $ \gamma>-1$ when $n=2... |
We consider the Euler-Korteweg system with space periodic boundary conditions $ x \in \mathbb T^d $. We prove a local in time existence result of classical solutions for irrotational velocity fields requiring natural minimal regularity assumptions on the initial data. |
We study nonlinear wave equations on $\mathbb R^{2+1}$ with quadratic derivative nonlinearities, which include in particular nonlinearities exhibiting a null form structure, with random initial data in $H_x^1\times L^2_x$. In contrast to the counterexamples of Zhou \cite{Zhou} and Foschi-Klainerman \cite{FK}, we obtai... |
We prove a $C^{1,\alpha}$ interior regularity theorem for fully nonlinear uniformly elliptic integro-differential equations without assuming any regularity of the kernel. We then give some applications to linear theory and higher regularity of a special class of nonlinear operators. |
Suppose $f(x,y) + \frac{\kappa}{2} \|x\|^2 - \frac{\sigma}{2}\|y\|^2$ is convex where $\sigma>0$, and the argmin function $\gamma(x) = \{ \gamma : \inf_y f(x,y) = f(x,\gamma)\}$ exists and is single valued. We will prove $\gamma$ is differentiable almost everywhere. |
We consider the two dimensional gravity water waves with nonzero constant vorticity in infinite depth. We show that for $s\geq \frac{3}{4}$, the water waves system is locally well-posed in $\mathcal{H}^{s}$, which is the nonzero constant vorticity counterpart of the breakthrough work of Ai-Ifrim-Tataru in [4]. |
In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. |
We prove the norm inflation phenomena for the Boussinesq system on $\mathbb T^3$. For arbitrarily small initial data $(u_0,\rho_0)$ in the negative-order Besov spaces $\dot{B}^{-1}_{\infty, \infty} \times \dot{B}^{-1}_{\infty, \infty}$, the solution can become arbitrarily large in a short time. |
In this chapter, we mainly review theoretical results on inverse source problems for diffusion equations with the Caputo time-fractional derivatives of order $\alpha\in(0,1)$. Our survey covers the following types of inverse problems: |
We consider the initial value problem (IVP) for the 2D generalized Zakharov-Kuznetsov (ZK) equation \begin{equation} \begin{cases} \partial_{t}u+\partial_{x}\Delta u+\mu \partial_{x}u^{k+1}=0, \,\;\; (x, y) \in \mathbb{R}^2, \, t \in \mathbb{R},\\ u(x,y,0)=u_0(x,y), \end{cases} \end{equation} where $\Delta=\partial_x^2... |
We consider a class of nonautonomous elliptic operators ${\mathscr A}$ with unbounded coefficients defined in $[0,T]\times\R^N$ and we prove optimal Schauder estimates for the solution to the parabolic Cauchy problem $D_tu={\mathscr A}u+f$, $u(0,\cdot)=g$. |
We prove interpolation inequalities of Gagliardo-Nirenberg type involving Fourier symbols that vanish on hypersurfaces in $\mathbb{R}^d$. |
This paper investigates the existence of weak solutions of biquasilinear boundary value problem for a coupled elliptic-parabolic system of divergence form with discontinuous leading coefficients. The mathematical framework addressed in the article considers the presence of an additional nonlinearity in the model which... |
We consider conservative and gradient flows for $N$-particle Riesz energies with mean-field scaling on the torus $\mathbb{T}^d$, for $d\geq 1$, and with thermal noise of McKean-Vlasov type. We prove global well-posedness and relaxation to equilibrium rates for the limiting PDE. |
We present new examples of complexes of differential operators of order $k$ (any given positive integer) that satisfy div-curl and/or $L^1$-duality estimates. |
We study the internal controllability of a wave equation with memory in the principal part, defined on the one-dimensional torus $\mathbb{T}=\mathbb{R}/2\pi\mathbb{Z}$. We assume that the control is acting on an open subset $\omega(t)\subset\mathbb{T}$, which is moving with a constant velocity $c\in\mathbb{R}\setminus... |
We consider a perturbed energy critical focusing Nonlinear Schrödinger Equation in three dimensions. We construct solitary wave solutions for focusing subcritical perturbations as well as defocusing supercritical perturbations. |
Consider weakly nonlinear complex Ginzburg--Landau (CGL) equation of the form: $$ u_t+i(-\Delta u+V(x)u)=\epsilon\mu\Delta u+\epsilon \mathcal{P}( u),\quad x\in {R^d}\,, \quad(*) <br>$$ under the periodic boundary conditions, where $\mu\geqslant0$ and $\mathcal{P}$ is a smooth function. Let $\{\zeta_1(x),\zeta_2(x),\d... |
In this paper, we consider the 3-D compressible isentropic MHD equations with infinity electric conductivity. The existence of unique local classical solutions is firstly established when the initial data is arbitrarily large, contains vacuum and satisfies some initial layer compatibility condition. |
We establish a global uniqueness result for an inverse boundary problem with partial data for the magnetic Schrödinger operator with a magnetic potential of class $W^{1,n}\cap L^\infty$, and an electric potential of class $L^n$. Our result is an extension, in terms of the regularity of the potentials, of the results [... |
We present a mathematical study for the development of Multiple Sclerosis in which a spatio-temporal kinetic { theory} model describes, at the mesoscopic level, the dynamics of a high number of interacting agents. We consider both interactions among different populations of human cells and the motion of immune cells, ... |
By applying Wiegner' method in \cite{Wiegner}, we first prove the large time decay estimate for the global solutions of a 2.5 dimensional Navier-Stokes system, which is a sort of singular perturbed 2-D Navier-Stokes system in three space dimension. As an application of this decay estimate, we give a simplified pro... |
The Dirichlet eigenvalues of the Laplace-Beltrami operator are larger on a flat disc than on any other surface of revoltuion immersed in Euclidean space with the same boundary. |
For initial data $f$ in a subcritical Lorentz space $L^{p,q}(\mathbb{R}^{n}) \hookrightarrow \dot B^{-\frac np}_{\infty,\infty}(\mathbb{R}^n)$ ($n<p<\infty$, $1\leq q \leq \infty$), we prove results which imply in particular that a local in time mild Navier-Stokes solution cannot become unbounded in the $L^{p,q}(... |
This paper studies the regularity of Villani solutions of the space homogeneous Landau equation with Coulomb interaction in dimension 3. Specifically, we prove that any such solution belonging to the Lebesgue space L_{t}^{\infty}L_{v}^{q} with q>3 in an open cylinder (0,S)\times B, where B is an open ball of \mathb... |
The goal of this paper is to prove a comparison principle for viscosity solutions of semilinear Hamilton-Jacobi equations in the space of probability measures. The method involves leveraging differentiability properties of the $2$-Wasserstein distance in the doubling of variables argument, which is done by introducing... |
The results on $\Gamma$-limits of sequences of free-discontinuity functionals with bounded cohesive surface terms are extended to the case of vector-valued functions. In this framework, we prove an integral representation result for the $\Gamma$-limit, which is then used to study deterministic and stochastic homogenis... |
We analyze the $\Gamma$-convergence of sequences of free-discontinuity functionals arising in the modeling of linear elastic solids with surface discontinuities, including phenomena as fracture, damage, or material voids. We prove compactness with respect to $\Gamma$-convergence and represent the $\Gamma$-limit in an ... |
By developing the d-bar approach to global "inverse scattering" at zero energy we give a principal effectivization of the global reconstruction method for the Gel'fand-Calderon inverse boundary value problem in three dimensions. This work goes back to results published by the author in 1987, 1988 and proce... |
In this paper, one considers the coupling of a Brinkman model and Stokes equations with jump embedded transmission conditions. In this model, one assumes that the viscosity in the porous region is very small. |
We are concerned with the so-called Boussinesq equations with partial viscosity. These equations consist of the ordinary incompressible Navier-Stokes equations with a forcing term which is transported {\it with no dissipation} by the velocity field. |
We describe high order accurate and stable finite difference schemes for the initial-boundary value problem associated with the magnetic induction equations. These equations model the evolution of a magnetic field due to a given velocity field. |
We study the qualitative properties of a limiting elliptic system arising in phase separation for Bose-Einstein condensates with multiple states: \Delta u=u v^2 in R^n, \Delta v= v u^2 in R^n, u, v>0\quad in R^n. When n=1, we prove uniqueness of the one-dimensional profile. |
We consider $L^{2}$-supercritical and $H^{1}$-subcritical focusing nonlinear Schrödinger equations. We introduce a subset $PW$ of $H^{1}(\mathbb{R}^{d})$ for $d\ge 1$, and investigate behavior of the solutions with initial data in this set. |
We consider the compressible Navier-Stokes system where the viscosity depends on density and the heat conductivity is proportional to a positive power of the temperature under stress-free and thermally insulated boundary conditions. Under the same conditions on the initial data as those of the constant viscosity and h... |
We consider the defocusing supercritical generalized Korteweg-de Vries (gKdV) equation $\partial_t u+\partial_x^3u-\partial_x(u^{k+1})=0$, where $k>4$ is an even integer number. We show that if the initial data $u_0$ belongs to $H^1$ then the corresponding solution is global and scatters in $H^1$. |
It is well known that, for fast rotating fluids with the axis of rotation being perpendicular to the boundary, the boundary layer is of Ekman-type, described by a linear ODE system. In this paper we consider fast rotating fluids, with the axis of rotation being parallel to the boundary. |
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