arXiv subjects
Publications and source records attributed to Tien-Tai Nguyen.
In this note, we are interested in the linear Rayleigh-Taylor instability problem for the incompressible fluid with an upper free surface. Using the spectral theory of self-adjoint and compact operator, we rigorously prove the existence of infinitely many normal mode solutions to the linearized equations. Some numerical computations are presented to support our theoretical study.
In this paper, we investigate the instability of the trivial steady states to the incompressible viscous fluid with Navier-slip boundary conditions. For the linear instability, the existence of finitely many normal mode solutions to the linearized equations is shown via the operator method of Lafitte and Nguyen (2022). Hence, we prove the nonlinear instability by adapting the framework of Desjardins and Grenier (2003) studying some classes of viscous boundary layers to obtain two separated solutions at escaping time. Our work performs a different approach from that of Ding, Li and Xin (2018).
Let $p$ be positive and $n \geq 3$ be an integer. Let $f(\cdot,\cdot): \mathbf{R}_+\times \mathbf{R}_+\to \mathbf{R}_+$ be a continuous function. In this paper, we are concerned with positive solutions to the following integral equation \[ u(x)= \int_{\mathbf{R}^n} |x-y|^p f(|y|,u(y)) dy \quad\text{in }\mathbf{R}^n\setminus\{\textbf{0}\}. \] By imposing some suitable conditions on $f$, we obtain the radially symmetry property of positive solutions to the above equation by using the method of moving spheres in integral form.
In this paper, we consider an incompressible viscous fluid in an infinitely deep ocean, being bounded above by a free moving boundary. The governing equations are the gravity-driven incompressible Navier-Stokes equations with variable density and no surface tension is taken into account on the free surface. After using the Lagrangian transformation, we write the main equations in a perturbed form in a fixed domain. In the first part, we describe a spectral analysis of the linearized equations around a hydrostatic equilibrium $(ρ_0(x_3), 0, P_0(x_3))$ for a smooth increasing density profile $ρ_0$. Precisely, we prove that there exist infinitely many normal modes to the linearized equations by following the operator method initiated by Lafitte and Nguyen. In the second part, we study the nonlinear Rayleigh-Taylor instability around the above profile by constructing a \textit{wide class} of initial data for the nonlinear perturbation problem departing from the equilibrium, based on the finding of infinitely many normal modes. Our nonlinear result follows the previous framework of Guo and Strauss and also of Grenier with a refinement.
Motivated by Bresch, Desjardins, Gisclon and Sart (2008), in this paper, we study the influence of capillary number on an instability result related to the Navier-Stokes-Korteweg equations. Precisely, we investigate the instability of a steady-state profile with a heavier fluid lying above a lighter fluid, i.e., to study the Rayleigh-Taylor instability problem if the capillary number is below the critical value. After writing the nonlinear equations in a perturbed form, the first part is to provide a spectral analysis showing that, there exist possibly multiple normal modes to the linearized equations by following the operator method of Lafitte-Nguyen (2022). Hence, we construct a wide class of initial data for which the nonlinear perturbation problem departs from the equilibrium, based on the finding of possibly multiple normal modes. Using a refined framework of Guo-Strauss (1995), we prove the nonlinear instability.
Let $n$ be an integer and $s$ be a real number such that $n > 2s \geq 2$. Inspired by the perturbation approach initiated by F. Hang and P. Yang (\textit{Int. Math. Res. Not. IMRN}, 2020), we are interested in non-negative, smooth solution $v$ to the following higher-order fractional equation \[ {\mathbf P}_n^{2s}(v) = Q_n^{2s}(\varepsilon v+v^α) \] on $\mathbf S^n$ with $0<α\leq (n+2s)/(n-2s)$, and $\varepsilon \geq 0$. Here ${\mathbf P}_n^{2s}$ is the fractional GJMS type operator of order $2s$ on $\mathbf S^n$ and $Q_n^{2s} ={\mathbf P}_n^{2s}(1)$ is constant. We show that if $\varepsilon >0$ and $0<α\leq (n+2s)/(n-2s)$, then any positive, smooth solution $v$ to the above equation must be constant. The same result remains valid if $\varepsilon=0$ but with $0<α< (n+2s)/(n-2s)$.As a by-product, with $0<α\leq (n+2s)/(n-2s)$, we compute the sharp constant of the subcritical/critical Sobolev inequalities \[ \int_{\mathbf S^n} v {\mathbf P}_n^{2s} (v) dμ_{g_{\mathbf S^n}} \geq \frac{Γ(n/2 + s)}{Γ(n/2 - s )} | \mathbf S^n|^\frac{α-1}{α+1} \Big( \int_{\mathbf S^n} v^{α+1} dμ_{g_{\mathbf S^n}} \Big)^\frac{2}{α+1}. \] for the GJMS operator ${\mathbf P}_n^{2s}$ on $\mathbf S^n$ and for all non-negative functions $v\in H^s(\mathbf S^n)$.
In this paper, we are interested in the nonlinear Rayleigh-Taylor instability for the gravity-driven incompressible Navier-Stokes equations with Navier-slip boundary conditions around a smooth increasing density profile $ρ_0(x_2)$ in a slab domain $2πL\mathbb{T} \times (-1,1)$ ($L>0$, $\mathbb{T}$ is the usual 1D torus). The linear instability study of the viscous Rayleigh-Taylor model amounts to the study of the following ODE on the finite interval $(-1,1)$, \begin{equation}\label{EqMain} λ^2 ( ρ_0 k^2 ϕ- (ρ_0 ϕ')')+ λμ(ϕ^{(4)} - 2k^2 ϕ'' + k^4 ϕ) = gk^2 ρ_0'ϕ, \end{equation} with the boundary conditions \begin{equation}\label{4thBound} \begin{cases} ϕ(-1)=ϕ(1)=0,\\ μϕ''(1) = ξ_+ ϕ'(1), \\ μϕ''(-1) =- ξ_- ϕ'(-1), \end{cases} \end{equation} where $λ>0$ is the growth rate in time, $g>0$ is the gravity constant, $k$ is the wave number and two Navier-slip coefficients $ξ_{\pm}$ are nonnegative constants. For each $k\in L^{-1}\mathbb{Z}$, we define a threshold of viscosity coefficient $μ_c(k,Ξ)$ for the linear instability. So that, in the $k$-supercritical regime, i.e. $μ>μ_c(k,Ξ)$, we describe a spectral analysis adapting the operator method initiated by Lafitte-Nguyen \cite{LN20} and prove that there are infinite nontrivial solutions $(λ_n, ϕ_n)_{n\geqslant 1} $ of \eqref{EqMain}-\eqref{4thBound} with $λ_n \to 0$ as $n\to \infty$ and $ϕ_n\in H^4((-1,1))$. Based on the existence of infinitely many normal modes of the linearized problem, we construct a wide class of initial data to the nonlinear equations, extending the previous framework of Guo-Strauss \cite{GS95} and of Grenier \cite{Gre00}, to prove the nonlinear Rayleigh-Taylor instability in a high regime of viscosity coefficient, namely $μ>3\sup_{k\in L^{-1}\mathbb{Z}\setminus\{0\}}μ_c(k,Ξ)$.
The linear instability study of the viscous Rayleigh-Taylor model in the neighborhood of a laminar smooth increasing density profile $ρ_0(x_3)$ amounts to the study of the following ordinary differential equation of order 4: \begin{equation}\label{MainEq} -λ^2 [ ρ_0 k^2 ϕ- (ρ_0 ϕ')'] = λμ(ϕ^{(4)} - 2k^2 ϕ" + k^4 ϕ) - gk^2 ρ_0'ϕ, \end{equation} where $λ$ is the growth rate in time, $k$ is the wave number transverse to the density profile. In the case of $ρ'_0\geq 0$ compactly supported, we provide a spectral analysis showing that in accordance with the results of \cite{HL03}, there is an infinite sequence of non trivial solutions $(λ_n, ϕ_n)$, with $λ_n\rightarrow 0$ when $n\rightarrow +\infty$ and $ϕ_n\in H^4(\mathbf{R})$. In the more general case where $ρ_0'>0$ everywhere and $ρ_0$ converges at $\pm\infty$ to finite limits $ρ_{\pm}>0$, we prove that there exist finitely non trivial solutions $(λ_n, ϕ_n)$. The line of investigation is to reduce both cases to the study of an operator on a compact set.
In this article, we are concerned with the following geometric equation \begin{equation}\label{MainEq} Δ^2 u = -u^{-q} \qquad \text{in } \mathbf{R}^3 \end{equation} for $q>0$. Recently in \cite{GWZ18}, Guo, Wei and Zhou have established the relationship between the radial symmetry and the exact growth rate at infinity of a positive entire solution of that equation as $1 3$ thanks to the method of moving plane.