Related papers: A symplectic non-squeezing theorem for BBM equatio…

In this article, we study the blowup phenomena of compressible Euler equations with non-vacuum initial data. Our new results, which cover a general class of testing functions, present new initial value blowup conditions. The corresponding…

We deal with the vanishing viscosity scheme for the transport/continuity equation $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$ drifted by a divergence-free vector field $\boldsymbol{b}$. Under general Sobolev assumptions on…

The initial-boundary value problem for an infinite one-dimensional chain of harmonic oscillators on the half-line is considered. The large time asymptotic behavior of solutions is studied. The initial data of the system are supposed to be a…

Let $(b,u)$ be a pair consisting of a symplectic form $b$ on a finite-dimensional vector space $V$ over a field $\mathbb{F}$, and of a $b$-alternating endomorphism $u$ of $V$ (i.e. $b(x,u(x))=0$ for all $x$ in $V$). Let $p$ and $q$ be…

We consider the (barotropic) Euler system describing the motion of a compressible inviscid fluid driven by a stochastic forcing. Adapting the method of convex integration we show that the initial value problem is ill-posed in the class of…

In this paper we study the initial boundary value problem for the system $\Delta v= u_{x_1},\ u_t-\mbox{div}\left(\left((a|\mathbf{q}|+m)I+(b-a)\frac{\mathbf{q}\otimes\mathbf{q}}{|\mathbf{q}|}\right)\nabla u\right)=-\nabla…

We study infinite time blow-up phenomenon for the half-harmonic map flow \begin{equation}\label{e:main00} \left\{\begin{array}{ll} u_t = -(-\Delta)^{\frac{1}{2}}u +…

In this paper, we consider the following non-local semi-linear parabolic equation with advection: for $1 \le p<1+\frac{2}{N}$, \begin{equation*} \begin{cases} u_t+v \cdot \nabla u-\Delta u=|u|^p-\int_{\mathbb T^N} |u|^p \quad & \textrm{on}…

The paper deals with the problem of stability for the flow of the 1D Burgers equation on a circle. Using some ideas from the theory of positivity preserving semigroups, we establish the strong contraction in the $L^1$ norm. As a…

We study nonnegative solutions of the boundary value problem $$-\Delta u = \lambda c(x)u + \mu(x)|\nabla u|^2 + h(x),\quad u\in H^1_0(\Omega)\cap L^\infty(\Omega), \leqno(P_\lambda)$$ where $\Omega$ is a smooth bounded domain, $\mu, c\in…

Although Lattice Boltzmann Method (LBM) is relatively straightforward, it demands a well-crafted framework to handle the complex partial differential equations involved in multiphase flow simulations. For the first time to our knowledge,…

In this paper, we develop a three-dimensional multiple-relaxation-time lattice Boltzmann method (MRT-LBM) based on a set of non-orthogonal basis vectors. Compared with the classical MRT-LBM based on a set of orthogonal basis vectors, the…

We show that the second iteration $T^2$ of the outer symplectic billiard map with respect to a convex domain $M$ in a symplectic vector space is approximated by an explicit Hamiltonian flow for points far away from $M$. More precisely,…

In this paper, we study the following semilinear Schr\"odinger equation with periodic coefficient: $$-\triangle u +V(x)u=f(x,u), u\in H^{1}(\mathbb{R}^{N}).$$ The functional corresponding to this equation possesses strongly indefinite…

We study solutions to the system $u_{tt}-u_{xx}+q(x)u=0, x>0,t>0$; $u|_{t=0}=u_t|_{t=0}=0, x>0$; $u|_{x=0}=g(t), t>0$, with a locally summable Hermitian matrix-valued potential $q$ and a $C^{\infty}$-smooth $\mathbb C^n$-valued boundary…

For the initial value problem (IVP) associated the generalized Korteweg-de Vries (gKdV) equation with supercritical nonlinearity, u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5, numerical evidence \cite{BDKM1, BSS1} shows that…

In this paper, we show the nondegeneracy of positive bubble solutions for generalized energy-critical Hartree equations (NLH) \begin{equation*} -{\Delta u}\sts{x} -{\bm\alpha}\sts{N,\lambda} \int_{\R^N} { \frac{…

We consider the initial boundary value problem \begin{eqnarray*}u_t=\mu u_x+\tfrac{1}{2}u_{xx}\qquad (t>0,x\ge0),\\u(0,x)=f(x)\qquad (x\ge0),\\u_t(t,0)=\nu u_x(t,0)\qquad (t>0)\end{eqnarray*} of Stroock and Williams [Comm. Pure Appl. Math.…

We study the initial-boundary value problem for a nonlinear wave equation given by u_{tt}-u_{xx}+\int_{0}^{t}k(t-s)u_{xx}(s)ds+ u_{t}^{q-2}u_{t}=f(x,t,u) , 0 < x < 1, 0 < t < T, u_{x}(0,t)=u(0,t), u_{x}(1,t)+\eta u(1,t)=g(t),…

Poincare's invariance principle for Hamiltonian flows implies Kelvin's principle for solution to Incompressible Euler Equation. Iyer-Constantin Circulation Theorem offers a stochastic analog of Kelvin's principle for Navier-Stokes Equation.…