We consider the initial value problem of a system of 2D nonlinear Schrödinger equations with quadratic nonlinearities in homogeneous Sobolev spaces which are close to the scale invariant spaces. We show the global solution, which is not necessarily in $\mathbf{L}^{2}( \mathbb{R}^{2})$, satisfies uniform time decay of order $|t|^{-1}$ in $\mathbf{L}^{\infty }( \mathbb{R}^{2})$.

We consider the linear and nonlinear cubic Schrödinger equations with periodic boundary conditions and their approximations by splitting methods. We prove that for a dense set of arbitrarily small time steps, there exist numerical solutions leading to strong numerical instabilities, preventing the energy conservation and regularity bounds obtained for the exact solution. We analyze rigorously these instabilities in the semi-discrete and fully discrete cases.

A class of semi-bounded solutions of the two-dimensional incompressible Euler equations, satisfying either periodic or Dirichlet boundary conditions, is examined. For smooth initial data, new blowup criteria in terms of the initial concavity profile is presented and the effects that the boundary conditions have on the global regularity of solutions is discussed. In particular, by deriving a formula for a general solution along Lagrangian trajectories, we describe how periodicity can prevent blow-up. This is in opposition to Dirichlet boundary conditions which, as we will show, allow for the formation of singularities in finite time. Lastly, regularity of solutions arising from non-smooth initial data is briefly discussed.

In this paper, we study some nonhomogeneous degenerate elliptic equation in dimension $ N,$ $ N \geq 3 $ with nonlinearity term having an exponential growth and does not satisfy the Ambrosetti-Rabinowitz (AR) condition. Using a variational tools, we establish the existence of at least two positive solutions.

We construct functions $u: \mathbb R^2 \to \mathbb C$ that satisfy an elliptic eigenvalue equation of the form $-\Delta u + W \cdot \nabla u + V u = \lambda u$, where $\lambda \in \mathbb C$, and $V$ and $W$ satisfy $| V ({x} ) | \lesssim \langle {x} \rangle ^{-N}$, and $ | W ({x})| \lesssim \langle{x} \rangle ^{-P}$, with $\min\{N, P\} = 1/2$. For $|{x}|$ sufficiently large, these solutions satisfy $|{u(x)}| \lesssim \exp ({- c|{x}|})$. In the author's previous work, examples of solutions over $\mathbb R^2$ were constructed for all $N, P$ such that $\min\{N,P\} \in [0, 1/2)$. These solutions were shown to have the optimal rate of decay at infinity. A recent result of Lin and Wang shows that the constructions presented in this note for the borderline case of $\min\{N, P\} = 1/2$ also have the optimal rate of decay at infinity.

In this paper, we study the spatial behavior of solutions to the equations obtained by taking formal Taylor approximations to the heat conduction dual-phase-lag and three-phase-lag theories, reflecting Saint-Venant's principle. Depending on the relative order of derivation, with respect to the time, we propose different arguments. One is inspired by the arguments for parabolic problems and the other is inspired by the arguments for hyperbolic problems. In the first case, we obtain a Phragmén-Lindelöf alternative for the solutions. In the second case, we obtain an estimate for the decay as well as a domain of influence result. The main tool to manage these problems is the use of an exponentially weighted Poincaré inequality.

In this paper, we give sharp extensions of convoluted solutions of wave equations in abstract Banach spaces. The main technique is to use the algebraic structure, for convolution products $\ast$ and $\ast_c$, of these solutions which are defined by a version of the Duhamel's formula. We define algebra homomorphisms, for the convolution product $\ast_c$, from a certain set of test-functions and apply our results to concrete examples of abstract wave equations.

The solution of the Chern-Simons-Higgs model in Lorenz gauge with data for the potential in $H^{s-1/2}$ and for the Higgs field in $H^s \times H^{s-1}$ is shown to be unique in the natural space $C([0,T];H^{s-1/2} \times H^s \times H^{s-1})$ for $s \ge 1$, where $s=1$ corresponds to finite energy. Huh and Oh recently proved local well-posedness for $s > 3/4$, but uniqueness was obtained only in a proper subspace $Y^s$ of Bourgain type. We prove that any solution in $C([0,T];H^{1/2} \times H^1 \times L^2)$ must in fact belong to the space $Y^{3/4+\epsilon}$. Hence, it is the unique solution obtained by Huh and Oh.

We consider a Neumann boundary problem like Gelfand's problem on polygonal domains and prove that a priori bounds of solutions fail for specific parameters. The same results are already proved by Senba-Suzuki in 2000 when domains are open sets in $\mathbb{R}^2$ with smooth boundaries. The novelty is that the foregoing specific parameters depend on angles of polygonal domains.

In this work, we study the Brinkman-Forchheimer equations for unsteady flows. We prove the continuous dependence of the solution on the Brinkman's and Forchheimer's coefficients as well as the initial data and external forces. Next, we propose and study a perturbed compressible system that approximate the Brinkman-Forchheimer equations. Finally, we propose a time discretization of the perturbed system by a semi-implicit Euler scheme and next a lowest-order Raviart-Thomas element is applied for spatial discretization. Some numerical results are given.

We consider critical points of the energy $$ E(v) := \int_{\mathbb R^n} |\nabla^s v|^{\frac{n}{s}}, $$ where $v$ maps locally into the sphere or $SO(N)$, and $\nabla^s = (\partial_1^s,\ldots,\partial_n^s)$ is the formal fractional gradient, i.e., $\partial_\alpha^s$ is a composition of the fractional Laplacian with the $\alpha$-th Riesz transform. We show that critical points of this energy are Hölder continuous. As a special case, for $s = 1$, we obtain a new, more stable proof of Fuchs and Strzelecki's regularity result of $n$-harmonic maps into the sphere [22, 9], which is interesting on its own.