In this paper we study weak continuity of the dynamical systems for the KdV equation in $H^{-3/4}(\mathbb{R})$ and the modified KdV equation in $H^{1/4}(\mathbb{R})$. This topic should have significant applications in the study of other properties of these equations such as finite time blow-up and asymptotic stability and instability of solitary waves. The spaces considered here are borderline Sobolev spaces for the corresponding equations from the viewpoint of the local well-posedness theory. We first use a variant of the method of [5] to prove weak continuity for the mKdV, and next use a similar result for an mKdV system and the generalized Miura transform to get weak continuity for the KdV equation.

In this paper we consider the Cauchy problem for the $3$-dimensional Navier-Stokes equations and we establish some Serrin-type regularity criterion in weak spaces involving the summability of the pressure or the gradient of the pressure.

We prove the existence of maximizers for Strichartz inequalities for the wave equation in dimensions $d\geq 3$. Our approach follows the scheme given by Shao in [21] which obtains the existence of maximizers in the context of the Schrödinger equation. The main tool that we use is the linear profile decomposition for the wave equation which we prove in $\mathbb{R}^d$, $d\geq 3$, extending the profile decomposition result of Bahouri and Gerard [1], previously obtained in $\mathbb{R}^3$.

We use the method of Strzelecki [Calc. Var. 1 (2003)] to generalize the Bethuel theorem [C. R. Acad. Sci. Paris 314 (1992)] to $n$-dimensional H-systems. We prove that if $u$ is a parameterization of an $n$-dimensional hypersurface in $\mathbb R^{n+1}$, weakly satisfies the system $\Delta_n u = H(u) u_{x_1} \times \cdots \times u_{x_n}$ and additionally has $n-1$ weak derivatives in $L^{n/(n-1)}$, then $u$ is Hölder continuous. Furthermore it is continuous up to the boundary, whenever it has continuous trace. We also give an example showing that the structure of the H-system is relevant and that the assumption that $u$ has $n-1$ weak derivatives does not trivialize the problem.

This note presents a simple method to generalize the Garding inequality to unbounded domains. By introducing a special partition of unity associated to some covering of unbounded domains, we show that the Garding inequality, known in the literature on bounded domains (see Garding, 1953), holds for more general domains. The method proposed here only requires weak smoothness assumptions which allows coverage of a large class of elliptic operators. It also gives a new tool to extend some local properties to global ones, in particular when we deal with functions vanishing on the boundary.

We study large-time behavior of solutions to the initial-value problem for the Burgers-Poisson equations introduced to analyze the behavior of high-temperature gas. It is known that the solution tends to a solution to the viscous Burgers equation. By a suitable choice of a linearized equation including the low-order term, we succeed in improving the asymptotic rate.

The following nonlinear eigenvalue problem for a pair of real parameters $(\lambda,\mu)$ is studied: $$ \begin{cases} - \Delta_p u = \lambda\, a(x)\, |u|^{\alpha_1} |v|^{\beta_1 - 1} v & \mbox{ in }\, \Omega; \\ - \Delta_q v = \mu\, b(x)\, |v|^{\alpha_2} |u|^{\beta_2 - 1} u & \mbox{ in }\, \Omega; \\ u = v = 0 & \mbox{ on }\, \partial\Omega. \end{cases} $$ Here, $p,q\in (1,\infty)$ are given numbers, $\Omega$ is a bounded domain in ${\mathbb{R}}^N$ with a $C^2$-boundary, $a,b\in L^{\infty}(\Omega)$ are given functions, both assumed to be strictly positive on compact subsets of $\Omega$, and the coefficients $\alpha_i, \beta_i$ are nonnegative numbers satisfying either the conditions $ \alpha_1 + \beta_1 = p-1 \,\mbox{ and }\, \alpha_2 + \beta_2 = q-1, $ or the condition $$ (p-1 - \alpha_1) (q-1 - \alpha_2) = \beta_1\beta_2. $$ A {\em smooth curve} of pairs $(\lambda,\mu)$ in $(0,\infty)\times (0,\infty)$ is found for which the quasilinear elliptic system possesses a solution pair $(u,v)$ consisting of nontrivial, nonnegative functions $u\in W_0^{1,p}(\Omega)$ and $v\in W_0^{1,q}(\Omega)$. Key roles in the proof are played by the strong comparison principle and a nonlinear Kreĭn-Rutman theorem obtained by the authors in earlier works. The main result is applied to some quasilinear elliptic systems related to the above system.

We consider the resolvent problem for the Oseen system and for the scalar Oseen equation in the whole space $\mathbb{R}^{3}$. It is shown that, for small values of the resolvent parameter, it is impossible to obtain an L$^{2}$-estimate analogous to the one that is valid for the Stokes resolvent, even if the resolvent parameter has positive real part.

We consider an estimate of the life span of solutions on a semilinear heat equation. In one dimension, we show that the life span may be estimated from above in terms of the average of two limits as $x\to\pm\infty$ of the initial data. In general dimensions, under some monotonicity conditions for initial data, an explicit representation of the uniform norm of the life span of solutions is obtained.

It is proved that the flow map for the Hunter-Saxton (HS) equation from the homogeneous Sobolev space $\dot{H}^s({\mathbb{T}})$ into the space $C([0,T], \dot{H}^s({\mathbb{T}}))$ is continuous but not uniformly continuous on bounded subsets. To demonstrate this sharpness of continuity, two sequences of bounded solutions to the HS equation are constructed whose distance at the initial time converges to zero and whose distance at any later time is bounded from below by a positive constant. To achieve this result, appropriate approximate solutions are chosen and then the actual solutions are found by solving the Cauchy problem with initial data taken to be the value of approximate solutions at time zero. Then, using well-posedness estimates, it is shown that the difference between solutions and approximate solutions is negligible.