In this work we prove that the symmetries of a Lagrangian function $L$ play an important role in the regularity of the solutions to its associated variational problem. More precisely, we prove that any absolutely continuous solution to $$\min \Big \{\int_a^b L(t,u(t),\dot u(t))\,dt: u\in{\bf W}_0^{1,1}(a,b)\Big \}$$ is regular in the sense of Tonelli, that is, has extended-values continuous derivative, if $L(t,u,\xi)$ is strictly convex in $\xi$, continuous in $t$ and $u$, and is invariant under a group of ${\bf C}^1$ transformations as in Noether's theorem. Our proof does not require $L$ to be Lipschitz continuous with respect to $u$, which is the standard hypothesis [2, 4, 5, 6, 8, 13] for proving this kind of regularity. As a corollary of our main result, we then obtain the Tonelli's partial regularity result without assuming any regularity of $L$ in $u$ (more than continuity) in the autonomous case, i.e., $L(t,u,\xi)=L(u,\xi)$.

In this paper, superposition operators, $ (N_gu ) (x )= g (u (x ) )$, between two arbitrary Sobolev spaces are studied. Sufficient conditions which ensure the well-definedness, the continuity, and the validity of the higher-order chain rule for such operators are given in the subcritical case (see Remark 1.1). As a consequence of these properties, it is proved that $N_g (W^{m,p} (\Omega )\cap W_0^{k,p} (\Omega ) )\subset W_0^{l,q} (\Omega )$.

The Abstract Bubble Vibration model ($\mathrm{ABV}$) is a system of two PDEs consisting of a transport equation and a Poisson equation. It has been derived in order to provide a better understanding of hyperbolic-elliptic couplings which are involved in low Mach number models. While a local existence theorem has already been proven in any dimension for the ($\mathrm{ABV}$) model, we get interested in this paper in the one-dimensional case, where we prove the existence of global-in-time solutions no matter how smooth the data. We also provide explicit expressions of these solutions thanks to the method of characteristics that we apply to the transport equation taking advantage of the coupling with the Poisson equation. We then illustrate numerically these results using two different schemes depending on the smoothness of data.

We study the dynamics of soliton solutions to the perturbed mKdV equation $\partial_t u = \partial_x(-\partial_x^2 u -2u^3) + \epsilon V u$, where $V\in \mathcal{C}^1_b(\mathbb{R})$ and $0 < \epsilon\ll 1$. This type of perturbation is non-Hamiltonian. Nevertheless, via symplectic considerations, we show that solutions remain $O(\epsilon {\langle} t\rangle^{1/2})$ close to a soliton on an $O(\epsilon^{-1})$ time scale. Furthermore, we show that the soliton parameters can be chosen to evolve according to specific exact ODEs on the shorter, but still dynamically relevant, time scale $O(\epsilon^{-1/2})$. Over this time scale, the perturbation can impart an $O(1)$ influence on the soliton position.

In this paper, the main goal is to prove the existence of blow-up solutions for a class of elliptic systems. Here, we combine variational methods with the existence of sub- and supersolution to get the blow-up solutions.

For a bounded, open set $\Omega\subset\mathbb{R}^N$ and depending on $\lambda>0$, we study the multiplicity of solutions of \begin{equation*} \begin{cases} u>0 \text{ in }\;\Omega\;, \\ -\div (M(x)\nabla u)=\frac{\lambda}{\;u^\gamma\;}+ u^{p} \text{ in }\;\Omega, \\ u=0 \text{ on }\;\partial\Omega, \end{cases} \end{equation*} where $M(x)$ is a symmetric, bounded, and elliptic matrix and $0 <\gamma <1 <p <\frac{N+2}{N-2}$.

We study the a priori symmetry of periodic gravity water waves with arbitrarily many crests and troughs, and general vorticity distributions. We show that if all the streamlines achieve their global minimum on the same vertical line, then the wave has exactly one trough and crest within a period, and is symmetric with respect to the vertical line containing the crest.

In this paper we consider a hypersurface of the mean curvature flow with transport term when it is expressed by a graph. The existence of the mean curvature flow with transport term was proved by Liu, Sato, and Tonegawa [19] by using geometric measure theory. We give a proof of the gradient estimates and the short time existence for the mean curvature flow with transport term by applying the backward heat kernel.

For Sobolev exponent $s>3/2$, it is shown that the data-to-solution map for the 2-component Camassa--Holm system is continuous from $H^s \times H^{s-1}(\mathbb {T})$ into $C([0,T];H^s \times H^{s-1}(\mathbb {T}))$ but not uniformly continuous. The proof of non-uniform dependence on the initial data is based on the method of approximate solutions, delicate commutator and multiplier estimates, and well-posedness results for the solution and its lifespan. Also, the solution map is Hölder continuous if the $H^s \times H^{s-1}(\mathbb {T})$ norm is replaced by an $H^r \times H^{r-1}(\mathbb {T})$ norm for $0 \leq r <s$.

In this paper, we are concerned with the multiplicity of solutions for semilinear elliptic problems with weight functions in exterior domains. We prove that, if the decay of the weight function at spatial infinity is sufficiently slow, then the equation admits at least three solutions and two of them escape away to the spatial infinity as the decay rate of the weight function tends to $0$. The result is proved via the variational method combined with the Ljusternik--Schnirelman-type multiplicity theorem.