Analysis of PDEs (math.AP)

Abstract: We focus on the general theory to the Cauchy problem for one dimensional nonlinear wave equations with small initial data. In the general theory, we aim to obtain the lower bound estimate of the lifespan of classical solution. In this paper, we improve it in some case related to the combined effect, which was expected complete more than 30 years ago.

Abstract: In this paper, we establish a priori estimates for the positive solutions to a higher-order fractional Laplace equation on a bounded domain by a blowing-up and rescaling argument. To overcome the technical difficulty due to the high-order and fractional order mixed operators, we divide the high-order fractional Laplacian equation into a system, and provide uniform estimates for each equation in the system. Finding a proper scaling parameter for the domain is the crux of rescaling argument to the above system, and the new idea is introduced in the rescaling proof, which may hopefully be applied to many other system problems. In order to derive a contradiction in the blowing-up proof, combining the moving planes method and suitable Kelvin transform, we prove a key Liouville-type theorem under a weaker regularity assumption in a half space.

Abstract: In this paper, we obtain a priori estimates for the set of anti-symmetric solutions to a fractional Laplacian equation in a bounded domain using a blowing-up and rescaling argument. In order to establish a contradiction to possible blow-ups, we apply a certain variation of the moving planes method in order to prove a monotonicity result for the limit equation after rescaling.

Abstract: In this paper, we consider the complex Ginzburg Landau equation $$ \partial_t u = (1 + i \beta ) \Delta u + (1 + i \delta) |u|^{p-1}u - \alpha u \text{ where } \beta, \delta, \alpha \in \mathbb R. $$ The study aims to investigate the finite time blowup phenomenon. In particular, for fixed $ \beta\in \mathbb R$, the existence of finite time blowup solutions for an arbitrary large $|\delta|$ is still unknown. Especially, Popp, Stiller, Kuznetsov, and Kramer formally conjectured in 1998 that there is no blowup (collapse) in such a case. In this work, considered as a breakthrough, we give a counter example to this conjecture. We show the existence of blowup solutions in one dimension with $\delta $ arbitrarily given and $\beta =0$. The novelty is based on two main contributions: an investigation of a new blowup scaling (flat blowup regime) and a suitable modulation.

Abstract: The aim of this work is to prove a compact embedding for a weighted fractional Sobolev spaces. As an application, we use this embedding to prove, via variational methods, the existence of solutions for the following Schr\"odinger equation $$ (-\Delta)^su + V(|x|)u = K(|x|)f(u), \quad \text{ in } \mathbb{R}^N, $$ where the two measurable functions $K > 0$ and $V \geq 0$ could vanish at infinity.

Abstract: This article surveys recent results aiming at obtaining refined mapping estimates for the X-ray transform on a Riemannian manifold with boundary, which leverage the condition that the boundary be strictly geodesically convex. These questions are motivated by classical inverse problems questions (e.g. range characterization, stability estimates, mapping properties on Hilbert scales), and more recently by uncertainty quantification and operator learning questions.

Abstract: In this work we prove that the initial value problem associated to the Schr\"odinger-Benjamin-Ono type system \begin{equation*} \left\{ \begin{array}{ll} \mathrm{i}\partial_{t}u+ \partial_{x}^{2} u= uv+ \beta u|u|^{2}, \partial_{t}v-\mathcal{H}_{x}\partial_{x}^{2}v+ \rho v\partial_{x}v=\partial_{x}\left(|u|^{2}\right) u(x,0)=u_{0}(x), \quad v(x,0)=v_{0}(x), \end{array} \right. \end{equation*} with $\beta,\rho \in \mathbb{R}$ is locally well-posed for initial data $(u_{0},v_{0})\in H^{s+\frac12}(\mathbb{R})\times H^{s}(\mathbb{R})$ for $s>\frac54$. Our method of proof relies on energy methods and compactness arguments. However, due to the lack of symmetry of the nonlinearity, the usual energy has to be modified to cancel out some bad terms appearing in the estimates. Finally, in order to lower the regularity below the Sobolev threshold $s=\frac32$, we employ a refined Strichartz estimate introduced in the Benjamin-Ono setting by Koch and Tzvetkov, and further developed by Kenig and Koenig.

Abstract: We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition, controlled by the exponent $p$ depending on the time and the spatial variable. We show the existence of a weak and an entropy solution to our system, as well as the uniqueness of an entropy solution, under the assumption of boundedness and log-H\"{o}lder continuity of the variable exponent~$p$ with respect to the spatial variable. On the other hand, we do not assume any smoothness of~$p$ with respect to the time variable.