Analysis of PDEs (math.AP)

Abstract: In this paper, we consider two types of damped wave equations: the weakly damped equation and the strongly damped equation and show that the initial velocity from the solution on the unit sphere. This inverse problem is related to Photoacoustic Tomography (PAT), a hybrid medical imaging technique. PAT is based on generating acoustic waves inside of an object of interest and one of the mathematical problem in PAT is reconstructing the initial velocity from the solution of the wave equation measured on the outside of object. Using the spherical harmonics and spectral theorem, we demonstrate a way to recover the initial velocity.

Abstract: We obtain some rigidity results for overdetermined boundary value problems for singular solutions in bounded domains.

Abstract: In this paper, we consider three components system of nonlinear Schr\"odinger equations related to the Raman amplification in a plasma. By using variational method, a new result on the existence of traveling wave solutions are obtained under the non-mass resonance condition. We also study the new global existence result for oscillating data. Both of our results essentially due to the absence of Galilean symmetry in the system.

Abstract: We study the nonexistence of multi-dimensional solitary waves for the Euler-Poisson system governing ion dynamics. It is well-known that the one-dimensional Euler-Poisson system has solitary waves that travel faster than the ion-sound speed. In contrast, we show that the two-dimensional and three-dimensional models do not admit nontrivial irrotational spatially localized traveling waves for any traveling velocity and for general pressure laws. We derive some Pohozaev type identities associated with the energy and density integrals. This approach is extended to prove the nonexistence of irrotational multi-dimensional solitary waves for the two-species Euler-Poisson system for ions and electrons.

Abstract: This paper concerns the large Reynold number limits and asymptotic behaviors of solutions to the 2D steady Navier-Stokes equations in an infinitely long convergent channel. It is shown that for a general convergent infinitely long nozzle whose boundary curves satisfy curvature-decreasing and any given finite negative mass flux, the Prandtl's viscous boundary layer theory holds in the sense that there exists a Navier-Stokes flow with no-slip boundary condition for small viscosity, which is approximated uniformly by the superposition of an Euler flow and a Prandtl flow. Moreover, the singular asymptotic behaviors of the solution to the Navier-Stokes equations near the vertex of the nozzle and at infinity are determined by the given mass flux, which is also important for the constructions of the Prandtl approximation solution due to the possible singularities at the vertex and non-compactness of the nozzle. One of the key ingredients in our analysis is that the curvature-decreasing condition on boundary curves of the convergent nozzle ensures that the limiting inviscid flow is pressure favourable and plays crucial roles in both the Prandtl expansion and the stability analysis.

Abstract: Blow-up solutions to a heat equation with spatial periodicity and a quadratic nonlinearity are studied through asymptotic analyses and a variety of numerical methods. The focus is on the dynamics of the singularities in the complexified space domain. Blow up in finite time is caused by these singularities eventually reaching the real axis. The analysis provides a distinction between small and large nonlinear effects, as well as insight into the various time scales on which blow up is approached. It is shown that an ordinary differential equation with quadratic nonlinearity plays a central role in the asymptotic analysis. This equation is studied in detail, including its numerical computation on multiple Riemann sheets, and the far-field solutions are shown to be given at leading order by a Weierstrass elliptic function.

Abstract: We consider single-phase flow in a fractured porous medium governed by Darcy's law with spatially varying hydraulic conductivity matrices in both bulk and fractures. The width-to-length ratio of a fracture is of the order of a small parameter $\varepsilon$ and the ratio $K_\mathrm{f}^\star / K_\mathrm{b}^\star$ of the characteristic hydraulic conductivities in the fracture and bulk domains is assumed to scale with $\varepsilon^\alpha$ for a parameter $\alpha \in \mathbb{R}$. The fracture geometry is parameterized by aperture functions on a submanifold of codimension one. Given a fracture, we derive the limit models as $\varepsilon \rightarrow 0$. Depending on the value of $\alpha$, we obtain five different limit models as $\varepsilon \rightarrow 0$, for which we present rigorous convergence results.

Abstract: We study a space-fractional Stefan problem with the Dirichlet boundary conditions. It is a model that describes superdiffusive phenomena. Our main result is the existence of the unique classical solution to this problem. In the proof we apply evolution operators theory and the Schauder fixed point theorem. It appears that studying fractional Stefan problem with Dirichlet boundary conditions requires a substantial modifications of the approach in comparison with the existing results for problems with different kinds of boundary conditions.

Abstract: We consider wave equations with a critically singular potential $\xi \cdot \sigma^{-2}$ diverging as an inverse square at a hypersurface $\sigma = 0$. Our aim is to construct counterexamples to unique continuation from $\sigma = 0$ for this equation, provided there exists a family of null geodesics trapped near $\sigma = 0$. This extends the classical geometric optics construction of Alinhac-Baouendi (i) to linear differential operators with singular coefficients, and (ii) over non-small portions of $\sigma = 0$ - by showing that such counterexamples can be further continued as long as this null geodesic family remains trapped and regular. As an application to relativity and holography, we construct counterexamples to unique continuation from the conformal boundaries of asymptotically Anti-de Sitter spacetimes for some Klein-Gordon equations; this complements the unique continuation results of the second author with Chatzikaleas, Holzegel, and McGill and suggests a potential mechanism for counterexamples to the AdS/CFT correspondence.

Abstract: We investigate the maximum principle for the weak solutions to the Cauchy problem for the hyperbolic fourth-order linear equations with constant complex coefficients in the plane bounded domain