Analysis and Partial Differential Equations

Abstract

In this talk, we study the Cauchy problem \begin {equation*} \begin {aligned} \partial _tu+ \mathcal {L}_{\Delta } u&=0 &&\text {in }(0,\frac N2) \times \mathbb R^N,\\ u(0,\cdot )&=0&&\text {in }\mathbb R^N\setminus \{0\}. \end {aligned} \end {equation*} where \(\mathcal {L}_\Delta \) is the logarithmic Laplacian operator, a singular integral operator with symbol \(2\log |\zeta |\).

We apply our results to give aclassification of the solutions of \begin {equation*} \begin {aligned} \partial _tu+\mathcal {L}_{\Delta } u&=0 &&\text {in }(0,T)\times \mathbb R^N\\ u(0,\cdot )&=f &&\text {in}\mathbb R^N \end {aligned} \end {equation*} and obtain an expression of the fundamental solution of the associated stationary equation in \(\mathbb R^N\), and of the fundamental solution in a bounded domain, i.e. \begin {equation*} \begin {aligned} \mathcal {L}_{\Delta } u&=k\delta _0&&\text {in }\mathcal {D}'(\Omega )\\ u&=0&&\text { in }\mathbb R^N\setminus \Omega . \end {aligned} \end {equation*} This is joint work with Laurent Véron.

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Abstract

Brascamp-Lieb inequalities generalise various inequalities from analysis, including Hölder'€™s inequality, Young'€™s convolution inequality, and the Loomis-Whitney inequality. They are often used in conjunction with phase-space decompositions to obtain estimates for solutions to partial differential equations. Recently there has been interest in extending these inequalities from euclidean spaces to more general contexts, and this talk describes some progress in understanding what '€œgeneral'€ Brascamp-Lieb inequalities should look like.

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Abstract

We consider the existence and uniqueness of solutions to hte abstract logistic equation \begin {equation*} -Au=\lambda u-m(x)g(u)u, \end {equation*} in \(L^p(\Omega )\), where \(-A\) is the generator of a compact irreducible positive analytic semigroup with some interior smoothing properties, \(\Omega \subset \mathbb R\) with no or very mild regularity properties, \(\lambda \geq 0\) a parameter, \(g\in C^1([0,\infty ))\) strictly increasing with \(g(0)=0\) and \(m(x)\geq 0\) bounded and not identically zero. In the classical theory, the usual sub-and super-solution methods relies on Hopf'€™s boundary maximum principle, essentially forcing \(C^2\)-regularity of \(\Omega \). The aim of this work is to replace that maximum principle by tools that do not rely on the boundary regularity, but only interior regularity. The tool is Kato'€™s inequality that allows to prove a comparison theorem that is independent of any boundary conditions.

This is joint work with Wolfgang Arendt.

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Abstract

A new phenomenon in nonlocal diffusion models is that accelerated propagation may happen, that is, the propagation speed could be

infinite, which never occurs in the corresponding local diffusion model with compactly supported initial data. In this talk we will look at such a phenomenon for the KPP equation with nonlocal diffusion and free boundaries. For several natural classes of kernel functions appearing in the nonlocal diffusion term, we will show how the exact rate of acceleration can be determined.

The talk is based on joint work with Dr Wenjie Ni.

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Abstract

We explore a linear inhomogeneous elasticity equation with random Lamé parameters. The latter are parameterized by a countably infinite number of terms in separated expansions. The main aim of this work is to estimate the expected values of linear functionals acting on the solution of the elasticity equation. To achieve this, the expansions of the random parameters are truncated, a high-order quasi-Monte Carlo (QMC) is combined with a sparse grid approach to approximate the high dimensional integral, and a Galerkin finite element method (FEM) is introduced to approximate the solution of the elasticity equation over the physical domain. Some reasonable assumptions on the expansions of the random coefficients are imposed to achieve our theoretical regularity and convergence results. Finally, some numerical results are delivered.

This is joint work with J. Dick, K. Mustapha, and T. Tran (UNSW, Sydney).

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Abstract

We consider the following three dimensional defocusing cubic nonlinear Schrödinger equation (NLS) with partial harmonic potential \begin {align*} i\partial _t u + \left (\Delta _{y}+\partial _x^2-x^2 \right ) u &= |u|^2 u,\\ u|_{t=0} &= u_0 \end {align*}

where \(u(t,y,x):\mathbb R\times \mathbb R^2\times \mathbb R\to \mathbb C\). Our main result shows that the solution \(u\) scatters for any given initial data \(u_0\) with finite mass and energy.

The main new ingredient in our approach is to approximate (NLS) in the large-scale case by a relevant dispersive continuous resonant (DCR) system: \begin {align*} i\partial _t v + \Delta _{\mathbb {R}^2} v &= \sum _{n=n_1+n_2+n_3}\Pi _n(\Pi _{n_1}v\overline {\Pi _{n_2}v}\Pi _{n_3}v)\\ v(0,y,x) &= \phi (y,x), \end {align*}

where \(\Pi _k\) denotes the orthogonal projector onto the \(k\)-th Hermite function in the \(x\) variable.

The proof of global well-posedness and scattering of the new (DCR) system is greatly inspired by the fundamental works of Dodson in his study of scattering for the mass-critical nonlinear Schrödinger equation. The analysis of (DCR) system allows us to utilize the additional regularity of the smooth nonlinear profile so that the celebrated concentration-compactness/rigidity argument of Kenig and Merle applies.

This is joint work with X. Cheng, C.-Y. Guo, X. Liao and J. Shen.

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Abstract

We study pseudodifferential operators on a compact hyperbolic surface using '€™Zelditch'€™ quantisation, which provides an exact correspondence between symbols and kernels of pseudodifferential operators on the hyperbolic surface. We motivate and study the trace of a certain time-varying family of operators, \((A_2)^*A_1(t)\) with \(A_2\) fixed and \(A_1(t)\) being the Zelditch quantisation of a symbol pulled back under the geodesic flow. We find such traces are related to the theory of decay of correlations in dynamical systems and prove conditions under which this trace decays as \(t\) approaches plus/minus infinity.

This is joint work with Andrew Hassell.

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Abstract

Curve shortening flow (while deeply fascinating in its own right) provides a useful playground for testing ideas which may have wider applicability in the context of more complicated geometric flow equations. It is also a great setting for students to cut their teeth on. I will present a some results on curve shortening flow with mixed (Dirichlet-Neumann) boundary condition obtained over the summer with the help of two outstanding undergraduate research students, Yuxing Liu and George McNamara. (We thank Julie Clutterbuck for suggesting the problem.)

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Abstract

Chow and Glickenstein considered a second order linear semi-discrete analogue of the curve shortening flow for closed curves formed by joining by straight line segments an ordered set of points in the plane or in higher dimensional Euclidean space. In this talk we consider similar flows with boundary conditions, along with a semi-discrete analogue of the Yau problem of flowing one curve to another by curvature flow.

This is joint work with Rohan St Hill.

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Abstract

Motivated by higher order polyharmonic evolution equations on smooth curves, this talk presents comparable semi-discrete geometric flows that evolve and continuously untangle closed polygons (that can be planar or exist in higher co-dimensions). Setup and properties will be introduced and behaviour of explicit solutions demonstrated and explored. As an application, a semi-discrete setting can be given to evolve any closed polygon to another via these geometric flows.

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Abstract

We analyze \(p\)-Laplace type operators with degenerate elliptic coefficients. This investigation includes Grušin type \(p\)-Laplace operators. We describe a separation phenomenon in elliptic and parabolic \(p\)-Laplace type equations, which provides an illuminating illustration of simple jump discontinuities of the corresponding weak solutions. Interestingly validity of an isoperimetric inequality for considered setting does not imply continuity of elliptic equations. On the other hand, we are able to establish global \(L^1\)-\(L^\infty \)-regularization and decay estimates of every mild solution of the parabolic Grušin type \(p\)-Laplace equation.

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Abstract

In this talk, we first give the Liouville type results, universal estimates and periodic solutions for the non-homogeneous parabolic system. On the other hand, we give the existence of multiple positive solutions of the parabolic system with singular initial data.

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