Analysis of PDEs (math.AP)

Abstract: We derive a relative entropy inequality for capillary compressible fluids with density dependent viscosity. Applications in the context of weak-strong uniqueness analysis, pressureless fluids and high-Mach number flows are presented.

Abstract: In this paper, we study the zero-viscosity limit of the compressible Navier-Stokes equations in a half-space with non-slip boundary condition. We justify the Prandtl boundary layer expansion for the analytic data: the compressible Navier-Stokes equations can be approximated by the compressible Euler equations away from the boundary, and by the compressible Prandtl equation near the boundary.

Abstract: Based on a combination of insights afforded by Rainer Picard and Serge Nicaise, we extend a set of abstract piezo-electromagnetic impedance boundary conditions. We achieve this by accommodating for the influence of heat with the inclusion of a new equation and additional boundary terms. We prove the evolutionary well-posedness of a known thermo-piezo-electromagnetic system under these boundary conditions. Evolutionary well-posedness here means unique solvability as well as continuous and causal dependence on given data.

Abstract: We establish fine bounds for best constants of the fractional subcritical Sobolev embeddings \begin{align*} W_{0}^{s,p}\left(\Omega\right)\hookrightarrow L^{q}\left(\Omega\right), \end{align*} where $N\geq1$, $0<s<1$, $p=1,2$, $1\leq q<p_{s}^{\ast}=\frac{Np}{N-sp}$ and $\Omega\subset\mathbb{R}^{N}$ is a bounded smooth domain or the whole space $\mathbb{R}^{N}$. Our results cover the borderline case $p=1$, the Hilbert case $p=2$, $N>2s$ and the so-called Sobolev limiting case $N=1$, $s=\frac{1}{2}$ and $p=2$, where a sharp asymptotic estimate is given by means of a limiting procedure. We apply the obtained results to prove existence and non-existence of solutions for a wide class of nonlocal partial differential equations.

Abstract: Evaluation of a product integral with values in the Lie group SU(1,1) yields the explicit solution to the impedance form of the Schr\"odinger equation. Explicit formulas for the transmission coefficient and $S$-matrix of the classical one-dimensional Schr\"odinger operator with arbitrary compactly supported potential are obtained as a consequence. The formulas involve operator theoretic analogues of the standard hyperbolic functions, and provide a new window on acoustic and quantum scattering in one dimension.

Abstract: We establish a number of results that reveal a form of irreversibility (distinguishing arbitrarily long from finite time) in 2d Euler flows, by virtue of twisting of the particle trajectory map. Our main observation is that twisting in Hamiltonian flows on annular domains, which can be quantified by the differential winding of particles around the center of the annulus, is stable to perturbations. In fact, it is possible to prove the stability of the whole of the lifted dynamics to non-autonomous perturbations (though single particle paths are generically unstable). These all-time stability facts are used to establish a number of results related to the long-time behavior of inviscid fluid flows. In particular, we show that nearby general stable steady states (i) all Euler flows exhibit indefinite twisting and hence "age", (ii) vorticity generically becomes filamented and exhibits wandering in $L^\infty$. We also give examples of infinite time gradient growth for smooth solutions to the SQG equation and of smooth vortex patch solutions to the Euler equation that entangle and develop unbounded perimeter in infinite time.

Abstract: In this paper, a class of nonlocal fractional Dirichlet problems is studied. By using a variational principle due to Ricceri (whose original version was given in J. Comput. Appl. Math. 113 (2000), 401-410), the existence of infinitely many weak solutions for these problems is established by requiring that the nonlinear term $f$ has a suitable oscillating behaviour either at the origin or at infinity.

Abstract: We present Oseen equations on Lipschitz domains in a port-Hamiltonian context. Such equations arise, for instance, by linearization of the Navier-Stokes equations. In our setup, the external port consists of the boundary traces of velocity and the normal component of the stress tensor, and boundary control is imposed by velocity and normal stress tensor prescription at disjoint parts of the boundary. We employ the recently developed theory of port-Hamiltonian system nodes for our formulation. An illustration is provided by means of flow through a cylinder.

Abstract: In this paper we address two boundary cases of the classical Kazdan-Warner problem. More precisely, we consider the problem of prescribing the Gaussian and boundary geodesic curvature on a disk of R^2, and the scalar and mean curvature on a ball in higher dimensions, via a conformal change of the metric. We deal with the case of negative interior curvature and positive boundary curvature. Using a Ljapunov-Schmidt procedure, we obtain new existence results when the prescribed functions are close to constants.