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This volume covers some of the most seminal research in the areas of mathematical analysis and numerical computation for nonlinear phenomena. Collected from the international conference held in honor of Professor Yoshikazu Giga’s 60th birthday, the featured research papers and survey articles discuss partial differential equations related to fluid mechanics, electromagnetism, surface diffusion, and evolving interfaces. Specific focus is placed on topics such as the solvability of the Navier-Stokes equations and the regularity, stability, and symmetry of their solutions, analysis of a living fluid, stochastic effects and numerics for Maxwell’s equations, nonlinear heat equations in critical spaces, viscosity solutions describing various kinds of interfaces, numerics for evolving interfaces, and a hyperbolic obstacle problem. Also included in this volume are an introduction of Yoshikazu Giga’s extensive academic career and a long list of his published work. Students and researchers in mathematical analysis and computation will find interest in this volume on theoretical study for nonlinear phenomena.



An Implicit Boundary Integral Method for Interfaces Evolving by Mullins-Sekerka Dynamics

We present an algorithm for computing the nonlinear interface dynamics of the Mullins-Sekerka model for interfaces that are defined implicitly (e.g. by a level set function) using integral equations. The computation of the dynamics involves solving Laplace’s equation with Dirichlet boundary conditions on multiply connected and unbounded domains and propagating the interface using a normal velocity obtained from the solution of the PDE at each time step. Our method is based on a simple formulation for implicit interfaces, which rewrites boundary integrals as volume integrals over the entire space. The resulting algorithm thus inherits the benefits of both level set methods and boundary integral methods to simulate the nonlocal front propagation problem with possible topological changes. We present numerical results in both two and three dimensions to demonstrate the effectiveness of the algorithm.
Chieh Chen, Catherine Kublik, Richard Tsai

Geometric Interfacial Motion: Coupling Surface Diffusion and Mean Curvature Motion

Mean curvature motion as well as surface diffusion both constitute geometric interfacial motions which have received considerable attention. However in many applications a complex coupling of surfaces occurs whose evolution may be described using both these types of motion. A variety of such physical problems are described. While sometimes an anisotropic formulation might seem to be preferable, often an isotropic formulation is helpful to consider. Some analytic and numerical results are being presented, with some supporting experimental evidence. Many open questions remain.
V. Derkach, J. McCuan, A. Novick-Cohen, A. Vilenkin

A Note on Regularity Criteria for Navier-Stokes System

Note on Navier-Stokes
We use some interpolation inequalities on Besov spaces to show a regularity criterion for n-dimensional Navier-Stokes system.
Jishan Fan, Tohru Ozawa

On Periodic and Almost Periodic Solutions to Incompressible Viscous Fluid Flow Problems on the Whole Line

It is shown that a large class of semilinear evolution equations on the whole line with periodic or almost periodic forces admit periodic or almost periodic mild solutions. The approach presented generalizes the method described in [28] to the case of the whole line and to forces which are almost periodic in the sense of H. Bohr. It relies on interpolation methods and on \(L^p-L^q\)-smoothing properties of the underlying linearized equation. Applied to incompressible fluid flow problems, the approach yields new results on (almost) periodic solutions to the Navier-Stokes-Oseen equations, to the flow past rotating obstacles, to the Navier-Stokes equations in the rotational setting as well as to Ornstein–Uhlenbeck type equations.
Matthias Hieber, Thieu Huy Nguyen, Anton Seyfert

Remarks on Viscosity Solutions for Mean Curvature Flow with Obstacles

Obstacle problems for mean curvature flow equations are concerned. Existence of Lipschitz continuous viscosity solutions are obtained under several hypotheses. Comparison principle globally in time is also discussed.
K. Ishii, H. Kamata, S. Koike

Remark on Stability of Scale-Critical Stationary Flows in a Two-Dimensional Exterior Disk

We prove the asymptotic stability of stationary flows in a two-dimensional exterior disk under suitable decay and smallness conditions on the stationary flows and the initial perturbations. The class of stationary flows considered in this paper includes some typical circular flows which decay in the scale-critical order \(O(|x|^{-1})\) as \(|x|\rightarrow \infty \).
Yasunori Maekawa

Stochastic Effects and Time-Filtered Leapfrog Schemes for Maxwell’s Equations

We consider Maxwell’s equations where the conductivity contains fast random fluctuations in time. Using an Ornstein–Uhlenbeck process, we study the effects of correlations between the random fluctuations of two different time scales, with one an order of magnitude smaller than the other. We show that this asymptotic regime gives rise to a limiting equation where the effects of the fluctuations in the conductivity are captured in additional terms containing deterministic and stochastic corrections. For deterministic dynamics, numerical solutions to the time dependent Maxwell’s equations using a new time stepping scheme are presented. This scheme, which is based on the leapfrog method and a fourth-order time filter, significantly reduces the short oscillations generated by numerical dispersion. It uses staggering in space only, allowing explicit treatment of the electric current density terms and application of numerical smoothers. Comparisons of simulation results where Maxwell’s equations are integrated in a presence of the scattering of an electromagnetic pulse by a perfectly conducting square and those obtained with the unfiltered leapfrog show that the developed method is robust and accurate.
Alex Mahalov, Austin McDaniel

Weak Solutions to the Navier–Stokes Equations with Data in

The paper concerns the existence of weak solutions to the 3d-Navier–Stokes initial boundary value problem in exterior domains. The problem is considered with an initial data belonging to \(\mathbb L(3,\infty )\) which is a special subspace of the Lorentz’s space \(L(3,\infty )\). The nature of the domain and the initial data in \(L(3,\infty )\) make the result of existence not comparable with the usual Leray-Hopf theory of weak solutions. However, we are able to prove both that the weak solutions enjoy the partial regularity in the sense of Leray’s structure theorem and the asymptotic limit of \(|u(t)|_{3\infty }\).
P. Maremonti

Energy Solutions to One-Dimensional Singular Parabolic Problems with Data are Viscosity Solutions

We study one-dimensional very singular parabolic equations with periodic boundary conditions and initial data in BV, which is the energy space. We show existence of solutions in this energy space and then we prove that they are viscosity solutions in the sense of Giga–Giga.
Atsushi Nakayasu, Piotr Rybka

Local Well-Posedness for the Cauchy Problem to Nonlinear Heat Equations of Fujita Type in Nearly Critical Besov Space

We show the local well-posedness of the Cauchy problem to a nonlinear heat equation of Fujita type in lower space dimensions. It is well known that the nonnegative solution corresponding to the Fujita critical exponent \(p=1+\frac{2}{n}\) does not exist in the critical scaling invariant space \(L^1(\mathbb R^n)\). We show if the initial data is in a modified Besov spaces, then the corresponding mild solution to the equation with the Fujita critical exponent \(p=1+\frac{2}{n}\) exists and the problem is locally well-posed in the same space of the initial data. Besides we also show the problem is ill-posed in the scaling invariant Besov and inhomogeneous Besov spaces. This is known in \(L^1\) space and extension of the result known in the Lebesgue spaces.
Takayoshi Ogawa, Yuuki Yamane

Spatial Lipschitz Continuity of Viscosity Solution to Level Set Equation for Evolving Spirals by Eikonal-Curvature Flow

Level set equation for evolving spirals by an eikonal-curvature flow is considered. In this paper, we prove the spatial Lipschitz continuity of viscosity solutions to the level set equation for the evolving spirals provided that a suitable approximation of initial data exists. It is established with Bernstein’s method for regularized equation approximating the level set equation, and limiting procedure of solutions tending the approximating parameter to zero.
Takeshi Ohtsuka

A Hyperbolic Obstacle Problem with an Adhesion Force

We will treat free boundary problems of a wave type in this section. Examples of the physical phenomena that we have in mind are a motion of a soap film attached to a water surface or a droplet motion on a planner surface. The surface acts as an obstacle and there may exist adhesion forces when the film or the droplet detach from the obstacle. We consider the case with a positive contact angle in an equilibrium state. We also calculate the moving contact angle according to a dynamical action functional.
Seiro Omata

Epi-Two-Dimensional Flow and Generalized Enstrophy

The conservation of the enstrophy (\(L^2\) norm of the vorticity \(\omega \)) plays an essential role in the physics and mathematics of two-dimensional (2D) Euler fluids. Generalizing to compressible ideal (inviscid and barotropic) fluids, the generalized enstrophy \(\int _{\varSigma (t)}\) f(\(\omega /\rho )\rho \mathrm {d}^2 x\) (f an arbitrary smooth function, \(\rho \) the density, and \(\varSigma (t)\) an arbitrary 2D domain co-moving with the fluid) is a constant of motion, and plays the same role. On the other hand, for the three-dimensional (3D) ideal fluid, the helicity \(\int _{M}\) V \(\cdot \varvec{\omega }\,\mathrm {d}^3x\) (\(\varvec{V}\) the flow velocity, \(\varvec{\omega }=\nabla \times \varvec{V}\), and M the three-dimensional domain containing the fluid) is conserved. Evidently, the helicity degenerates in a 2D system, and the (generalized) enstrophy emerges as a compensating constant. This transition of the constants of motion is a reflection of an essential difference between 2D and 3D systems, because the conservation of the (generalized) enstrophy imposes stronger constraints, than the helicity, on the flow. In this paper, we make a deeper inquiry into the helicity-enstrophy interplay: the ideal fluid mechanics is cast into a Hamiltonian form in the phase space of Clebsch parameters, generalizing 2D to a wider category of epi-2D flows (2D embedded in 3D has zero-helicity, while the converse is not true – our epi-2D category encompasses a wider class of zero-helicity flows); how helicity degenerates and is substituted by a new constant is delineated; and how a further generalized enstrophy is introduced as a constant of motion applying to epi-2D flow is described.
Zensho Yoshida, Philip J. Morrison

Analysis of a Living Fluid Continuum Model

Generalized Navier–Stokes equations which were proposed recently to describe active turbulence in living fluids are analyzed rigorously. Results on wellposedness and stability in the \(L^2(\mathbb {R}^n)\)-setting are derived. Due to the presence of a Swift-Hohenberg term, global wellposedness in a strong setting for arbitrary initial data in \(L^2_\sigma (\mathbb {R}^n)\) is available. Based on the existence of global strong solutions, results on linear and nonlinear (in-) stability for the disordered steady state and the manifold of ordered polar steady states are derived, depending on the involved parameters.
Florian Zanger, Hartmut Löwen, Jürgen Saal
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