SPH modelling of hydrodynamic lubrication: laminar fluid flow-structure
interaction with no-slip **conditions** for slider bearings

**conditions**. Expand abstract.

**conditions**and laminar regimes for the simulation of hydrodynamic lubrication. The code is herein validated in relation to a uniform slider bearing (i.e., for a constant lubricant film depth) and a linear slider bearing (i.e., for a film depth with a linear profile variation along the main flow direction). Validations refer to comparisons with analytical solutions, herein generalized to consider any Dirichlet

**boundary**

**condition**. Further, this study allows a first code validation of the fluid-fixed frontier interactions under no-slip

**conditions**. With respect to the most state-of-the-art models (2D codes based on Reynolds' equation for fluid films), the following distinctive features are highlighted: (i) 3D formulation on all the terms of the Navier-Stokes equations for incompressible fluids with uniform viscosity; (ii) validations on both local and global quantities (pressure and velocity profiles; load-bearing capacity); (iii) possibility to simulate any 3D topology. This study also shows the advantages of using a CFD-SPH code in simulating the inertia and 3D effects close to the slider edges, and it opens new research directions overcoming the limitations of the codes for hydrodynamic lubrication based on the Reynolds' equation for fluid films. This study finally allows SPHERA to deal with hydrodynamic lubrication and empowers the code for other relevant application fields involving fluid-structure interactions (e.g., transport of solid bodies by floods and earth landslides; rock landslides). SPHERA is developed and distributed on a GitHub public repository.

4/10 relevant

arXiv

Nonlocal diffusion equations with dynamical **boundary** **conditions**

**boundary**

**conditions**. Expand abstract.

**boundary**

**conditions**. We deal both with smooth and with singular kernels and show existence and uniqueness of solutions and study their asymptotic behaviour as t goes to infinity.

10/10 relevant

arXiv

Spectral analysis for the class of integral operators arising from
well-posed **boundary** value problems of finite beam deflection on elastic
foundation: characteristic equation

**boundary**value problem for the deflection of a beam with finite length resting on an elastic foundation and subject to a vertical loading. For each well-posed two-point

**boundary**

**condition**represented by a $4 \times 8$ matrix $\mathbf{M}$, we explicitly construct the corresponding Green's function $G_\mathbf{M}$, from which we construct a one-to-one correspondence $\Gamma$ from the set of equivalent well-posed

**boundary**

**conditions**to $\mathrm{gl}(4,\mathbb{C})$. Using $\Gamma$, we derive, for each well-posed

**boundary**

**condition**$\mathbf{M}$, an eigencondition and characteristic equations for the spectrum $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$ of the integral operator $\mathcal{K}_\mathbf{M}$ with $G_\mathbf{M}$ as its kernel. Special features of our eigencondition and characteristic equations include, (1) they isolate the effect of the

**boundary**

**condition**$\mathbf{M}$ on $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$, and (2) they connect $\mathrm{Spec}\,\mathcal{K}_\mathbf{M}$ with the spectrum of a particular integral operator $\mathcal{K}_{l,\alpha,k}$, whose structure has been well understood. Using our characteristic equations, we show that, for each nonzero real number $\lambda$ not in $\mathrm{Spec}\,\mathcal{K}_{l,\alpha,k}$, there exists a real well-posed

**boundary**

**condition**$\mathbf{M}$ such that $\lambda$ is an eigenvalue of $\mathcal{K}_\mathbf{M}$. This in particular shows that the integral operators $\mathcal{K}_\mathbf{M}$ arising from well-posed

**boundary**conditions, may not be positive nor contractive in general, as opposed to $\mathcal{K}_{l,\alpha,k}$.

10/10 relevant

arXiv

Non-linear Stability of Double Bubbles under Surface Diffusion

**boundary**

**conditions**and problems with the (non-local) tangential part. Expand abstract.

**boundary**

**conditions**derived by Garcke and Novick-Cohen as the singular limit of a Cahn-Hilliard equation with degenerated mobility. These

**conditions**are the concurrency of the triple junction, angle

**conditions**between the hypersurfaces, continuity of the chemical potentials and a flux-balance. For this system we show stability of its energy minimizers, i.e., standard double bubbles.The main argument relies on a Lojasiewicz-Simon gradient inequality. The proof of it differs from others works due to the fully non-linear

**boundary**

**conditions**and problems with the (non-local) tangential part.

5/10 relevant

arXiv

Quantum Control at the **Boundary**: an application to quantum circuits

**boundary**

**conditions**will be proven. Expand abstract.

**boundary**

**conditions**will be proven. This establishes a first theoretical proof of the feasibility of the quantum control at the

**boundary**paradigm. A simplified version of this results can be found in arXiv:1811.09541.

4/10 relevant

arXiv

On the equality between the infimum obtained by solving various Plateau's problem

**boundary**

**conditions**, we also compare that with the problem of minimizing the size of integral currents with given boundary. Expand abstract.

**boundary**conditions, we also compare that with the problem of minimizing the size of integral currents with given

**boundary**. Finally we will get the agreement of the infimum for those Plateau's problem.

4/10 relevant

arXiv

**Boundary** Lebesgue mixed-norm estimates for non-stationary Stokes systems
with VMO coefficients

**boundary**

**conditions**. We show that for the Lions conditions, in contrast to the Dirichlet

**boundary**conditions, local

**boundary**mixed-norm $L_{s,q}$-estimates of the spatial second-order derivatives of solutions hold, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with $s=q=2$, the result is still new and provides local

**boundary**Caccioppoli-type estimates, which are important in applications. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the

**boundary**with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

5/10 relevant

arXiv

On periodic solutions for one-phase and two-phase problems of the Navier-Stokes equations

**boundary**

**conditions**or transmission

**conditions**in fixed domains by using the so-called Hanzawa transform. We separate solutions into the stationary part and the oscillatory part. The linearized equations for the stationary part have eigen-value $0$, which is avoided by changing the equations with the help of the necessary

**conditions**for the existence of solutions to the original problems. To treat the oscillatory part, we establish the maximal $L_p$-$L_q$ regularity theorem of the periodic solutions for the system of parabolic equations with non-homogeneous

**boundary**

**conditions**or transmission conditions, which is obtained by the systematic use of $\mathcal{R}$-solvers to the resolvent problem for the linearized equations and the transference theorem for the $L_p$ boundedness of operator-valued Fourier multipliers. These approaches are the novelty of this paper.

4/10 relevant

arXiv

Quantitative Estimates on Reiterated Homogenization of Linear Elliptic operators Using Fourier Transform Methods

**boundary**

**conditions**. We obtain error estimates $O(\varepsilon)$ for a bounded $C^{1,1}$ domain for this equation as well as the interior Lipschitz estimates at (very) large scale. Compared to the general homogenization problems, the difficulty in the reiterated homogenization is that we need to handle different scales of $x$. To overcome this difficulty, we firstly introduce the Fourier transform in the homogenization theory to separate these different scales. We also note that this method may be adapted to the following reiterated homogenization problem: $-\frac{\partial}{\partial x_{i}} \left(a_{i j} \left(\frac{x}{\varepsilon},\cdots, \frac{x}{\varepsilon^{N}}\right) \frac{\partial u_\varepsilon}{\partial x_{j}} \right) = f$ in $\Omega$ with Dirichlet

**boundary**

**conditions**. Moreover, our results may be extended to the related Neumann

**boundary**problems without any real difficulty.

4/10 relevant

arXiv

Order reduction and how to avoid it when Lawson methods integrate
reaction-diffusion **boundary** value problems

**boundary**

**conditions**. Expand abstract.

**boundary**value problems where the solutions are not periodic in space or do not satisfy enough

**conditions**of annihilation on the

**boundary**. However, in a previous paper, a modification of Lawson quadrature rules has been suggested so that no order reduction turns up when integrating linear problems subject to even time-dependent

**boundary**

**conditions**. In this paper, we describe and thoroughly analyse a technique to avoid also order reduction when integrating nonlinear problems. This is very useful because, given any Runge-Kutta method of any classical order, a Lawson method can be constructed associated to it for which the order is conserved.

5/10 relevant

arXiv