arXiv subjects
Publications and source records attributed to Sebastian Schwarzacher.
We provide a higher integrability result for the gradient of positive solutions to Trudinger's equation (also known as the doubly non-linear equation) for the range $p\in [2,\infty)$. The estimate is achieved by refining a construction of intrinsic cylinders from the vectorial setting by incorporating estimates only available in the scalar case.
We study the motion of a compressible heat-conducting fluid in three dimensions interacting with a non-linear flexible shell. The fluid is described by the full Navier--Stokes--Fourier system. The shell constitutes an unknown part of the boundary of the physical domain of the fluid and is changing in time. The solid is described as an elastic non-linear shell of Koiter type; in particular it possesses a non-convex elastic energy. Its deformation is modelled by a (non-linear) elastic Koiter shell. We show the existence of a weak solution to the corresponding system of PDEs which exists until the moving boundary approaches a self-intersection or the non-linear elastic energy of the shell degenerates. It is achieved by compactness results (in highest order spaces) for the solid-deformation and fluid-density. Our solutions comply with the first and second law of thermodynamics: the total energy is preserved and the entropy balance is understood as a variational inequality.
We consider the physical setup of a three-dimensional fluid-structure interaction problem. A viscous compressible gas or liquid interacts with a nonlinear, visco-elastic, three-dimensional bulk solid. The latter is described by a hyperbolic evolution with a non-convex elastic energy functional. The fluid is modelled by the compressible Navier--Stokes equations with a barotropic pressure law. Due to the motion of the solid, the fluid domain is time-changing. Our main result is the long-time existence of a weak solution to the coupled system until the time of a collision. The nonlinear coupling between the motions of the two different matters is established via the method of minimising movements. The motion of both the solid and the fluid is chosen via an incrimental minimization with respect to dissipative and static potentials. These variational choices together with a careful construction of an underlying flow map for our approximation then directly result in the pressure gradient and the material time derivatives.
We consider the homogenization limit of the compressible barotropic Navier-Stokes equations in a three-dimensional domain perforated by periodically distributed identical particles. We study the regime of particle sizes and distances such that the volume fraction of particles tends to zero but their resistance density tends to infinity. Assuming that the Mach number is increasing with a certain rate, the rescaled velocity and pressure of the microscopic system converges to the solution of an effective equation which is given by Darcy's law. The range of sizes of particles we consider are exactly the same which lead to Darcy's law in the homogenization limit of incompressible fluids. Unlike previous results for the Darcy regime we estimate the deficit related to the pressure approximation via the Bogovski\u{ı} operator This allows for more flexible estimates of the pressure in Lebesgue and Sobolev spaces and allows to proof convergence results for all barotropic exponents $γ> \frac32$.
We provide existence of very weak solutions and new a-priori estimates for steady flows of non-Newtonian fluids when the right-hand sides are not in the natural existence class. To obtain the a-priori estimates we make use of a newly developed solenoidal Lipschitz truncation that preserves zero boundary values. We provide also estimates in (Muckenhoupt) weighted spaces which permit us to regain a duality pairing. Our estimates are valid even in the presence of the convective term. They are obtained via a newly developed comparison method that allows to "cut out" the singularities of the right hand side such that the skew symmetry of the convective term can be used for large parts of the right hand side.
In this paper, we investigate the phenomenon of particle rebound in a viscous incompressible fluid environment. We focus on the important case of no-slip boundary conditions, for which it is by now classical that, under certain assumptions, collisions cannot occur in finite time. Motivated by the desire to understand this fascinating yet counterintuitive fluid-structure interaction, we introduce a reduced model which we study both analytically and numerically. In this simplified framework, we provide conditions which allow to prove that rebound is possible even in the absence of a topological contact. Our results lead to conjecture that a qualitative change in the shape of the solid is necessary for obtaining a physically meaningful rebound. We support the conjecture by also comparing numerical simulations performed for the reduced model with the finite element solutions obtained for the corresponding well-established PDE system.
We show weak-strong uniqueness and stability results for the motion of a two or three dimensional fluid governed by the Navier-Stokes equation interacting with a flexible, elastic plate of Koiter type. The plate is situated at the top of the fluid and as such determines the variable part of a time changing domain (that is hence a part of the solution) containing the fluid. The uniqueness result is a consequence of a stability estimate where the difference of two solutions is estimated by the distance of the initial values and outer forces. For that we introduce a methodology that overcomes the problem that the two (variable in time) domains of the fluid velocities and pressures are not the same. The estimate holds under the assumption that one of the two weak solutions possesses some additional higher regularity. The additional regularity is exclusively requested for the velocity of one of the solutions resembling the celebrated Ladyzhenskaya-Prodi-Serrin conditions in the given framework.
We introduce a two time-scale scheme which allows to extend the method of minimizing movements to hyperbolic problems. This method is used to show the existence of weak solutions to a fluid-structure interaction problem between a nonlinear, visco-elastic, $n$-dimensional bulk solid governed by a hyperbolic evolution and an incompressible fluid governed by the ($n$-dimensional) Navier-Stokes equations for $n\geq 2$.
In this paper we introduce a numerical scheme for fluid-structure interaction problems in two or three space dimensions: A flexible elastic plate is interacting with a viscous, compressible barotropic fluid. Hence the physical domain of definition (the domain of Eulerian coordinates) is changing in time. We introduce a fully discrete scheme that is stable, satisfies geometric conservation, mass conservation and the positivity of the density. We also prove that the scheme is consistent with the definition of continuous weak solutions.
We consider evolutionary PDE inclusions of the form \[ -λ\dot{u}_λ+ Δu - \mathrm{D} W_0(u) + f \ni \partial \mathcal{R}_1(\dot{u}) \qquad\text{in $(0,T) \times Ω$,} \] where $\mathcal{R}_1$ is a positively $1$-homogeneous rate-independent dissipation potential and $W_0$ is a (generally) non-convex energy density. This work constructs solutions to the above system in the slow-loading limit $λ\downarrow 0$. Our solutions have more regularity both in space and time than those that have been obtained with other approaches. On the "slow" time scale we see strong solutions to a purely rate-independent evolution. Over the jumps, we obtain a detailed description of the behavior of the solution and we resolve the jump transients at a "fast" time scale, where the original rate-dependent evolution is still visible. Crucially, every jump transient splits into a (possibly countable) number of rate-dependent evolutions, for which the energy dissipation can be explicitly computed. This in particular yields a global energy equality for the whole evolution process. It also turns out that there is a canonical slow time scale that avoids intermediate-scale effects, where movement occurs in a mixed rate-dependent / rate-independent way. In this way, we obtain precise information on the impact of the approximation on the constructed solution. Our results are illustrated by examples, which elucidate the effects that can occur.
We establish higher integrability up to the boundary for the gradient of solutions to porous medium type systems, whose model case is given by \begin{equation*} \partial_t u-Δ(|u|^{m-1}u)=\mathrm{div}\,F\,, \end{equation*} where $m>1$. More precisely, we prove that under suitable assumptions the spatial gradient $D(|u|^{m-1}u)$ of any weak solution is integrable to a larger power than the natural power $2$. Our analysis includes both the case of the lateral boundary and the initial boundary.
We study the generalized stationary Stokes system in a bounded domain in the plane equipped with perfect slip boundary conditions. We show natural stability results in oscillatory spaces, i.e. Hölder spaces and Campanato spaces including the border-line spaces of bounded mean oscillations (BMO) and vanishing mean oscillations (VMO). In particular, we show that under appropriate assumptions gradients of solutions are globally continuous. Since the stress tensor is assumed to be governed by a general Orlicz function, our theory includes various cases of (possibly degenerate) shear thickening and shear thinning fluids; including the model case of power law fluids. The global estimates seem to be new even in case of the linear Stokes system.
An optimal first-order global regularity theory, in spaces of functions defined in terms of oscillations, is established for solutions to Dirichlet problems for the $p$-Laplace equation and system, with right-hand side in divergence form. The exact mutual dependence among the regularity of the solution, of the datum on the right-hand side, and of the boundary of the domain in these spaces is exhibited. A comprehensive formulation of our results is given in terms of Campanato seminorms. New regularity results in customary function spaces, such as Hölder, $\mathrm{BMO}$ and $\mathrm{VMO}$ spaces, follow as a consequence. Importantly, the conclusions are new even in the linear case when $p=2$, and hence the differential operator is the plain Laplacian. Yet in this classical linear setting, our contribution completes and augments the celebrated Schauder theory in Hölder spaces, and complements the Jerison-Kenig gradient theory in Lebesgue spaces with a parallel in the oscillation spaces realm. The sharpness of our results is demonstrated by apropos examples.
We consider the isentropic Euler equations of gas dynamics in the whole two-dimensional space and we prove the existence of a $C^\infty$ initial datum which admits infinitely many bounded admissible weak solutions. Taking advantage of the relation between smooth solutions to the Euler system and to the Burgers equation we construct a smooth compression wave which collapses into a perturbed Riemann state at some time instant $T > 0$. In order to continue the solution after the formation of the discontinuity, we adjust and apply the theory developed by De Lellis and Székelyhidi and we construct infinitely many solutions. We introduce the notion of an admissible generalized fan subsolution to be able to handle data which are not piecewise constant and we reduce the argument to finding a single generalized subsolution.
We develop a methodology for proving well-posedness in optimal regularity spaces for a wide class of nonlinear parabolic initial-boundary value systems, where the standard monotone operator theory fails. A motivational example of a problem accessible to our technique is the following system \[ \partial_tu-\mathrm{div} ( ν(|\nabla u|) \nabla u )= -\mathrm{div} f \] with a given {strictly} positive bounded function $ν$, {such that $\lim_{k\to \infty} ν(k)=ν_\infty$} and $f \in L^q$ with $q\in (1,\infty)$. The {existence, uniqueness and regularity} results for $q\ge 2$ are by now standard. However, even if a priori estimates are available, the existence in case $q\in (1,2)$ was essentially missing. We overcome the related crucial difficulty, namely the lack of a standard duality pairing, by resorting to proper weighted spaces and consequently provide existence, uniqueness and optimal regularity in the entire range $q\in (1,\infty)$.
We show that the gradient of the $m$-power of a solution to a singular parabolic equation of porous medium-type (also known as fast diffusion equation), satisfies a reverse Hölder inequality in suitable intrinsic cylinders. Relying on an intrinsic Calderón-Zygmund covering argument, we are able to prove the local higher integrability of such a gradient for $m\in\left(\frac{(n-2)_+}{n+2},1\right)$. Our estimates are satisfied for a general class of growth assumptions on the non linearity. In this way, we extend the theory for $m\geq 1$ (see [GS16] in the list of references) to the singular case. In particular, an intrinsic metric that depends on the solution itself is introduced for the singular regime.
A pointwise bound for local weak solutions to the p-Laplace system is established in terms of data on the right-hand side in divergence form. The relevant bound involves a Havin-Maz'ya- Wulff potential of the datum, and is a counterpart for data in divergence form of a classical result of [KiMa], that has recently been extended to systems in [KuMi2]. A local bound for oscillations is also provided. These results allow for a unified approach to regularity estimates for broad classes of norms, including Banach function norms (e.g. Lebesgue, Lorentz and Orlicz norms), and norms depending on the oscillation of functions (e.g. Holder, BMO and, more generally, Campanato type norms). In particular, new regularity properties are exhibited, and well-known results are easily recovered.
We study the local regularity of $p$-caloric functions or more generally of $ϕ$-caloric functions. In particular, we study local solutions of non-linear parabolic systems with homogeneous right hand side, where the leading terms has Uhlenbeck structure of Orlicz type. This paper closes the gap of [22] where Liebermann proved that if the gradient of a solution is bounded, it is Hölder continuous. The crucial step is a novel local estimates for the gradient of the solutions, which generalize and improve the pioneering estimates of DiBenedetto and Friedman [12,10] for the $p$-Laplace heat equation.