Introduction to the special issue on hydrodynamic model equations
Adrian Constantin Joachim Escher
Communications on Pure & Applied Analysis 2012, 11(4): i-iii doi: 10.3934/cpaa.2012.11.4i
The increased interest in water wave theory over the last decade has been motivated, arguably, by two themes: rst, noticeable progress in the investigation of the governing equations for water waves (well-posedness issues, as well as in-depth qualitative studies of regular wave patterns{see the discussion and the list of references in [1, 17], respectively in [9]), and secondly, by the derivation and study of various model equations that, although simpler, capture with accuracy the prominent features of the governing equations in a certain physical regime. The two themes are intertwined with one another.
Two-component higher order Camassa-Holm systems with fractional inertia operator: A geometric approach
Joachim Escher Tony Lyons
Journal of Geometric Mechanics 2015, 7(3): 281-293 doi: 10.3934/jgm.2015.7.281
In the following we study the qualitative properties of solutions to the geodesic flow induced by a higher order two-component Camassa-Holm system. In particular, criteria to ensure the existence of temporally global solutions are presented. Moreover in the metric case, and for inertia operators of order higher than three, the flow is shown to be geodesically complete.
keywords: global solutions. Diffeomorphism group geodesic flow
Adrian Constantin Joachim Escher
Discrete & Continuous Dynamical Systems - B 2009, 12(3): i-iii doi: 10.3934/dcdsb.2009.12.3i
Over the period 12-23 October 2009, the program "Recent advances in integrable systems of hydrodynamic type'' was organized by us and held at the Erwin Schrödinger International Institute for Mathematical Physics (Vienna, Austria).

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The surface diffusion flow on rough phase spaces
Joachim Escher Piotr B. Mucha
Discrete & Continuous Dynamical Systems - A 2010, 26(2): 431-453 doi: 10.3934/dcds.2010.26.431
The surface diffusion flow is the gradient flow of the surface functional of compact hypersurfaces with respect to the inner product of $H^{-1}$ and leads to a nonlinear evolution equation of fourth order. This is an intrinsically difficult problem, due to the lack of an maximum principle and it is known that this flow may drive smoothly embedded uniformly convex initial surfaces in finite time into non-convex surfaces before developing a singularity [15, 16]. On the other hand it also known that singularities may occur in finite time for solutions emerging from non-convex initial data, cf. [10].
   Combining tools from harmonic analysis, such as Besov spaces, multiplier results with abstract results from the theory of maximal regularity we present an analytic framework in which we can investigate weak solutions to the original evolution equation. This approach allows us to prove well-posedness on a large (Besov) space of initial data which is in general larger than $C^2$ (and which is in the distributional sense almost optimal). Our second main result shows that the set of all compact embedded equilibria, i.e. the set of all spheres, is an invariant manifold in this phase space which attracts all solutions which are close enough (which respect to the norm of the phase space) to this manifold. As a consequence we are able to construct non-convex initial data which generate global solutions, converging finally to a sphere.
keywords: maximal regularity centre manifold stability Besov spaces Surface diffusion flow free boundary problem. global existence geometric evolution equation
Well-posedness and stability analysis for a moving boundary problem modelling the growth of nonnecrotic tumors
Joachim Escher Anca-Voichita Matioc
Discrete & Continuous Dynamical Systems - B 2011, 15(3): 573-596 doi: 10.3934/dcdsb.2011.15.573
We study a moving boundary problem describing the growth of nonnecrotic tumors in different regimes of vascularisation. This model consists of two decoupled Dirichlet problem, one for the rate at which nutrient is added to the tumor domain and one for the pressure inside the tumor. These variables are coupled by a relation which describes the dynamic of the boundary. By re-expressing the problem as an abstract evolution equation, we prove local well-posedness in the small Hölder spaces context. Further on, we use the principle of linearised stability to characterise the stability properties of the unique radially symmetric equilibrium of the problem.
keywords: Tumor growth Stability. Moving boundary problem Well-posedness
Right-invariant Sobolev metrics of fractional order on the diffeomorphism group of the circle
Joachim Escher Boris Kolev
Journal of Geometric Mechanics 2014, 6(3): 335-372 doi: 10.3934/jgm.2014.6.335
In this paper, we study the geodesic flow of a right-invariant metric induced by a general Fourier multiplier on the diffeomorphism group of the circle and on some of its homogeneous spaces. This study covers in particular right-invariant metrics induced by Sobolev norms of fractional order. We show that, under a certain condition on the symbol of the inertia operator (which is satisfied for the fractional Sobolev norm $H^{s}$ for $s \ge 1/2$), the corresponding initial value problem is well-posed in the smooth category and that the Riemannian exponential map is a smooth local diffeomorphism. Paradigmatic examples of our general setting cover, besides all traditional Euler equations induced by a local inertia operator, the Constantin-Lax-Majda equation, and the Euler-Weil-Petersson equation.
keywords: Sobolev metrics of fractional order. Euler equation diffeomorphism group of the circle
A survey on second order free boundary value problems modelling MEMS with general permittivity profile
Joachim Escher Christina Lienstromberg
Discrete & Continuous Dynamical Systems - S 2017, 10(4): 745-771 doi: 10.3934/dcdss.2017038

In this survey we review some recent results on microelectromechanical systems with general permittivity profile. Different systems of differential equations are derived by taking various physical modelling aspects into account, according to the particular application. In any case an either semi-or quasilinear hyperbolic or parabolic evolution problem for the displacement of an elastic membrane is coupled with an elliptic moving boundary problem that determines the electrostatic potential in the region occupied by the elastic membrane and a rigid ground plate. Of particular interest in all models is the influence of different classes of permittivity profiles.

The subsequent analytical investigations are restricted to a dissipation dominated regime for the membrane's displacement. For the resulting parabolic evolution problems local well-posedness, global existence, the occurrence of finite-time singularities, and convergence of solutions to those of the so-called small-aspect ratio model, respectively, are investigated. Furthermore, a topic is addressed that is of note not till non-constant permittivity profiles are taken into account -the direction of the membrane's deflection or, in mathematical parlance, the sign of the solution to the evolution problem. The survey is completed by a presentation of some numerical results that in particular justify the consideration of the coupled problem by revealing substantial qualitative differences of the solutions to the widely-used small-aspect ratio model and the coupled problem.

keywords: MEMS free boundary value problem general permittivity profile finite-time singularity
The geometry of a vorticity model equation
Joachim Escher Boris Kolev Marcus Wunsch
Communications on Pure & Applied Analysis 2012, 11(4): 1407-1419 doi: 10.3934/cpaa.2012.11.1407
We show that the modified Constantin-Lax-Majda equation modeling vortex and quasi-geostrophic dynamics [27] can be recast as the geodesic flow on the subgroup $\mathrm{Diff}_{1}^{\infty}(\mathbb{S})$ of orientation-preserving diffeomorphisms $\varphi \in \mathrm{Diff}^{\infty}(\mathbb{S})$ such that $\varphi(1) = 1$ equipped with the right-invariant metric induced by the homogeneous Sobolev norm $\dot H^{1/2}$. On the extended group of diffeomorphisms of Sobolev class $H^{k}$ with $k\ge 2$, this induces a weak Riemannian structure. We establish that the geodesic spray is smooth and we obtain local existence and uniqueness of the geodesics.
keywords: Euler equation on diffeomorphisms group of the circle. CLM equation
Global existence and blow-up phenomena for a weakly dissipative Degasperis-Procesi equation
Shuyin Wu Joachim Escher Zhaoyang Yin
Discrete & Continuous Dynamical Systems - B 2009, 12(3): 633-645 doi: 10.3934/dcdsb.2009.12.633
This paper is concerned with the long time behaviour of a weakly dissipative Degasperis-Procesi equation. Our analysis discloses the co-existence of global in time solutions and finite time break down of strong solutions. Our blow-up criterion for the initial profile generalizes considerably results obtained earlier in [32].
keywords: Weakly dissipative Degasperis-Procesi equation; Global existence; Blow-up.
Euler equations on a semi-direct product of the diffeomorphisms group by itself
Joachim Escher Rossen Ivanov Boris Kolev
Journal of Geometric Mechanics 2011, 3(3): 313-322 doi: 10.3934/jgm.2011.3.313
The geodesic equations of a class of right invariant metrics on the semi-direct product $Diff(\mathbb{S}^1)$Ⓢ$Diff(\mathbb{S}^1)$ are studied. The equations are explicitly described, they have the form of a system of coupled equations of Camassa-Holm type and possess singular (peakon) solutions. Their integrability is further investigated, however no compatible bi-Hamiltonian structures on the corresponding dual Lie algebra $(Vect(\mathbb{S}^1)$Ⓢ$Vect(\mathbb{S}^1))^{*}$ are found.
keywords: diffeomorphism group of the circle. peakons integrable systems Euler equation

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