Classical well-posedness in dispersive equations with nonlinearities of mild regularity, and a composition theorem in Besov spaces

For both localized and periodic initial data, we prove local existence in classical energy space $H^s, s>\frac{3}{2}$, for a class of dispersive equations $u_{t}+(n(u))_{x}+Lu_{x}=0$ with nonlinearities of mild regularity. Our results are valid for symmetric Fourier multiplier operators $L$ whose symbol is of temperate growth, and $n(\cdot)$ in local Sobolev space $H^{s+2}_{\mathrm{loc}}(\mathbb{R})$. In particular, the results include non-smooth and exponentially growing nonlinearities. Our proof is based on a combination of semi-group methods and a new composition result for Besov spaces. In particular, we extend a previous result for Nemytskii operators on Besov spaces on $\mathbb{R}$ to the periodic setting by using the difference-derivative characterization of Besov spaces

Trimodal steady water waves

We construct three-dimensional families of small-amplitude gravity-driven rotational steady water waves on finite depth. The solutions contain counter-currents and multiple crests in each minimal period. Each such wave generically is a combination of three different Fourier modes, giving rise to a rich and complex variety of wave patterns. The bifurcation argument is based on a blow-up technique, taking advantage of three parameters associated with the vorticity distribution, the strength of the background stream, and the period of the wave.Comment: The final publication is available at Springer via http://dx.doi.org/10.1007/s00205-014-0812-

Small-amplitude fully localised solitary waves for the full-dispersion Kadomtsev--Petviashvili equation

The KP-I equation $$(u_t-2uu_x+\tfrac{1}{2}(\beta-\tfrac{1}{3})u_{xxx})_x -u_{yy}=0$$ arises as a weakly nonlinear model equation for gravity-capillary waves with strong surface tension (Bond number $\beta>1/3$). This equation admits --- as an explicit solution --- a `fully localised' or `lump' solitary wave which decays to zero in all spatial directions. Recently there has been interest in the \emph{full-dispersion KP-I equation} $$u_t + m({\mathrm D}) u_x + 2 u u_x = 0,$$ where $m({\mathrm D})$ is the Fourier multiplier with symbol $$m(k) = \left( 1 + \beta |k|^2|\right)^{\frac{1}{2}} \left( \frac{\tanh |k|}{|k|} \right)^{\frac{1}{2}} \left( 1 + \frac{2k_2^2}{k_1^2} \right)^{\frac{1}{2}},$$ which is obtained by retaining the exact dispersion relation from the water-wave problem. In this paper we show that the FDKP-I equation also has a fully localised solitary-wave solution. The existence theory is variational and perturbative in nature. A variational principle for fully localised solitary waves is reduced to a locally equivalent variational principle featuring a perturbation of the variational functional associated with fully localised solitary-wave solutions of the {KP-I} equation. A nontrivial critical point of the reduced functional is found by minimising it over its natural constraint set.Comment: 31 page

Steady water waves with multiple critical layers

We construct small-amplitude periodic water waves with multiple critical layers. In addition to waves with arbitrarily many critical layers and a single crest in each period, two-dimensional sets of waves with several crests and troughs in each period are found. The setting is that of steady two-dimensional finite-depth gravity water waves with vorticity.Comment: 16 pages, 2 figures. As accepted for publication in SIAM J. Math. Ana

The Whitham Equation as a Model for Surface Water Waves

The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation. In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to two standard free surface models: the KdV and the BBM equation. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than both the KdV and BBM equations.Comment: 14 pages, 4 figure

Traveling waves for the Whitham equation

The existence of traveling waves for the original Whitham equation is investigated. This equation combines a generic nonlinear quadratic term with the exact linear dispersion relation of surface water waves on finite depth. It is found that there exist small-amplitude periodic traveling waves with sub-critical speeds. As the period of these traveling waves tends to infinity, their velocities approach the limiting long-wave speed c0, and the waves approach a solitary wave. It is also shown that there can be no solitary waves with velocities much greater than c0. Finally, numerical approximations of some periodic traveling waves are presented

Forestry Extractivism in Uruguay

This chapter assesses how prominent definitions of (agro)extractivism are suited to explain forestry extractivism, and what are the shared and particular qualities of forestry extractivism as it manifests itself in large-scale tree monocultures for pulp production in Uruguay. Different definitions of (agro)extractivism are assessed. Forestry extractivism is one distinct form of agroextractivism, with some notable differences. Key features of forestry extractivism include: 1) Specific trade deals, as pulp investments are costly. 2) Long-term setting-up through stages: master plans; enclosures; establishing pulp mills; managing rising conflicts after the building. 3) Mills and plantations. 4) Ecological and carbon impacts. 5) Massive legitimization campaigns. This analysis should be accompanied by a global political economic and resource geopolitics analysis of particular global extractivisms, such as forestry. This should also be tied to particular contexts, polities and lived environments, which significantly influence especially the politics through which global extractivisms of different types are birthed and resisted. In here, what constitutes forestry extractivism in the context, polity and sector of Uruguayan pulpwood tree plantation expansion is analyzed. World-ecological and political ontological analyses are important for defining what activities should be called extractivist, and what types of extractivisms of what are involved in each activity.Peer reviewe