We are interested in dispersive equations of the Korteweg-de Vries type of the form `u_t + f(u)_x - (-\Delta)^{\alpha/2}u_x = 0`, for `\alpha \geq 1`. For example, one can cite the Korteweg-de Vries equation when `\alpha = 2` and the Benjamin-Ono equation when `\alpha = 1`. These equations have been used, among others, to describe the wave propagation in shallow water and can be derived from Newton's second law. Because of the nonlinearity `f(u)_x` depending on the spatial derivative, and thus the difficulty to prove the Lispchitz property, to study the initial value problem, the global existence and the continuity with respect to the initial datum, remains difficult.
Theorem: Let `\alpha \geq 1` and `u_0 \in H^{\alpha/2}(\mathbb{R})`. Then the initial value problem is globally well posed in `H^{s}(\mathbb{R}), s \geq \alpha/2`. Moreover pour all `t \in \mathbb{R}`, `||u(t)||_{\alpha/2} = ||u_0(t)||_{\alpha/2}, ||u(t)||_s \leq C e^{e^|t|}`, while for `|t| \leq 1/||u_0(t)||_s, ||u(t)||_s \leq C ||u_0(t)||_s`.

When the solution is defined in a bounded domain, change his point of view may be useful. When `f(u) = u^2/2`, there exists a locally invertible change of variables `v = \Lambda u = u - \frac{1}{2}\sum_{k\neq k1 \in \mathbb{Z}^*} \frac{e^{ikx} {ik}/(1+|k|^\alpha)}{\sigma(k_1)+\sigma(k-k_1)-\sigma(k)}\hat u(k_1) \hat u (k-k_1)`, where `\sigma(k) = {ik}/(1+k^\alpha)` is the dispersion relation, and `v` is solution of the normal form `v_t + \tilde{f}(v)_x - (-\Delta)^{\alpha/2}v_x = 0`. Here `\tilde{f}` is a trilinear function that allows to extend the existence of solution. This method is based on the theory of Poincaré's normal forms.
Theorem: Let `s > \alpha/2`. There exists `\varepsilon > 0` such that if `|| u_0 || \leq \varepsilon`, then the unique solution `u \in \mathcal{C}([-T^2,T^2]; H^s(\mathbb{T}))` with `T \~ 1/\varepsilon` satisfies for `|t| \leq T^2, ||u(t)||_s \leq C ||u_0(t)||_s.`

Another important question concerns the uniqueness of the solutions. It can be put in the following form: let `u(x,t)=0` for all `(t,x) \in I \times \Omega` with `I \times \Omega \subset [0,T] \times \mathbb{R}`, then `u = 0?` When `I = [0,T]` and `\Omega` is compact, an adequate analytical development has been proposed by Bourgain. Thanks to the Paley-Wiener theorem, it is possible to write `\hat u (\xi + i \sigma, t) = e^{i(\xi + i \sigma)^{\alpha+1}} \hat u_0 (\xi + i \sigma).` Adding a decay condition on the initial data `u_0` allows to conclude positively.
Theorem: Let `u \in \mathcal{C}([0,T];H^s)` with `s\geq 4` such that for all `t \in [0,T], u(t)` is compactly supported. Then `u` is identically null.

This assumption can be weakened by taking `I \times \Omega` an open set of `[0,T] \times \mathbb{R}`. It is then necessary to determine a Carleman inequality `\int |u|^2 e^{\psi(x,t)} dx dt \leq \frac{C}{\tau^{\alpha+1}} \int |u_t + f(u)_x - (-\Delta)^{\alpha/2}u_x|^2 e^{\psi(x,t)} dx dt,` where `\psi` is a convex function. Thus, when the right-hand-side is bounded, it is sufficient to `\tau \to \infty` so that u vanishes.
Theorem: Let `u \in \mathcal{C}([0,T];H^s)` with `s\geq 4` such that `u=0`in an open set of `[0,T] \times \mathbb{R}`. Then `u`is identically null.

One method is to use dispersive (or Strichartz) inequalities of the type `||u(t)||_\infty \leq {||u_0||_{L^1 \cap H^s}}/(1+|t|)^{1/{\alpha+1}}.` This reflects the decay of the solutions, due to dispersion, i.e. the separation of the different Fourier modes that move at distinct speeds. Such an inequality is deduced from the dispersion relation. The solution of the equation linearized around `0` is written as an oscillating integral (via the Fourier transform) `u(x,t) = 1/{2\pi} \int e^{i(\xi^{\alpha+1}t + x \xi)} \hat u_0{\xi} d\xi,` and a stationary phase lemma allows us to conclude.
Theorem: Let `\rho \geq \alpha +2.` Then for `u_0 \in L^p \cap H^s` small enough, there exists a unique global in time solution `u \in \mathcal{C}(\mathbb{R};L^p \cap H^s)` and for all `t \in \mathbb{R}`, `||u(t)||_q \leq {||u_0||_{L^p \cap H^s}}/(1+|t|)^{{1-2/q}/{\alpha+1}}.`
Spectral methods are very suitable for the discretization of such equations, nonlinearity and derivatives being easily treated. To preserve the conservation laws, high order schemes are required. The resulting nonlinear system is then solved using a fixed point method. To avoid periodic boundary conditions while preserving conservation of mass and energy, we have recently developed compact schemes.

We are interested in the stochastic oscillating integral `u(w, x,t) = 1/{2\pi} \int e^{i(\sigma(\xi) W_t + x \xi)} \hat u_0{\xi} d\xi.` If `\sigma` is the symbol of the operator `A`, then this function is a solution of the linear stochastic PDE `du + Au \circ dW = 0` or in Itô form `du + Au dW - 1/2 A^2u dt = 0.` Following the methods developed in the deterministic case above, we established a stationary phase lemma that yields an average decay result. If `\sigma(\xi) = \xi/(1+\xi^2)`, we obtain global well-posedness with a nonlinearity `A u^{p+1}`.
Theorem: Let `\rho \geq 9`. Then for `u_0 \in L^2 \cap H^4` small enough, there exists a unique global in time solution with path in `\mathcal{C}(\mathbb{R}; H^1)` and for all `t \in \mathbb{R}`, `\mathbb{E}(||u(t)||_\infty) \leq {C(||u_0||)}/(1+|t|)^{1/6}.`
This stochastic PDE can be discretized by spectral methods. To preserve mass and energy, a midpoint-in-time scheme has been chosen. Note that in general the order of convergence of a scheme is lower for a stochastic equation than for its deterministic version. Because of the speed of convergence of the Stratonovich integral, one expects a (strong) order `1/2`. A proper writing of the global error and the consistency error allows to show that the order is `1`.
Theorem: Let `T>0` and `u_0 \in H^1`. For `N>0` large enough and `\delta t = T/N`, we have for all `n\leq N`, `\lim_{C \to \infty} \mathbb{P} ( Sup_{\delta t} {||u(t_n)-u_n||_{H^1}}/{\delta t} \geq C ) = 0.`

The dissipation in the equations reflects the damping of waves due for example to the Coriolis force, or the mangrove forests... From a mathematical point of view, the dissipation can be written `u_t + f(u)_x - \delta (-\Delta)^{\alpha/2}u_x + \varepsilon \mathcal{A}u = 0,` with `\langle \mathcal{A}u, u \rangle > 0`. Thus, multiplying the equation by u and integrating in space, we obtain `d/{dt} ||u||^2 = - \varepsilon \langle \mathcal{A}u, u \rangle < 0,` and the `L^2-`norm decreases whereas it was preserved in the purely dispersive case above. The study of the space `\mathcal{H} := \{ u:\mathbb{R} \to \mathbb{R}, \langle \mathcal{A}u, u \rangle < + \infty \}` is then of primary importance. Relevant choices of damping, e.g. `\hat{\mathcal{A} u} = -\gamma(\xi) \hat u(\xi)`, provide continuous and compact injection properties in `L^\infty`. In particular it allows `\mathcal{H}` to be an algebra. This may clearly define the nonlinearity `f(u)_x.`
Theorem: Let `s>0`. If `1/\gamma^s \in L^1(\mathbb{R}),` then `\mathcal{H}_{\gamma^s} ` is continuously injected in `L^\infty(\mathbb{R})`. If `\gamma_1(\xi) \geq \gamma_2(\xi),` then `\mathcal{H}_{\gamma_1^s}` is compactly injected in `\mathcal{H}_{\gamma_2^s}.` If `1/\gamma^s \in L^1(\mathbb{R}),` and for all `\xi, \xi_1 \in \mathbb{R}, \sqrt{\gamma(\xi)} \leq \sqrt{\gamma(\xi-\xi_1)} + \sqrt{\gamma(\xi_1)},` then `\mathcal{H}_{\gamma^s} ` is an algebra.
From then on, we can build other operators that, in addition to damp the `L^2-`norm, stabilize the energy. We were interested more specifically on operators with memory as `\mathcal{A}u = - \int_0^{\infty} \mu(s) u_{x x}(t-s) ds.`
Theorem: Let `\mu` being such that there exists `\delta >0`, for almost every `s\in \mathbb{R}^+` `\mu'(s) +\delta \mu(s) \leq 0.` Then there exists `\omega >0` and a positive increasing function `\mathcal{Q}`, depending on `\mu` and the domain, such that `||u(t)||^2 \leq \mathcal{Q}(u_0) e^{-\omega t}.`

Formally, when `\varepsilon` and `\delta` go `0`, the equation becomes the hyperbolic conservation law `u_t + f(u)_x = 0,` for which there is a unique entropy solution. It is possible to prove this convergence as soon as the nonlinearity `f` is convex and the dissipation is large compared to the dispersion (`\delta ≪ \varepsilon`). This allows to attenuate the oscillations due to the dispersion. Consider `u_t + f(u)_x = \varepsilon u_{x x} - \delta u_{x x x}.` Three cases are possible :

If `\delta` is large compared to `\varepsilon`, threre is no convergence.

If `\delta= o(\varepsilon^2)`, the sequence converges to the entropy solution.

If `\delta = O(\varepsilon^2)`, the sequence converges but not necessary to the entropy solution.

We obtained similar result for a nonlinear dispersion `g(u_{x x})_x`.
Theorem: Let `varepsilon>0` and `f:\mathbb{R} \to \mathbb{R}` a convex function with `f''(u) \leq C (1+|u|^\beta)`.

If `g(u) = -u^2, \delta = o(\varepsilon^{5/2})` and `0\leq \beta < 1/2`.

If `g(u) = -|u|, \delta = o(\varepsilon^2)` and `0\leq \beta < 3`.

If `g(u) = -|u|^n` with `1< n < 2`, and `\delta = o(\varepsilon^{{3n-1}/2}, \varepsilon^{{5n-1}/{2(2n-1)}})` and `0\leq \beta < {4-n}/n^2`.

Then the sequence converges to the entropy solution.