Section 1
Among the most widely studied dispersive equations are those of Korteweg–de Vries type, which describe in particular the propagation of waves at the surface of water. These equations, derived from the fundamental principle of dynamics, take the general form
or in regularised form
They exhibit a nonlinearity acting on a spatial derivative, which makes the analysis of the Cauchy problem particularly delicate. Establishing the global existence of solutions and their continuity with respect to initial data represents a genuine mathematical challenge.
In this setting, one can show that for initial data in the Sobolev space $H^{\alpha/2}(\mathbb{R})$, the problem is globally well-posed. The $H^{\alpha/2}$ norm is conserved in time, whereas higher-regularity norms grow at most doubly exponentially globally in time, but only linearly on any bounded interval.
Theorem 1.1
Let $\alpha \geq 1$ and $u_0 \in H^{\alpha/2}(\mathbb{R})$. Then the Cauchy problem associated with (KdV) is globally well-posed in $H^s(\mathbb{R})$ for all $s \geq \alpha/2$. Moreover, for all $t \in \mathbb{R}$,
\[ \|u(t)\|_{\alpha/2} = \|u_0\|_{\alpha/2} \quad \text{and} \quad \|u(t)\|_s \lesssim e^{e^{t}}, \]whilst for $t \in [-T,T]$,
\[ \|u(t)\|_s \lesssim \|u_0\|_s. \]When solutions are defined on a periodic domain, a change of variables inspired by the theory of Poincaré normal forms transforms the original equation into a new equation whose nonlinearity is trilinear. For the PDE (BBM) and $f(u) = u^2/2$, this locally invertible change of variables $v = \Lambda u$ is defined by
where $\sigma(k) = ik/(1+k^\alpha)$ is the dispersion relation. It leads to existence results on substantially longer time intervals: for sufficiently small initial data in $H^s$, the solution exists up to time $T^2$, whereas classical methods only gave existence up to time $T \sim \|u_0\|_s^{-1}$ (Mammeri, 2009, 2012).
Theorem 1.2
Let $s > \alpha/2$. There exists $\varepsilon > 0$ such that if $u_0 \in H^s(\mathbb{T})$ with $\|u_0\|_s \leq \varepsilon$, then there exists a unique solution $u \in \mathcal{C}([-T^2, T^2]; H^s(\mathbb{T}))$ with $T \simeq \varepsilon^{-1}$, satisfying
\[ \|u(t)\|_s \lesssim \|u_0\|_s \qquad \text{for all } |t| \leq T^2. \]The question of uniqueness may be posed in a finer form: if two solutions coincide on an open subset of space-time, are they necessarily identical everywhere? When this open set is $[0,T] \times \Omega$ with $\Omega$ compact, an analytic argument based on the Paley–Wiener theorem gives an affirmative answer, provided the solution has compact support in space (Mammeri, 2009, 2010).
Theorem 1.3 — Unique Continuation
Let $u \in \mathcal{C}(0,T; H^s(\mathbb{R}))$ with $s \geq 4$, such that for all $t \in [0,T]$, $u(t)$ has compact support. Then $u$ is identically zero.
In the case of an arbitrary open set, the answer relies on establishing a Carleman inequality (Mammeri, 2013, 2013, Esfahani and Mammeri, 2021):
where $\psi$ is a convex function. By letting the parameter $\tau$ tend to infinity, the solution is forced to vanish as soon as the right-hand side is bounded.
Dispersion manifests itself not only through the conservation of certain norms, but also through the decay of solutions over long times. This phenomenon is explained by the progressive separation of the different Fourier modes (Mammeri, 2008, 2010, 2015). It is quantified by Strichartz-type inequalities:
The key to the proof lies in writing the solution as an oscillatory integral:
followed by an application of the stationary phase lemma. These estimates allow the construction of global solutions of the nonlinear problem with $f(u) = u^{\rho+1}$ satisfying quantitative decay bounds.
Theorem 1.4 — Quantitative Decay
Let $\rho \geq \alpha + 2$. There exists $\varepsilon > 0$ such that for $u_0 \in L^p \cap H^s(\mathbb{R})$ with norm at most $\varepsilon$, there exists a unique global solution $u \in \mathcal{C}(\mathbb{R}, L^p \cap H^s(\mathbb{R}))$ satisfying, for all $t \in \mathbb{R}$,
\[ \|u(t)\|_{L^q} \lesssim \frac{\|u_0\|_{L^p \cap H^s}}{(1 + |t|)^{\frac{1 - 2/q}{\alpha+1}}}. \]These techniques were also applied to the Whitham–Broer–Kaup equation with negative dispersion, for which the first well-posedness result since the introduction of this system in the 1970s was established, by constructing a symmetriser and using the vanishing viscosity method (Bedjaoui et al., 2022; Bedjaoui and Mammeri, 2023; Liverani et al., 2025).
More recently, this work was extended to the stochastic setting. The solution of the linear equation perturbed by multiplicative noise is written as a stochastic oscillatory integral:
solution of the linear stochastic PDE (Itô form): $du + Au\,dW - \frac{1}{2} A^2 u\,dt = 0$. A stochastic analogue of the stationary phase lemma then establishes decay in expectation of the $L^\infty$ norm (Chen et al., 2017; Dumont et al., 2021).
Theorem 1.5 — Stochastic Decay
Let $\rho \geq 8$. There exists $\varepsilon > 0$ such that for $u_0 \in L^2 \cap H^4(\mathbb{R})$ with norm at most $\varepsilon$, there exists a unique global solution whose trajectories satisfy $u \in \mathcal{C}(0, +\infty; H^1(\mathbb{R}))$ almost surely, and for all $t > 0$,
\[ \mathbb{E}\bigl(\|u(t)\|_{L^\infty}\bigr) \leq \frac{C(u_0)}{(1+t)^{1/6}}. \]Theorem 1.6 — Numerical Analysis (Midpoint Scheme)
Let $T > 0$ and $u_0 \in H^1$. For $N > 0$ sufficiently large and $\delta t = T/N$, for all $n \leq N$,
\[ \lim_{C \to +\infty} \mathbb{P}\!\left( \sup_{\delta t > 0} \frac{\|u(t_n) - u_n\|_{H^1}}{\delta t} \geq C \right) = 0. \]The introduction of a dissipative term profoundly alters the dynamics. In mathematical form:
Multiplying by $u$ and integrating in space gives $\frac{d}{dt}\|u\|^2 = -\varepsilon\langle \mathcal{A}u, u\rangle < 0$, so the $L^2$ norm decreases. The analysis rests on the functional space $\mathcal{H} := \{ u : \mathbb{R} \to \mathbb{R},\; \langle \mathcal{A}u, u\rangle < +\infty \}$ (Chehab et al., 2014).
Theorem 1.7 — Structure of the Damping Space
Let $s > 0$. Setting $\widehat{\mathcal{A}u} = \gamma(\xi)\,\hat{u}(\xi)$:
Memory operators of the form $\mathcal{A}u = -\int_0^\infty \mu(s)\,u_{xx}(t-s)\,ds$ allow the energy to be stabilised exponentially in time, under a reasonable decay assumption on the memory kernel $\mu$ (Dell’Oro et al., 2015, 2019; Dell’Oro and Mammeri, 2021).
Theorem 1.8 — Exponential Stabilisation
Let $\mu$ be such that there exists $\delta > 0$ satisfying $\mu'(s) + \delta\mu(s) \leq 0$ a.e. on $\mathbb{R}^+$. Then there exist $\omega > 0$ and a positive increasing function $\mathcal{Q}$, depending on $\mu$ and the domain, such that
\[ \|u(t)\|^2 \leq \mathcal{Q}(u_0)\,e^{-\omega t}. \]When the dispersion parameter $\delta$ and the dissipation parameter $\varepsilon$ both tend to zero, the equation formally approaches the hyperbolic conservation law $u_t + f(u)_x = 0$. For the equation $u_t + f(u)_x = \varepsilon u_{xx} - \delta u_{xxx}$, three cases arise:
Theorem 1.9 — Convergence to the Entropy Solution
Let $\varepsilon > 0$ and $f : \mathbb{R} \to \mathbb{R}$ convex with $f''(u) \leq C(1 + |u|^\beta)$. The sequence converges to the entropy solution in each of the following cases:
Section 2
The transport of sap in plant phloem obeys a hydrostatic flow mechanism first described by Münch in 1930. Sugars are synthesised and loaded into the phloem at the leaves, where their concentration is high; water is then drawn in by osmosis, creating a strong turgor pressure that drives the sap downwards towards the roots. The distribution of sucrose $(C)$ coupled with the phloem pressure $(P)$ is described by (Mammeri and Sellier, 2017; Sellier and Mammeri, 2019; Sellier et al., 2025):
Theorem 2.1 — Positivity of Solutions
Let $(P_0, C_0) \in W^{2,\infty} \times W^{1,\infty}$ with $P_0 \geq 0$ and $C_0 \geq 0$. Then for all $t > 0$ and almost every $x \in \Omega$,
\[ P(t,x) \geq 0 \qquad \text{and} \qquad C(t,x) \geq 0. \]The spread of fungal diseases in crops results from short- and long-range spore dispersal, in interaction with plant growth dynamics. We derived a mixed PDE-ODE model. Fick's law describes the dispersal of short-range spores $(U_S)$ and long-range spores $(U_L)$ (Mammeri, 2014):
Theorem 2.2 — Global Well-Posedness
Assume the jump conditions $[U_S] = [U_L] = 0$ and $[D_S\,\nabla U_S] = [U_L\,\nabla U_L] = 0$. Then the system is globally well-posed. Moreover, $(U_S, U_L, S, L, I, R, T)$ converges to the disease-free equilibrium $(0, 0, 0, 0, 0, R^\star, T^\star)$, where $R^\star(x) + T^\star(x) = k(x)$.
In the particular case of P. infestans, we showed the importance of considering interaction dynamics through longitudinal monitoring and of using mechanistic models describing pathogen development to analyse phenotypic data (Leclerc et al., 2023; Permanes et al., 2025). The recent rise of imaging-based phenotyping in plant pathology now enables non-destructive in vivo monitoring of infected tissue, and thus the study of the spatio-temporal dynamics of host–pathogen interactions at the organ scale. We consider the dynamics of a reaction-diffusion system on a deformable domain described by the level-set method, combining a top-down approach (image classifier) and a bottom-up approach (Fisher–KPP equation).
Pest insect populations are strongly influenced by the characteristics of the agricultural landscape. We derived a multi-scale model accounting for the mechanisms of resource perception and energy reserve management. The movement of adult pests $(A)$ across the landscape is described by:
where the juvenility index $Y$, derived from the pest life cycle, satisfies an integro-differential equation (Bourhis et al., 2015, 2017; Poggi et al., 2021).
Theorem 2.3 — Existence and Uniqueness
Let $Y_0 \in L^\infty_x(\Omega)$ and $A_0 \in L^\infty_x(\Omega) \times L^\infty_U([0,1])$, both non-negative. Then there exists a unique solution
\[ (A,Y) \in L^\infty_t\!\left([0,T];\, L^\infty_x(\Omega) \times L^\infty_U([0,1])\right) \times L^\infty_t\!\left([0,T];\, L^\infty_x(\Omega)\right). \]Section 3
Sorafenib was originally developed as an inhibitor of the oncogenic RAF kinase family, essential regulators of the RAS-RAF-MEK-ERK signalling pathway in hepatocellular carcinoma (HCC) cells. We explored the regulation of this cascade through mathematical modelling, using Michaelis–Menten kinetics:
We analysed the dynamic regulation of the core components of the RAF-MEK-ERK cascade in three human HCC cell lines exhibiting heterogeneous responses to sorafenib. Morris and Sobol sensitivity analyses were carried out, revealing an unexpected mode of action of sorafenib and opening new avenues for understanding mechanisms of therapeutic resistance (Saidak et al., 2017, 2017; Lottin et al. 2022).
We developed a reaction-diffusion model providing an explicit description of the physical environment, accounting for the mean daily movements of susceptible, exposed, and asymptomatic individuals (Arcede et al., 2022, 2020; Caga-anan et al, 2021; Lanoix et al., 2021; Mammeri, 2020 ).
Theorem 3.1 — Global Existence and Epidemic Threshold
Let $0 \leq S_0, E_0, I_{a,0}, I_{s,0}, R_0 \leq N_0$ be the initial data. Then there exists a unique global weak solution $(S, E, I_a, I_s, R) \in \bigl(L^\infty_t(\mathbb{R}_+; L^\infty(\Omega))\bigr)^5$, non-negative and bounded above by $N_0$.
Moreover, if the basic reproduction number satisfies
\[ \mathcal{R}_0 := \omega_0 \left( \frac{\beta_e}{\delta} + \frac{(1-p)\,\beta_a}{\gamma} + \frac{p\,\beta_s}{\gamma + \mu} \right) \frac{S_0}{N_0} > 1, \]then the disease $(E, I_a, I_s)$ initially grows exponentially.
We derived models for lithium-sulphur and lithium-oxide battery cells, supported by a multi-scale description of the composite electrode microstructure. At the macroscopic scale, the equations are derived from the conservation law for chemical species, written for the concentration $C_i$ of species $i$ as ( Thangavel et al., 2016; Yin et al., 2017; Gaya et al., 2018; Maiza et al., 2019 ):
where the electrochemical potential is given by the Nernst–Planck equation:
The originality of our approach lies in the fact that, at each iteration, we update the effective chemical diffusion, calculated at the microscopic scale using a kinetic Monte Carlo method. A finite-volume scheme with a mesh size of around one micron has been developed, which preserves the chemical species whilst providing access to the microscopic scale.