Discrete and Continuous Dynamical Systems

Stability analyses and error estimates are carried out for a number of commonly usednumerical schemes for the Allen-Cahn and Cahn-Hilliard equations. It is shown thatall the schemes we considered are either unconditionally energy stable, orconditionally energy stable with reasonable stability conditions in thesemi-discretized versions. Error estimates for selected schemes with aspectral-Galerkin approximation are also derived. The stability analyses and errorestimates are based on a weak formulation thus the results can be easily extended toother spatial discretizations, such as Galerkin finite element methods, which arebased on a weak formulation.

In this paper we study the existence of non-trivial solutions forequations driven by a non-local integrodifferentialoperator $\mathcal L_K$ with homogeneous Dirichlet boundaryconditions. More precisely, we consider the problem$$ \left\{\begin{array}{ll}\mathcal L_K u+\lambda u+f(x,u)=0 in Ω \\u=0 in \mathbb{R}^n \backslash Ω ,\end{array} \right.$$where $\lambda$ is a real parameter and the nonlinear term $f$satisfies superlinear and subcritical growth conditions at zero andat infinity. This equation has a variational nature, and so itssolutions can be found as critical points of the energy functional$\mathcal J_\lambda$ associated to the problem. Here we get suchcritical points using both the Mountain Pass Theorem and the LinkingTheorem, respectively when $\lambda<\lambda_1$ and $\lambda\geq\lambda_1$\,, where $\lambda_1$ denotes the first eigenvalue of theoperator $-\mathcal L_K$.As a particular case, we derive an existence theorem for thefollowing equation driven by the fractional Laplacian$$ \left\{\begin{array}{ll}(-\Delta)^s u-\lambda u=f(x,u) in Ω \\u=0 in \mathbb{R}^n \backslash Ω.\end{array} \right.$$Thus, the results presented here may be seen as the extensionof some classical nonlinear analysis theorems to the case of fractionaloperators.

To understand the impact of spatial heterogeneity of environment and movement of individuals on the persistence and extinction of a disease, a spatial SIS reaction-diffusion model is studied, with the focus on the existence, uniqueness and particularly the asymptotic profile of the steady-states. First, the basic reproduction number $\R_{0}$ is defined for this SIS PDE model. It is shown that if $\R_{0} 1$, the disease-free equilibrium is unstable and there is a unique endemic equilibrium. A domain is called high (low) risk if the average of the transmission rates is greater (less) than the average of the recovery rates. It is shown that the disease-free equilibrium is always unstable $(\R_{0} > 1)$ for high-risk domains. For low-risk domains, the disease-free equilibrium is stable $(\R_{0} < 1)$ if and only if infected individuals have mobility above a threshold value. The endemic equilibrium tends to a spatially inhomogeneous disease-free equilibrium as the mobility of susceptible individuals tends to zero. Surprisingly, the density of susceptibles for this limiting disease-free equilibrium, which is always positive on the subdomain where the transmission rate is less than the recovery rate, must also be positive at some (but not all) places where the transmission rates exceed the recovery rates.

The existence of a global attractor in the natural energy space is proved for the semilinear wave equation $u_{t t}+\beta u_t -\Delta u + f(u)=0$ on a bounded domain $\Omega\subset\mathbf R^n$ with Dirichlet boundary conditions. The nonlinear term $f$ is supposed to satisfy an exponential growth condition for $n=2$, and for $n\geq 3$ the growth condition $|f(u)|\leq c_0(|u|^{\gamma}+1)$, where $1\leq\gamma\leq\frac{n}{n-2}$. No Lipschitz condition on $f$ is assumed, leading to presumed nonuniqueness of solutions with given initial data. The asymptotic compactness of the corresponding generalized semiflow is proved using an auxiliary functional. The system is shown to possess Kneser's property, which implies the connectedness of the attractor. In the case $n\geq 3$ and $\gamma>\frac{n}{n-2}$ the existence of a global attractor is proved under the (unproved) assumption that every weak solution satisfies the energy equation.

We prove the global well-posedness of the viscousincompressible Boussinesq equations in two spatial dimensions for generalinitial data in $H^m$ with $m\ge 3$.It is known that when both the velocity and thedensity equations have finite positive viscosity,the Boussinesq system does not develop finite time singularities.We consider here the challenging case when viscosity enters only in the velocityequation, but there is no viscosity in the density equation.Using sharp and delicate energy estimates, we prove global existenceand strong regularity of this viscous Boussinesq system for generalinitial data in $H^m$ with $m \ge 3$.

The course aims at presenting an introduction to the subject of singularity formation in nonlinear evolution problems usually known as blowup. In short, we are interested in the situation where, starting from a smooth initial configuration, and after a first period of classical evolution, the solution (or in some cases its derivatives) becomes infinite in finite time due to the cumulative effect of the nonlinearities. We concentrate on problems involving differential equations of parabolic type, or systems of such equations. A first part of the course introduces the subject and discusses the classical questions addressed by the blow-up theory. We propose a list of main questions that extends and hopefully updates on the existing literature. We also introduce extinction problems as a parallel subject. In the main bulk of the paper we describe in some detail the developments in which we have been involved in recent years, like rates of growth and pattern formation before blow-up, the characterization of complete blow-up, the occurrence of instantaneous blow-up (i.e., immediately after the initial moment) and the construction of transient blow-up patterns (peaking solutions), as well as similar questions for extinction. In a final part we have tried to give an idea of interesting lines of current research. The survey concludes with an extensive list of references. Due to the varied and intense activity in the field both aspects are partial, and reflect necessarily the authors' tastes.

In this paper, we consider a non-autonomous stochasticLotka-Volterra competitive system $ dx_i (t) = x_i(t)$[($b_i(t)$-$\sum_{j=1}^{n} a_{ij}(t)x_j(t))$$dt$$+\sigma_i(t) d B_i(t)]$, where $B_i(t)$($i=1 ,\ 2,\cdots,\ n$) areindependent standard Brownian motions. Some dynamical properties arediscussed and the sufficient conditions for the existence of globalpositive solutions, stochastic permanence, extinction as well asglobal attractivity are obtained. In addition, the limit of theaverage in time of the sample paths of solutions is estimated.

Here we established the partial regularity of suitable weak solutions to the dynamical systems modelling the flow of liquid crystals. It is a natural generalization of an earlier work of Caffarelli-Kohn-Nirenberg on the Navier-Stokes system with some simplifications due to better estimates on the pressure term.

After some remarks on a possible zero-curvature formulation we first establish local well-posedness for the 2-component Camassa-Holm equation. Then precise blow-up scenarios for strong solutions to the system are derived. Finally we present two blow-up results for strong solutions to the system.

In this paper we study the evolutions of the interfaces betweengases and the vacuum for both inviscid and viscous one dimensionalisentropic gas motions. The local (in time) existence of solutionsfor both inviscid and viscous models with initial data containingvacuum states is proved and some singular properties on the freesurfaces separating the gas and the vacuum are obtained. It isfound that the Euler equations are better behaved near the vacuum thanthe compressible Navier-Stokes equations. The Navier-Stokesequations with viscosity depending on density are introduced, which is shown to be well-posed (at leastlocally) and yield the desired solutions near vacuum.

We prove the persistence of boundary smoothness of vortex patches for the quasi-geostrophic shallow-water (QGSW) equations. The QGSW equations generalize the Euler equations by including an additional parameter, the Rossby radius $ \varepsilon^{-1} $, which modifies the relationship between the streamfunction and the (potential) vorticity. Furthermore, for compactly supported initial data in the little Hölder space $ {c^{\gamma}}_c $, $ 0<\gamma<1 $, we prove that solutions of the QGSW equations converge in the $ {\mathcal{C}^\gamma} $ norm to the corresponding two-dimensional Euler solution as $ \varepsilon\to0 $, uniformly on a time interval independent of $ \varepsilon $.

We study the optimal regularizing effect for scalar balance laws with strictly convex fluxes and time-dependent nonlinear source terms. In a recent work, we proved that a suitable transversality condition between the source term and the degeneracy set of the flux restores full $BV_{\mathrm{loc}}$ regularity. In this note, we show that this condition is optimal. If the flux has polynomial degeneracy and the source vanishes along a degenerate state on a nontrivial time interval, then the full $BV_{\mathrm{loc}}$ regularizing effect cannot be expected in general. We construct an entropy solution showing that the fractional threshold $BV_{\mathrm{loc}}^{1/p}$, already known from the regularizing theory, cannot be improved. The same construction also yields the corresponding sharp obstruction in the Sobolev scale.

In this work we study homoclinic classes for some classes of flows, extending to flows certain results previously obtained, by other authors, for diffeomorphisms. More precisely, we prove that given a conservative $ C^2 $ flow $ \varphi_t:M\to M $ over a smooth compact Riemannian manifold $ M $, if both the stable and unstable homoclinic classes of a certain periodic hyperbolic orbit $ \gamma $ have positive Lebesgue measure, then their intersection is an ergodic component of the flow. We also obtain analogous results for invariant SRB measures.

This paper aims to give a refined wave breaking description of the Cauchy problem for the one-dimensional nonlinear shallow water equations under a non-cavitation condition which excludes compactly supported water depth. We construct smooth, bounded initial data with first spatial derivatives in $ H^4(\mathbb R) $ such that the corresponding solution develops gradient blow-up at a single spatial point in finite time, and we give a precise description of the blow-up profile together with a quantitative estimate of the wave breaking time in terms of the initial compressive slope.

Motivated by the introduction of the Zakharov-Kuznetsov equation as a higher dimensional generalization of the Korteweg-de Vries equation, in this paper we introduce the modified Zakharov-Kuznetsov (mZK) equation as a 2-dimensional generalization of the complex-valued modified Korteweg-de Vries equation. We initiate the mathematical analysis of the mZK equation on $ \mathbb{T}^2 $ by proving a local well-posedness result in Sobolev spaces and by establishing a failure of uniform continuity. We also note that the real-valued version of the mZK equation has physical significance.

Suppose that $ M $ is a closed manifold of dimension greater than two and $ r \ge 2 $. We show that there exists a $C^r$-diffeomorphism $f:M\to M$ with a wild affine blender-horseshoe $\Lambda_f$ such that $f$ is $C^r$-robustly strongly pluripotent, in the sense that after arbitrarily small perturbations a positive-Lebesgue-measure family of orbits can shadow a prescribed orbit in time average, for the dense subset $\Lambda_f^{(\mathrm{mj})}$ consisting of itineraries satisfying the majority condition, namely, those for which the symbol 0 occupies at least half of certain growing terminal blocks, but not for the whole horseshoe $\Lambda_f$.

Peter Lax's 1968 discovery that certain nonlinear wave equations possess operator pairs (Lax pairs) opened the theory of completely integrable systems, yielding exact solutions, infinite conservation hierarchies, and perfectly elastic soliton collisions. When a nonlinear dispersive equation is provably non-integrable (admitting no Lax pair and only finitely many conservation laws), what mathematical structure governs its dynamics?The K(m,m) compacton equations, $ u_t + (u^m)_x + (u^m)_{xxx} = 0 $ for $ m \geq 2 $, provide a testing ground: they support traveling waves with compact support, yet fail every known integrability test. We prove universal algebraic formulas determining the shape, width, and speed of every K(m,m) compacton. We develop a constraint-deficit mechanism explaining how the five independent conservation laws constrain emergent amplitudes and suppress the fundamental Fourier mode of the collision debris, while higher radiation modes remain unconstrained. The mechanism's conclusions rest on explicit hypotheses supported by numerical evidence. A phase-plane analysis reveals the compacton as a singular limit of smooth solitary waves through simultaneous regularity loss and support compactification. These results delineate intermediate territory between integrable order and generic nonlinear dynamics, organized by scaling universality and by the constraint deficit of a finite conservation structure.

In this paper mathematical models for the evolutionary conserved Notch–Delta pathway are developed and analyzed in order to better understand how two neighboring biological cells can become different. We pursue a structure-based stoichiometric type of approach, such that no specific reaction kinetics have to be defined. Only their dependencies on the relevant species participating in the model network are taken into account. Reaction networks and their related systems of ODEs are presented and analyzed with respect to their capacity for symmetry-induced bifurcations. The possibility to obtain a singular Jacobian is analyzed symbolically. This approach is valid for parameter-rich kinetics, where the parametrizations of the steady-state fluxes and of the first derivatives of the reaction rates evaluated at the steady state are independent. In this context, also with the help of abstract minimal models, we could mathematically identify some of the Notch pathway's features as being more relevant than others.

We study the Cauchy problem for the following Majda-Biello system in the case $ \alpha = 4 $, where the resonance effect is the most significant, on the real line.$ \left\{ \begin{array}{rcl} u_{t} + u_{xxx} & = & - v v_x, \\ v_{t} + \alpha v_{xxx} & = & - (uv)_{x}, \\(u, v)|_{t = 0} & = & (u_0, v_0) \in H^{s}(\mathbb{R}) \times H^{s}(\mathbb{R}), \end{array} \right. \quad x \in \mathbb{R}, \, t \in \mathbb{R}. $For Sobolev regularity $ s\in[\frac34, 1) $, we establish global well-posedness by refining the I-method. Previously, the critical index for local well-posedness was known to be $ \frac34 $, while global well-posedness was only obtained for $ s\geq 1 $. Our global well-posedness result bridges the gap and matches the threshold in the local theory. The main novelty of our approach is to introduce a pair of distinct $ I $-operators, tailored to the resonant structure of the Majda-Biello system with $ \alpha = 4 $. This dual-operator framework allows for pointwise control of the multipliers in the modified energies constructed via the multilinear correction technique. These modified energies are almost conserved and provide effective control over the Sobolev norm of the solution for all time. This new approach has potential applications to other coupled dispersive systems exhibiting strong resonant interactions.

In this paper, we accelerate Langevin dynamics with Gibbs measure $ \pi\propto \exp(-U) $ by adding a large drift that preserves the invariant measure. For warm-start initial data, we characterize the sharp asymptotic decay rate of the relative entropy and introduce asymptotically relaxation enhancing flows: sequences that achieve arbitrarily fast decay. We construct such flows on the torus by scaling cellular flows and pushing them forward via diffeomorphisms, and we extend the construction to the full space using a Lyapunov function method to control behavior at infinity without periodization, obtaining explicit finite energy flows that guarantee arbitrarily fast convergence under natural growth conditions on $ U $.