We prove short-time existence for the negative L^2-gradient flow of the p-elastic energy of curves via a minimising movement scheme. In order to account for the degeneracy caused by the energy’s invariance under curve reparametrisations, we write the evolving curves as approximate normal graphs over a fixed smooth curve. This enables us to establish short-time existence and give a lower bound on the solution’s lifetime that depends only on the W^{2,p}-Sobolev norm of the initial data.

We study the Cauchy problem for the Schrödinger-improved Boussinesq system in a two-dimensional domain. Under natural assumptions on the data without smallness, we prove the existence and uniqueness of global strong solutions. Moreover, we consider the vanishing “improvement” limit of global solutions as the coefficient of the linear term of the highest order in the equation of ion sound waves tends to zero. Under the same smallness assumption on the data as in the Zakharov case, solutions in the vanishing “improvement” limit are shown to satisfy the Zakharov system.

We study semigroups generated by general fractional Ornstein–Uhlenbeck operators acting on \(L^2({\mathbb {R}}^n)\). We characterize geometrically the partial Gevrey-type smoothing properties of these semigroups, and we sharply describe the blow-up of the associated seminorms for short times, generalizing the hypoelliptic and the quadratic cases. As a byproduct of this study, we establish partial subelliptic estimates enjoyed by fractional Ornstein–Uhlenbeck operators on the whole space by using interpolation theory.

We introduce a generalized concept of solutions for reaction–diffusion systems and prove their global existence. The only restriction on the reaction function beyond regularity, quasipositivity and mass control is special in that it merely controls the growth of cross-absorptive terms. The result covers nonlinear diffusion and does not rely on an entropy estimate.

In this paper we provide sufficient conditions which ensure that the nonlinear equation mathrm{d}y(t)=Ay(t)mathrm{d}t+sigma (y(t))mathrm{d}x(t), tin (0,T], with y(0)=psi and A being an unbounded operator, admits a unique mild solution such that y(t)in D(A) for any tin (0,T], and we compute the blow-up rate of the norm of y(t) as trightarrow 0^+. We stress that the regularity of y is independent of the smoothness of the initial datum psi , which in general does not belong to D(A). As a consequence we get an integral representation of the mild solution y which allows us to prove a chain rule formula for smooth functions of y.

The purpose of this paper is twofold. First, we use a classical method to establish Gaussian bounds of the fundamental matrix of a generalized parabolic Lamé system with only bounded and measurable coefficients. Second, we derive a maximal $$L^1$$ regularity result for the abstract Cauchy problem associated with a composite operator. In a concrete example, we also obtain maximal $$L^1$$ regularity for the Lamé system, from which it follows that the Lipschitz seminorm of the solutions to the Lamé system is globally $$L^1$$ -in-time integrable. As an application, we use a Lagrangian approach to prove a global-in-time well-posedness result for a viscous pressureless flow in a perturbation framework, but with possibly discontinuous densities. The method established in this paper might be a useful tool for studying many issues arising from viscous fluids with truly variable densities.

We consider the long-time behavior of solutions to a fourth-order nonlinear Schrödinger (NLS) equation with a derivative nonlinearity. By using the method of testing by wave packets, we construct an approximate solution and show that the solution for the fourth-order NLS has the same decay estimate for linear solutions. We prove that the self-similar solution is the leading part of the asymptotic behavior.

In this paper, we examine regularity and stability issues for two damped abstract elastic systems. The damping involves the average velocity and a fractional power $$\theta $$ , with $$\theta $$ in $$[-1,1]$$ , of the principal operator. The matrix operator defining the damping mechanism for the coupled system is degenerate. First, we prove that for $$\theta $$ in (1/2, 1], the underlying semigroup is not analytic, but is differentiable for $$\theta $$ in (0, 1); this is in sharp contrast with known results for a single similarly damped elastic system, where the semigroup is analytic for $$\theta $$ in [1/2, 1]; this shows that the degeneracy dominates the dynamics of the interacting systems, preventing analyticity in that range. Next, we show that for $$\theta $$ in (0, 1/2], the semigroup is of certain Gevrey classes. Finally, we show that the semigroup decays exponentially for $$\theta $$ in [0, 1], and polynomially for $$\theta $$ in $$[-1,0)$$ . To prove our results, we use the frequency domain method, which relies on resolvent estimates. Optimality of our resolvent estimates is also established. Two examples of application are provided.

Given a complex, elliptic coefficient function we investigate for which values of p the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on \(\mathrm {L}^p(\Omega )\). Additional properties like analyticity of the semigroup, \(\mathrm {H}^\infty \)-calculus and maximal regularity are also discussed. Finally, we prove a perturbation result for real coefficients that gives the whole range of p’s for small imaginary parts of the coefficients. Our results are based on the recent notion of p-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients.