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Lorenzo Brandolese

Publications and source records attributed to Lorenzo Brandolese.

35 records · Page 2Linked to original sources
arXiv2016

Characterization of solutions to dissipative systems with sharp algebraic decay

We characterize the set of functions $u\_0\in L^2(R^n)$ such that the solution of the problem $u\_t=\mathcal{L}u$ in $R^n\times(0,\infty)$ starting from $u\_0$ satisfy upper and lower bounds of the form $c(1+t)^{-γ}\le \|u(t)\|\_2\le c'(1+t)^{-γ}$.Here $\mathcal{L}$ is in a large class of linear pseudo-differential operator with homogeneous symbol (including the Laplacian, the fractional Laplacian, etc.). Applications to nonlinear PDEs will be discussed: in particular our characterization provides necessary and sufficient conditions on $u\_0$ for a solution of the Navier--Stokes system to satisfy sharp upper-lower decay estimates as above.In doing so, we will revisit and improve the theory of \emph{decay characters} by C. Bjorland, C. Niche, and M.E. Schonbek, by getting advantage of the insight provided by the Littlewood--Paley analysis and the use of Besov spaces.

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arXiv2015

A Liouville theorem for the Degasperis-Procesi equation

We prove that the only global, strong, spatially periodic solution to the Degasperis-Procesi equation, vanishing at some point (t0, x0), is the identically zero solution. We also establish the analogue of such Liouville-type theorem for the Degasperis-Procesi equation with an additional dispersive term.

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arXiv2014

On permanent and breaking waves in hyperelastic rods and rings

We prove that the only global strong solution of the periodic rod equation vanishing in at least one point $(t_0,x_0)$ is the identically zero solution. Such conclusion holds provided the physical parameter $γ$ of the model (related to the finger deformation tensor) is outside some neighborhood of the origin and applies in particular for the Camassa--Holm equation, corresponding to $γ=1$. We also establish the analogue of this unique continuation result in the case of non-periodic solutions defined on the whole real line with vanishing boundary conditions at infinity. Our analysis relies on the application of new local-in-space blowup criteria and involves the computation of several best constants in convolution estimates and weighted Poincaré inequalities.

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arXiv2014

Blowup issues for a class of nonlinear dispersive wave equations

In this paper we consider the nonlinear dispersive wave equation on the real line, $u_t-u_{txx}+[f(u)]_x-[f(u)]_{xxx}+\bigl[g(u)+\frac{f''(u)}{2}u_x^2\bigr]_x=0$, that for appropriate choices of the functions $f$ and $g$ includes well known models, such as Dai's equation for the study of vibrations inside elastic rods or the Camassa--Holm equation modelling water wave propagation in shallow water. We establish a local-in-space blowup criterion (i.e., a criterion involving only the properties of the data $u_0$ in a neighbourhood of a single point) simplifying and extending earlier blowup criteria for this equation. Our arguments apply both to the finite and infinite energy case, yielding the finite time blowup of strong solutions with possibly different behavior as $x\to+\infty$ and $x\to-\infty$.

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arXiv2014

On the effect of external forces on incompressible fluid motions at large distances

We study incompressible Navier--Stokes flows in~$\R^d$ with small and well localized data and external force~$f$. We establish pointwise estimates for large~$|x|$ of the form \hbox{$c_t|x|^{-d}\le |u(x,t)|\le c'_t|x|^{-d}$}, where $c_t>0$ whenever $\int_0^t\!\!\int f(x,s)\,dx\,ds\not=\vec 0$. This sharply contrasts with the case of the Navier--Stokes equations without force, studied in [Brandolese, Vigneron, J. Math. Pures Appl. 88, 64--86 (2007)], where the spatial asymptotic behavior was $|u(x,t)|\simeq C_t|x|^{-d-1}$. In particular, this shows that external forces with non-zero mean, no matter how small and well localized (say, compactly supported in space-time), increase the velocity of fluid particles at {\it all times~$t$} and at \emph{at all points~$x$} in the far-field. As an application of our analysis on the pointwise behavior, we deduce sharp upper and lower bounds of weighted $L^p$-norms for strong solutions, extending the results obtained in [Bae, Brandolese, Jin, Asymptotic behavior for the Navier--Stokes equations with nonzero external forces, Nonlinear analysis, doi:10.1016/j.na.2008.10.074] for weak solutions, by considering here a larger (and in fact, optimal) class of weight functions.

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arXiv2013

Local-in-space criteria for blowup in shallow water and dispersive rod equations

We unify a few of the best known results on wave breaking for the Camassa--Holm equation (by R. Camassa, A. Constantin, J. Escher, L. Holm, J. Hyman and others) in a single theorem: a sufficient condition for the breakdown is that $u_0'+|u_0|$ is strictly negative in at least one point $x_0$ of the real line. Such blowup criterion looks more natural than the previous ones, as the condition on the initial data is purely local in the space variable. Our method relies on the introduction of two families of Lyapunov functions. Contrary to McKean's necessary and sufficient condition for blowup, our approach applies to other equations that are not integrable: we illustrate this fact by establishing new local-in-space blowup criteria for an equation modeling nonlinear dispersive waves in elastic rods.

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arXiv2012

Breakdown for the Camassa-Holm Equation Using Decay Criteria and Persistence in Weighted Spaces

We exhibit a sufficient condition in terms of decay at infinity of the initial data for the finite time blowup of strong solutions to the Camassa--Holm equation: a wave breaking will occur as soon as the initial data decay faster at infinity than the solitons. In the case of data decaying slower than solitons we provide persistence results for the solution in weighted $L^p$-spaces, for a large class of moderate weights. Explicit asymptotic profiles illustrate the optimality of these results.

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arXiv2010

Large time decay and growth for solutions of a viscous Boussinesq system

In this paper we analyze the decay and the growth for large time of weak and strong solutions to the three-dimensional viscous Boussinesq system. We show that generic solutions blow up as $t\to\infty$ in the sense that the energy and the $L^p$-norms of the velocity field grow to infinity for large time, for $1\le p<3$. In the case of strong solutions we provide sharp estimates both from above and from below and explicit asymptotic profiles. We also show that solutions arising from $(u_0,θ_0)$ with zero-mean for the initial temperature $θ_0$ have a special behavior as $|x|$ or $t$ tends to infinity: contrarily to the generic case, their energy dissipates to zero for large time.

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arXiv2009

On the parabolic-elliptic limit of the doubly parabolic Keller--Segel system modelling chemotaxis

We establish new convergence results, in strong topologies, for solutions of the parabolic-parabolic Keller--Segel system in the plane, to the corresponding solutions of the parabolic-elliptic model, as a physical parameter goes to zero. Our main tools are suitable space-time estimates, implying the global existence of slowly decaying (in general, nonintegrable) solutions for these models, under a natural smallness assumption.

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arXiv2008

Concentration-diffusion effects in viscous incompressible flows

Given a finite sequence of times $0<t_1<...<t_N$, we construct an example of a smooth solution of the free nonstationnary Navier--Stokes equations in $\R^d$, $d=2,3$, such that: (i) The velocity field $u(x,t)$ is spatially poorly localized at the beginning of the evolution but tends to concentrate until, as the time $t$ approaches $t_1$, it becomes well-localized. (ii) Then $u$ spreads out again after $t_1$, and such concentration-diffusion phenomena are later reproduced near the instants $t_2$, $t_3$, ...

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arXiv2008

Fine properties of self-similar solutions of the Navier-Stokes equations

We study the solutions of the nonstationary incompressible Navier--Stokes equations in $\R^d$, $d\ge2$, of self-similar form $u(x,t)=\frac{1}{\sqrt t}U\bigl(\frac{x}{\sqrt t}\bigr)$, obtained from small and homogeneous initial data $a(x)$. We construct an explicit asymptotic formula relating the self-similar profile $U(x)$ of the velocity field to its corresponding initial datum $a(x)$.

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arXiv2008

Far field asymptotics of solutions to convection equation with anomalous diffusion

The initial value problem for the conservation law $\partial_t u+(-Δ)^{α/2}u+\nabla \cdot f(u)=0$ is studied for $α\in (1,2)$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as $|x|\to \infty$ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity.

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arXiv2007

New Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System

We show that solutions $u(x,t)$ of the non-stationnary incompressible Navier--Stokes system in $\R^d$ ($d\geq2$) starting from mild decaying data $a$ behave as $|x|\to\infty$ as a potential field: u(x,t) = e^{tΔ}a(x) + γ_d\nabla_x(\sum_{h,k} \frac{δ_{h,k}|x|^2 - d x_h x_k}{d|x|^{d+2}} K_{h,k}(t))+\mathfrak{o}(\frac{1}{|x|^{d+1}}) where $γ_d$ is a constant and $K_{h,k}=\int_0^t(u_h| u_k)_{L^2}$ is the energy matrix of the flow. We deduce that, for well localized data, and for small $t$ and large enough $|x|$, c t |x|^{-(d+1)} \le |u(x,t)|\le c' t |x|^{-(d+1)}, where the lower bound holds on the complementary of a set of directions, of arbitrary small measure on $\mathbb{S}^{d-1}$. We also obtain new lower bounds for the large time decay of the weighted-$L^p$ norms, extending previous results of Schonbek, Miyakawa, Bae and Jin.

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arXiv2006

On the localization of the magnetic and the velocity fields in the equations of magnetohydrodynamics

We study the behavior at infinity, with respect to the space variable, of solutions to the magnetohydrodynamics equations in ${\bf R}^d$. We prove that if the initial magnetic field decays sufficiently fast, then the plasma flow behaves as a solution of the free nonstationnary Navier--Stokes equations when $|x|\to +\infty$, and that the magnetic field will govern the decay of the plasma, if it is poorly localized at the beginning of the evolution. Our main tools are boundedness criteria for convolution operators in weighted spaces.

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arXiv2006

Global existence versus blow up for some models of interacting particles

We study the global existence and space-time asymptotics of solutions for a class of nonlocal parabolic semilinear equations. Our models include the Nernst-Planck and the Debye-Hukel drift-diffusion systems as well as parabolic-elliptic systems of chemotaxis. In the case of a model of self-gravitating particles, we also give a result on the finite time blow up of solutions with localized and oscillating complex-valued initial data, using a method by S. Montgomery-Smith.

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arXiv2003

Space-time decay of Navier-Stokes flows invariant under rotations

We show that the solutions to the non-stationary Navier-Stokes equations in $R^d$, $d=2,3$ which are left invariant under the action of discrete subgroups of the orthogonal group $O(d)$ decay much faster as $|x|\to\infty$ or $ t \to\infty$ than in the generic case and we compute, for each subgroup, the precise decay rates in space-time of the velocity field.

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