arXiv subjects
Publications and source records attributed to Daniele Del Santo.
We consider a parabolic equation whose coefficients are Log-Lipschitz continuous in $t$ and Lipschitz continuous in $x$. Combining a recent conditional stability result with a well posed variational problem, we reconstruct the initial condition of an unknown solution from a rough measurement at the final time.
In the present paper, we consider second order strictly hyperbolic linear operators of the form $Lu\,=\,\partial_t^2u\,-\,{\rm div}\big(A(t,x)\nabla u\big)$, for $(t,x)\in[0,T]\times\mathbb{R}^n$. We assume the coefficients of the matrix $A(t,x)$ to be smooth in time on $\,]0,T]\times\mathbb{R}^n$, but rapidly oscillating when $t\to 0^+$; they match instead minimal regularity assumptions (either Lipschitz or log-Lipschitz regularity conditions) with respect to the space variable. Correspondingly, we prove well-posedness results for the Cauchy problem related to $L$, either with no loss of derivatives (in the Lipschitz case) or with a finite loss of derivatives, which is linearly increasing in time (in the log-Lipschitz case).
We prove logarithmic conditional stability up to the final time for backward-parabolic operators whose coefficients are Log-Lipschitz continuous in $t$ and Lipschitz continuous in $x$. The result complements previous achievements of Del Santo and Prizzi (2009) and Del Santo, Jaeh and Prizzi (2015), concerning conditional stability (of a type intermediate between Hoelder and logarithmic), arbitrarily closed, but not up to the final time.
We prove some $C^\infty$ and Gevrey well-posedness results for hyperbolic equations whose coefficients lose regularity at one point.
In the present paper, we study a multiscale limit for the barotropic Navier-Stokes system with Coriolis and gravitational forces, for vanishing values of the Mach, Rossby and Froude numbers ($\rm Ma$, $\rm Ro$ and $\rm Fr$, respectively). The focus here is on the effects of gravity: albeit remaining in a low stratification regime ${\rm Ma}/{\rm Fr}\,\rightarrow\,0$, we consider scaling for the Froude number which go beyond the "critical" value $\rm Fr\,=\,\sqrt{\rm Ma}$. The rigorous derivation of suitable limiting systems for the various choices of the scaling is shown by means of a compensated compactness argument. Exploiting the precise structure of the gravitational force is the key to get the convergence.
We prove uniqueness for backward parabolic equations whose coefficients are Osgood continuous in time for $t>0$ but not at $t=0$.
In the present paper, we study the combined incompressible and fast rotation limits for the full Navier-Stokes-Fourier system with Coriolis, centrifugal and gravitational forces, in the regime of small Mach, Froude and Rossby numbers and for general ill-prepared initial data. We consider both the isotropic scaling (where all the numbers have the same order of magnitude) and the multi-scale case (where some effect is predominant with respect to the others). In the case when the Mach number is of higher order than the Rossby number, we prove that the limit dynamics is described by an incompressible Oberbeck-Boussinesq system, where the velocity field is horizontal (according to the Taylor-Proudman theorem), but vertical effects on the temperature equation are not negligible. Instead, when the Mach and Rossby numbers have the same order of magnitude, and in absence of the centrifugal force, we show convergence to a quasi-geostrophic equation for a stream function of the limit velocity field, coupled with a transport-diffusion equation for a new unknown, which links the target density and temperature profiles. The proof of the convergence is based on a compensated compactness argument. The key point is to identify some compactness properties hidden in the system of acoustic-Poincaré waves. Compared to previous results, our method enables first of all to treat the whole range of parameters in the multi-scale problem, and also to consider a low Froude number regime with the somehow critical choice $Fr=\sqrt{Ma}$, where $Ma$ is the Mach number. This allows us to capture some (low) stratification effects in the limit.
In this note we prove a well-posedness result, without loss of derivatives, for strictly hyperbolic wave operators having coefficients which are Zygmund-continuous in the time variable and Lipschitz-continuous in the space variables. The proof is based on Tarama's idea of introducing a lower order corrector in the energy, in order to produce special algebraic cancellations when computing its time derivative, combined with paradifferential calculus with parameters, in order to handle the low regularity of the coefficients with respect to $x$.
We prove continuous dependence on initial data for a backward parabolic operator whose leading coefficients are Osgodd continuous in time. This result fills the gap between uniqueness and continuity results obtained so far.
The interest of the scientific community for the existence, uniqueness and stability of solutions to PDE's is testified by the numerous works available in the literature. In particular, in some recent publications on the subject an inequality guaranteeing stability is shown to hold provided that the coefficients of the principal part of the differential operator are Log-Lipschitz continuous. Herein this result is improved along two directions. First, we describe how to construct an operator, whose coefficients in the principal part are not Log-Lipschitz continuous, for which the above mentioned inequality does not hold. Second, we show that the stability of the solution is guaranteed, in a suitable functional space, if the coefficients of the principal part are Osgood continuous.
The present paper concerns the well-posedness of the Cauchy problem for microlocally symmetrizable hyperbolic systems whose coefficients and symmetrizer are log-Lipschitz continuous, uniformly in time and space variables. For the global in space problem we establish energy estimates with finite loss of derivatives, which is linearly increasing in time. This implies well-posedness in $H^\infty$, if the coefficients enjoy enough smoothness in $x$. From this result, by standard arguments (i.e. extension and convexification) we deduce also local existence and uniqueness. A huge part of the analysis is devoted to give an appropriate sense to the Cauchy problem, which is not evident a priori in our setting, due to the very low regularity of coefficients and solutions.
In this paper we study the backward uniqueness for parabolic equations with non-Lipschitz coefficients in time and space. The result presented here improves an old uniqueness theorem due to Lions and Malgrange [Math. Scand. ${\bf 8}$ (1960), 277--286] and some more recent results of Del Santo and Prizzi [J. Math. Pures Appl. ${\bf 84}$ (2005), 471--491; Ann. Mat. Pura Appl., to appear].
In this paper we study the well-posedness of the Cauchy problem for first order hyperbolic systems with constant multiplicities and with low regularity coefficients depending just on the time variable. We consider Zygmund and log-Zygmund type assumptions, and we prove well-posedness in $H^\infty$ respectively without loss and with finite loss of derivatives. The key to obtain the results is the construction of a suitable symmetrizer for our system, which allows us to recover energy estimates (with or without loss) for the hyperbolic operator under consideration. This can be achievied, in contrast with the classical case of systems with smooth (say Lipschitz) coefficients, by adding one step in the diagonalization process, and building the symmetrizer up to the second order.
In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coefficients both in time and in space. In particular, this estimate implies the well-posedness for the related Cauchy problem. On the one hand, this result is quite surprising, because it allows to consider coefficients which are not Lipschitz continuous in time. On the other hand, it holds true only in the very special case of initial data in $H^{1/2}\times H^{-1/2}$. Paradifferential calculus with parameters is the main ingredient to the proof.
In this paper we will study the Cauchy problem for strictly hyperbolic operators with low regularity coefficients in any space dimension $N\geq1$. We will suppose the coefficients to be log-Zygmund continuous in time and log-Lipschitz continuous in space. Paradifferential calculus with parameters will be the main tool to get energy estimates in Sobolev spaces and these estimates will present a time-dependent loss of derivatives.
Using Bony's paramultiplication we improve a result obtained in in a previous paper for operators having coefficients non-Lipschitz-continuous with respect to $t$ but ${\mathcal C}^2$ with respect to $x$, showing that the same result is valid when ${\mathcal C}^2$ regularity is replaced by Lipschitz regularity in $x$.
We prove continuous dependence on Cauchy data for a backward parabolic operator whose coefficients are Log-Lipschitz continuous in time.
We give sharp regularity conditions, ensuring the backward uniquess property to a class of parabolic operators.