IRIS Università degli Studi di Bolognahttps://cris.unibo.itIl sistema di repository digitale IRIS acquisisce, archivia, indicizza, conserva e rende accessibili prodotti digitali della ricerca.Wed, 25 Nov 2020 12:14:59 GMT2020-11-25T12:14:59Z10351Modulus of continuity and decay at infinity in evolution equations with real characteristicshttp://hdl.handle.net/11585/132223Titolo: Modulus of continuity and decay at infinity in evolution equations with real characteristics
Abstract: In the hyperbolic Cauchy problem, the well-posedness in Sobolev
spaces and the modulus of continuity of the coefficients are deeply connected.
This holds true in the more general framework of p-evolution equations with
real characteristics
where a sharp
scale of Hoelder continuity, with respect to the time variable has
been established.
We show that, for p>1, a lack of regularity in t can be compensated by a
decay as the space variable x tends to infinity
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/11585/1322232012-01-01T00:00:00ZSome remarks to Gevrey well-posedness for degenarate Schr\"odinger equationshttp://hdl.handle.net/11585/428367Titolo: Some remarks to Gevrey well-posedness for degenarate Schr\"odinger equations
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/11585/4283672015-01-01T00:00:00ZSchrödinger equations
of higher orderhttp://hdl.handle.net/11585/37957Titolo: Schrödinger equations
of higher order
Mon, 01 Jan 2007 00:00:00 GMThttp://hdl.handle.net/11585/379572007-01-01T00:00:00ZThe Cauchy problem for a class of Kovalevskian pseudo-differential operators.http://hdl.handle.net/11585/4460Titolo: The Cauchy problem for a class of Kovalevskian pseudo-differential operators.
Abstract: We prove the $H^{infty}$ well-posedness of the forward Cauchy problem for a $Psi$-do differential operator $P$ of order $mgeq 2$ with Log-Lipschitz continuous symbol in the time variable. The characteristic roots $lambda_k$ of $P$ are distinct and satisfy the necessary Lax-Mizohata condition Im$lambda_kgeq 0$. The Log-Lipschitz regularity has been tested as the optimal one for $H^{infty}$ well-posedness in the case of second order hyperbolic operators. Our main aim is to present a simple proof which
needs only a little of the basic calculus of standard $Psi$-do differential operators.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11585/44602004-01-01T00:00:00ZCoefficients with unbounded derivatives in hyperbolic equationshttp://hdl.handle.net/11585/4578Titolo: Coefficients with unbounded derivatives in hyperbolic equations
Abstract: We are concerned with the problem of determining the sharp regularity of the coefficients with respect to the time variable $t$ in order to have a well posed Cauchy problem in $H^infty$ or in Gevrey classes for linear or quasilinear hyperbolic operators of higher order.
We use and mix two different scales of regularity of global and local type: the modulus of H"older continuity and/or the behaviour with respect to $|t-t_1|^{-q}, qgeq 1,$ of the first derivative as $t$ tends to a point $t_1$. Both are ways to weaken the Lipschitz regularity.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11585/45782004-01-01T00:00:00ZOn the Nonlinear Cauchy ProblemOperator Theory: Advances and
Applicationshttp://hdl.handle.net/11585/4600Titolo: On the Nonlinear Cauchy ProblemOperator Theory: Advances and
Applications
Abstract: Our aim is to describe how to obtain, with the same procedure, several results of local existence, uniqueness and propagation of regularity for the solution of a quasilinear hyperbolic Cauchy Problem.
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/11585/46002005-01-01T00:00:00ZOperators of $p$-evolution with non regular coefficients in the time variablehttp://hdl.handle.net/11585/4580Titolo: Operators of $p$-evolution with non regular coefficients in the time variable
Abstract: We study the Cauchy problem for a class of $p$-evolution operators $P(t,x,D_t,D_x)$ in $[0,T]times {bf R}^n$, $p>1$, with less than ${mathcal C}^1$ coefficients with respect to the time variable.
According to Lipschitz, Log-Lipschitz or H"older regularity we find well posedness in Sobolev spaces or in Gevrey classes.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11585/45802004-01-01T00:00:00ZOptimal Well-posedness of The Cauchy Problem for Evolution Equations with $C^{N}$ Coefficientshttp://hdl.handle.net/11585/4589Titolo: Optimal Well-posedness of The Cauchy Problem for Evolution Equations with $C^{N}$ Coefficients
Abstract: We deal with the Cauchy problem for a $2$-evolution operator of Schr"odinger type with $C^N$ coefficients in the time variable, $N>2$. We find the Levi conditions for well-posedness in Gevrey classes of index $1/2 + N/4$ which is the best possible as we show by means of counterexamples.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11585/45892004-01-01T00:00:00ZStudies in Phase Space Analysis with Applications to PDEshttp://hdl.handle.net/11585/134366Titolo: Studies in Phase Space Analysis with Applications to PDEs
Abstract: This collection of original articles and surveys, emerging from a 2011 conference in Bertinoro, Italy, addresses recent advances in linear and nonlinear aspects of the theory of partial differential equations (PDEs). Phase space analysis methods, also known as microlocal analysis, have continued to yield striking results over the past years and are now one of the main tools of investigation of PDEs. Their role in many applications to physics, including quantum and spectral theory, is equally important. Key topics addressed in this volume include: *general theory of pseudodifferential operators *Hardy-type inequalities *linear and non-linear hyperbolic equations and systems *Schrödinger equations *water-wave equations *Euler-Poisson systems *Navier-Stokes equations *heat and parabolic equations Various levels of graduate students, along with researchers in PDEs and related fields, will find this book to be an excellent resource
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11585/1343662013-01-01T00:00:00Z$H^infty$ well-posedness for a 2-evolution Cauchy problem with complex coefficientshttp://hdl.handle.net/11585/134089Titolo: $H^infty$ well-posedness for a 2-evolution Cauchy problem with complex coefficients
Abstract: In this paper we deal with a 2-evolution Cauchy problem coming from the Euler–Bernoulli model for vibrating beams and plates. The leading coefficient, corresponding to the modulus of elasticity, is time-dependent and may vanish at t=0 . We prove a well-posedness result in the scale of Sobolev spaces using a C1 -approach, in this way we have H∞ well-posedness with an (at most) finite loss of regularity. We take special interest in the space and time-dependence of a complex coefficient of the extended principle part, related to the shear force, and in the assumptions we pose on that coefficient in order to get H∞ well-posedness.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/11585/1340892013-01-01T00:00:00Z