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.Sat, 04 Dec 2021 02:46:39 GMT2021-12-04T02:46:39Z10361Energy estimate and fundamental solution for degnerate hyperbolic Cauchy problemhttp://hdl.handle.net/11585/4612Titolo: Energy estimate and fundamental solution for degnerate hyperbolic Cauchy problem
Abstract: The aim of this paper is to give an uniform approach to different kinds of degenerate hyperbolic Cauchy
problems. We prove that a weakly hyperbolic equation, satisfying an intermediate condition between effective hyperbolicity and the $C^{infty}$ Levi condition, and a strictly hyperbolic equation with non-regular coefficients with respect to the time variable can be reduced to first order systems of the same type. For such a kind of systems, we prove an energy estimate in Sobolev spaces
(with a loss of derivatives) which gives the well-posedness of the Cauchy problem in $C^{infty}$. In the strictly hyperbolic case,
we also construct the fundamental solution and we describe the propagation of the space singularities of the solution which is influenced by the non-regularity of the coefficients with respect to the time variable.
Sat, 01 Jan 2005 00:00:00 GMThttp://hdl.handle.net/11585/46122005-01-01T00:00:00ZModulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problemhttp://hdl.handle.net/11585/4615Titolo: Modulus of continuity of the coefficients and loss of derivatives in the strictly hyperbolic Cauchy problem
Abstract: We deal with the Cauchy problem for a strictly hyperbolic second order operator with non-regular coefficients in the time variable. It is well-known that the problem
is well-posed in $L^{2}$ in case of Lipschitz continuous coefficients and that the
Log-Lipschitz continuity is the natural threshold for the well-posedness in Sobolev spaces which, in this case, holds with a loss of derivatives.
Here we prove that any intermediate modulus of continuity between the Lipschitz and the Log-Lipschitz one leads to an energy estimate with arbitrary small loss of derivatives.
We also provide counterexamples to show that the following classification
$$text{modulus of continuity $rightarrow$ loss of derivatives}$$
is sharp:
[begin{array}{l}
text{Lipschitz $rightarrow$ no loss}
text{intermediate $rightarrow$ arbitrary small loss}
text{Log-Lipschitz $rightarrow$ finite loss}
end{array}]
Sun, 01 Jan 2006 00:00:00 GMThttp://hdl.handle.net/11585/46152006-01-01T00:00:00ZStudies 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/11585/1343662013-01-01T00:00:00ZSchrödinger 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/11585/379572007-01-01T00:00:00ZModulus 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/11585/1322232012-01-01T00:00:00ZTHE CAUCHY PROBLEM FOR p-EVOLUTION EQUATIONShttp://hdl.handle.net/11585/89938Titolo: THE CAUCHY PROBLEM FOR p-EVOLUTION EQUATIONS
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/11585/899382010-01-01T00:00:00ZThe Cauchy problem for evolution equations with non-regular coefficientshttp://hdl.handle.net/11585/4585Titolo: The Cauchy problem for evolution equations with non-regular coefficients
Abstract: We deal with the Cauchy problem for a class of evolution operators of Schr"odinger type. We find the sharp regularity of the coefficients in the time variable for the well-posedness in Gevrey classes of the homogeneous problem, then we obtain the Levi conditions in the general case.
Thu, 01 Jan 2004 00:00:00 GMThttp://hdl.handle.net/11585/45852004-01-01T00:00:00ZThe 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/11585/44602004-01-01T00:00:00ZOperators 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/11585/45892004-01-01T00:00:00Z