1,721,065 research outputs found
Hyperbolic problems with totally characteristic boundary
No full textAbstract
We study first-order symmetrizable hyperbolic
N
×
N
systems in a spacetime cylinder whose lateral boundary is totally characteristic. In local coordinates near the boundary at
x
=
0
, these systems take the form
∂
t
u
+
A
(
t
,
x
,
y
,
x
D
x
,
D
y
)
u
=
f
(
t
,
x
,
y
)
,
(
t
,
x
,
y
)
∈
(
0
,
T
)
×
R
+
×
R
d
,
where
A
(
t
,
x
,
y
,
x
D
x
,
D
y
)
is a first-order differential operator with coefficients smooth up to
x
=
0
and the derivative with respect to
x
appears in the combination
x
D
x
. No boundary conditions are required in such a situation and corresponding initial-boundary value problems are effectively Cauchy problems. We introduce a certain scale of Sobolev spaces with asymptotics and show that the Cauchy problem for the operator
∂
t
+
A
(
t
,
x
,
y
,
x
D
x
,
D
y
)
is well-posed in that scale. More specifically, solutions
u
exhibit formal asymptotic expansions of the form
u
(
t
,
x
,
y
)
∼
∑
(
p
,
k
)
(
-
1
)
k
k
!
x
-
p
log
k
x
u
pk
(
t
,
y
)
as
x
→
+
0
where
(
p
,
k
)
∈
C
×
N
0
and
ℜ
p
→
-
∞
as
|
p
|
→
∞
, provided that the right-hand side
f
and the initial data
u
|
t
=
0
admit asymptotic expansions as
x
→
+
0
of a similar form, with the singular exponents
p
and their multiplicities unchanged. In fact, the coefficients
u
pk
are, in general, not regular enough to write the terms appearing in the asymptotic expansions as tensor products. This circumstance requires an additional analysis of the function spaces. In addition, we demonstrate that the coefficients
u
pk
solve certain explicitly known first-order symmetrizable hyperbolic systems in the lateral boundary. Especially, it follows that the Cauchy problem for the operator
∂
t
+
A
(
t
,
x
,
y
,
x
D
x
,
D
y
)
is well-posed in the scale of standard Sobolev spaces
H
s
(
(
0
,
T
)
×
R
+
1
+
d
)
.National Natural Science Foundation of China http://dx.doi.org/10.13039/501100001809Georg-August-Universität Göttingen 50110000338
Energy estimates for weakly hyperbolic systems of the first order
No full textFor a class of first-order weakly hyperbolic pseudo-differential systems with finite time degeneracy, well-posedness of the Cauchy problem is proved in an adapted scale of Sobolev spaces. These Sobolev spaces are constructed in correspondence to the hyperbolic operator under consideration, making use of ideas from the theory of elliptic boundary value problems on manifolds with singularities. In addition, an upper bound for the loss of regularity that occurs when passing from the Cauchy data to the solutions is established. In many examples, this upper bound turns out to be sharp.</p
Discrete Morse flow for Ricci flow and Porous Media equation
No full textIn this paper, we study the discrete Morse flow for the Ricci flow on football, which is the 2-sphere with removed north and south poles and with the metric we have a weak approximated discrete Morse flow for the approximated Ricci flow and Porous media equation on any time intervals
Discrete Morse flow for Ricci flow and Porous Media equation
No full textIn this paper, we study the discrete Morse flow for the Ricci flow on football, which is the 2-sphere with removed north and south poles and with the metric we have a weak approximated discrete Morse flow for the approximated Ricci flow and Porous media equation on any time intervals
Liouville theorem for the nonlinear Poisson equation on manifolds
No full textIn this note, we study a Modica type gradient estimate for smooth solutions to general non-linear Poisson equation Delta(u) - f(u) = 0, in M-n, u : M-n -> R where (M, g) is a complete Riemannian manifold with bounded geometry and non-negative Ricci curvature and f is the derivative of the non-negative smooth function F(u) on R. Then we use this gradient estimate to conclude a Lionville theorem. (C) 2014 Elsevier Inc. All rights reserved
BLOWUP TIME AND BLOWUP MECHANISM OF SMALL DATA SOLUTIONS TO GENERAL 2-D QUASILINEAR WAVE EQUATIONS
No full textFor the 2-D quasilinear wave equation Sigma(2)(i,j)= 0 (gij) (del u) partial derivative(2)(i j)u = 0, whose coe ffi cients are independent of the solution u, the blowup result of small data solution has been established in [1, 2] when the null condition does not hold as well as a generic nondegenerate condition of initial data is assumed. In this paper, we are concerned with the more general 2-D quasilinear wave equation Sigma(2)(i,j) = 0 (gij) (u, del u) partial derivative(2)(ij) u = 0, whose coe ffi cients depend on u and del u simultaneously. When the fi rst weak null condition is not ful filled and a suitable nondegenerate condition of initial data is assumed, we shall show that the small data smooth solution u blows up in fi nite time, moreover, an explicit expression of lifespan and blowup mechanism are also established
The small data solutions of general 3-D quasilinear wave equations. II
No full textThis paper is a continuation of the work in [8], where the authors established the global existence of 3 smooth small data solutions to the general 3-D quasilinear wave equation [GRAPHICS] when the weak null condition holds. In the present paper, we show that the smooth small data solutions of equations [GRAPHCS] will blow up in finite time when the weak null condition does not hold and a generic nondegenerate condition on the initial data is satisfied, moreover, a precise blowup time is completely determined. Therefore, collecting the main results in this paper and [8], we have given a basically complete study on the blowup or global existence of small data solutions to the 3-D quasilinear wave equation [GRAPHICS] . (C) 2016 Elsevier Inc. All rights reserved
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