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Results tagged with ap.analysis-of-pdes
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user 12898
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
2
votes
Variational formulation of abstract Cauchy problem, possible?
Another possible reference is Theorem 3.1.7 in
Curtain, Ruth F.; Zwart, Hans, An introduction to infinite-dimensional linear systems theory, Texts in Applied Mathematics. 21. New York, NY: Springer-Ve …
- 4,729
5
votes
Accepted
Generation of strict contraction semigroups
Your conditions is for contraction semigroups equivalent to have uniform exponential stability, i.e., to have growth bound less than zero, see Proposition V.1.7. in
Engel, Klaus-Jochen; Nagel, Rainer, …
- 4,729
4
votes
Accepted
Pointwise convergence in functional calculus
There are easyer and more direct ways to prove it, but this follows immediatelly as a special case from the Trotter-Kato approximation theorem, see Theorem III.4.8 in
Engel, Klaus-Jochen; Nagel, Rai …
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2
votes
Accepted
Laplacian dissipative?
See the example of Section II.3.30 in
Engel, Klaus-Jochen; Nagel, Rainer, One-parameter semigroups for linear evolution equations, Graduate Texts in Mathematics. 194. Berlin: Springer. xxi, 586 p. (2 …
2
votes
Accepted
Estimate of semigroup with dual norm?
First, if $A$ is symmetric, then $X$ should be a Hilbert space, but I remain in a Banach space.
If
$$\|T(t)x\| \leq C\|A^{-1} x\|$$
holds for all $x\in X$, then using the substitution $y=A^{-1}x$, …
- 4,729
5
votes
Generator of an analytic semigroup of operators
As you write it, this is just a bounded perturbation of the sectorial operator
$$\begin{pmatrix} 0 & 0 & 0 \\ 0 & \Delta & 0 \\ 0 & 0 & 0 \end{pmatrix}.$$
It is quite standard that bounded perturbat …
- 4,729
4
votes
Accepted
Characterization of the interpolation space $(X,D(A^\alpha))_{\theta,p}$ with semigroup $A$ ...
Yes, there is a characterization like this. See Theorems 1-3 (p.182) in
Markus Haase, MR 2183483 A functional calculus description of real interpolation spaces for sectorial operators, Studia Math. 1 …
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3
votes
Airy's equation on $\mathbb R_-$
This is a very interesting question and I do not know the answer. I would start at something like
N. A. Larkin, Correct initial boundary value problems for dispersive equations, J. Math. Anal. Appl. …
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5
votes
Historical developement of analysis and partial differential equations (especially in the 20...
If you are interested in the history of Banach space geometry, then the monograph
Pietsch: History of Banach spaces and linear operators
is a good reference, even if it reflects at places the per …
4
votes
Real analysis on vector-valued spaces, $L^{p}(\mathbb{R}^N,E)$ ,$H^{s}(\mathbb{R}^N,E)$
You should brows the papers of Amann and his students for this type of results. You will find a lots of interesting results in the paper
http://user.math.uzh.ch/amann/files/cevvss.pdf
about embeddin …
- 4,729
3
votes
Accepted
PDEs on torus $\mathbb T$
It is a bit old, but I would check Bourgain first.
A more recent reference is (among many others) the preprint of Strunk.
- 4,729
2
votes
Accepted
$C_0$ semigroups on parameterized Banach spaces or moving domains
You will not find many things about such operator families in the literature, but what there is is for the non-autonomous case. See for example
M. Geissert & A. Lunardi Invariant Measures and Maxima …
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4
votes
When does $\{u\in H^1_0: \Delta_{\mu}u\in L^2\}=H_0^1\cap H^2$.
Renardy-Rogers, Theorem 9.53 states that $C^2$ boundary is sufficient. Example 9.52 before shows that this cannot be heavily relaxed.
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7
votes
Accepted
Abstract ODE; PDE; uniqueness of solution
No, this is not true. There is no backward uniqueness in general.
What you need is the theory of operator semigroups, and here is a simple example.
Consider the operator $Af=f'$ in the space $X=L …
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3
votes
Techniques to show existence for a PDE with dynamic boundary condition
I would take a look at the paper by Ciprian Gal and Martin Meyries about elliptic problems with nonlinear time dependent boundary conditions. They treat a similar equation like the one you have and us …
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