Questions tagged [semigroups-of-operators]
(Usually one-parameter) semigroups of linear operators and their applications to partial differential equations, stochastic processes such as Markov processes and other branches of mathematics.
231 questions
2 votes
1 answer
152 views
A cosine family that exponentially decays
Let $X$ be a Banach space. A strongly continuous family $(C(t))_{t\in\mathbb{R}}$ of bounded operators on $X$ is called a cosine family if $C(0) = I$, $C(t+s) + C(t-s) = 2C(t)C(s)$ for all $t,s\in\...
- 21
0 votes
0 answers
91 views
Evolution operators reference request
Recently, I've been studying non-autonomous parabolic problems and, as a consquence, the non-autonomous version of semigroups, evolution operators, has appeared. I'm very interested in how resolvent ...
- 145
2 votes
0 answers
80 views
Inhomogeneous version of $L^\infty_x L^2_t$ local smoothing estimate for general dispersive symbols?
Consider the group $e^{t (i \partial_x)^{2j+1}}$. We have a well-known homogeneous local smoothing estimate $$ \| \partial_x^j e^{t (i \partial_x)^{2j+1}} u_0 \|_{L^\infty_x L^2_t} \leq C \|u_0\|_{L^2}...
- 389
0 votes
0 answers
62 views
Uniqueness for inhomogeneous PDE semigroup theory
I’m reading nonlinear evolution equations by Song-Mu Zheng and I have some questions regarding the following statement THEOREM 2.4.1 ; COROLLARY 2.4.1 ; COROLLARY 2.4.2 ; COROLLARY 2.4.3 All are ...
- 181
1 vote
0 answers
90 views
How is the time decay of the heat semigroup in sectorial domains obtained?
I am having trouble understanding Lemma 20.10 from Souplet's book Superlinear Parabolic Problems, which provides a time decay estimate for the semigroup in a sectorial domain. I will first write the ...
- 687
0 votes
0 answers
52 views
Asking for an example for a classic estimate in the theory of Gaussian kernel and heat equations
Is there a concrete example that illustrates the righthand side of the inequality of functions on $\mathbb{R}^d$ $$ \|\mathrm{e}^{t\Delta}f\|_{\infty} \leq \frac{C}{t^{\frac{d}{2}}}\|f\|_{1} $$ can ...
- 19
0 votes
0 answers
59 views
Effective 'generator' of semigroup restricted to non-invariant subspace
Let $H: \mathcal{D} \rightarrow \mathcal{H}$ be a densely defined, self-adjoint, non-negative operator. Let $P: \mathcal{H} \rightarrow \mathcal{H} $ be an orthogonal projection onto a subspace. We ...
- 143
3 votes
1 answer
359 views
Eigenvalues of Laplace operator and Schrödinger operator
When reading the paper Control for Schrödinger operators on tori by N.Burq and M.Zworski (Math. Res. Lett., 19(2):309–324, 2012), an inequality confused me: Define the flat torus $\mathbb{T}^2=\mathbb{...
- 31
0 votes
1 answer
258 views
Is the evolution family self-adjoint?
$ \newcommand{\bR}{\mathbb{R}} \newcommand{\diff}{\mathop{}\!\mathrm{d}} \newcommand{\qtext}[1]{\quad\text{#1}} \newcommand{\qtextq}[1]{\quad\text{#1}\quad} $ I am reading Roland Schnaubelt's survey ...
- 1,163
2 votes
1 answer
385 views
Self-adjointness of generator and semigroup of an SDE
$ \newcommand{\bR}{\mathbb{R}} \newcommand{\bE}{\mathbb{E}} \newcommand{\bT}{\mathbb{T}} \newcommand{\bP}{\mathbb{P}} \newcommand{\bF}{\mathbb{F}} \newcommand{\cF}{\mathcal{F}} \newcommand{\eps}{\...
- 1,163
1 vote
0 answers
129 views
Studying flows on $L^2(\mathbb{R}^2)$ given by vector fields using unitary operators
Background: In the $xy$-plane (or a 2-sphere if you are concerned about compactness), we can think about three different types of flow given by vector fields. (1) pushes everything towards the origin, ...
- 1,133
4 votes
0 answers
129 views
Well-posedness for linear transport equations with fractional diffusion term
I have a rather applied problem where I consider an equation of the form $$ \partial_{t} u + V\cdot \nabla u = -(-\Delta)^s u, \quad (t,x) \in (0,\infty)\times \mathbf{R}^d, \quad u(0,x) = u_0. {}$$ ...
- 103
5 votes
1 answer
308 views
When does an Itô diffusion give a semigroup on $L^2$
I would like a reference for when an Itô diffusion generates a strongly continuous semigroup on $L^2(\mathbb{R}^n)$. I have a time-homogeneous Itô diffusion of the form $$dX_t=b(X_t)dt+\sigma(X_t)dB_t$...
- 323
2 votes
1 answer
126 views
There is some initial data such that the decay of the semigroup in it is faster than $t^{-n/2}$?
Lee and Ni show in their work Link Here that the heat semigroup $e^{t \Delta}u_0$ has decay as $t^{-\min \{a, n\} /2}$, $t \to \infty$ if $u_0 = C(1+|x|^2)^{a/2}$ if $a \neq n$. I'm trying to ...
- 687
1 vote
0 answers
56 views
Is it possible to manipulate heat kernel on H-type groups?
In Nathaniel Eldredge's work see here, he uses the explicit expression of the Heat Kernel on H-type groups (for example the Heisenberg group is an H-type group): $$p_t(x,z)= (2\pi )^{-m} (4 \pi )^{-n}...
- 687
3 votes
0 answers
78 views
Infinitesimal generators of random evolutions
Consider two state spaces $X$ and $Y$ and infinitesimal generators of Markov processes $(A_y)_{y\in Y}$ and $B$, on $X$ and $Y$ respectively. We assume that $A_y$ share the same domain $D(A)$, and ...
- 51
2 votes
1 answer
163 views
The contractivity of the time derivative of the heat semigroup in $L^p$ spaces
Let $M$ be a complete manifold. The heat semigroup $e^{-tL}$ is bounded on $L^p(M)$, for any $1 \leq p \leq \infty$; see this for instance. It seems that we can deduce the time derivative of the heat ...
- 147
2 votes
0 answers
84 views
Semigroup property in SPDEs
In fact, we know that a bounded linear operators on a Banach space $X$ satisfies the semigroup property, i.e. $$S(t+s)=S(t)S(s), \text{for every}\ t,s\geq 0.$$ However, in various literatures, I ...
- 57
1 vote
0 answers
93 views
The derivative of semigroup in the weak sense imply strong sense
Suppose $X$ is a Banach space, and $T(t)$ $t\ge0$ is a strongly continuous semigroup with generator $A$. Assume $\frac{T(t)-I}{t}x$ weakly converges to $y\in X$ when $t\to 0$, then I need to prove $x\...
- 1,051
2 votes
0 answers
156 views
interchange of integrals and semigroup without the semigroup being an integral operator
In Cazenave's book: BREZIS, HAIM.; CAZENAVE, T. Nonlinear evolution equations. IM-UFRJ, Rio, v. 1, p. 994, 1994. The following corollary appears The formula (1.5.2) is Duhamel formula: $$u(t) = T(t)u(...
- 687
2 votes
1 answer
217 views
Domain of the infinitesimal generator of a composition $C_0$-semigroup
In the paper [1] the following $C_0$-group is presented, $$ T(t)f(x) = f(e^{-t} x) , \quad x \in (0,\infty) \quad f \in E $$ where $E$ is an ($L^1,L^\infty$)-interpolation space. In mi case, I'm just ...
1 vote
0 answers
39 views
Analyticity of the semigroup generated by the sublaplacian on unimodular Lie group
Let $G$ a connected unimodular Lie group, endowed with Haar measure $X={X_1,\cdots,X_k}$ a Hörmander system of left-invariant vector fields. The sublaplacian $\Delta = - \sum_{i=1}^k X_i^2$ generates ...
- 687
6 votes
1 answer
187 views
Error estimates for projection onto the Wiener chaos expansion for stochastic Sobolev spaces (stochastic Rellich–Kondrachov theorem)
Let $n$ be a positive integer, $s\in \mathbb{R}$, $(\Omega,\mathcal{F},(\mathcal{F}_t)_{t\ge 0},\mathbb{P})$ be a filtered probability space whose filtration supports and is generated by an $n$-...
- 4,942
2 votes
1 answer
253 views
Koopman operators on $L^p(X)$
On spaces $L^p(X)$ the Koopman operator is defined as $T=T_\varphi: L^p(X) \rightarrow L^p(X)$, where $(X,\varphi)$ is a measure preserving system. As $\varphi$ is measure preserving we have that $T$ ...
1 vote
1 answer
182 views
Is the heat kernel of a manifold $p$-integrable?
If $M$ is a separable, oriented Riemannian manifold, without any other assumption on its geometry, and $h$ is its heat kernel, it is known that $h(t,x,\cdot)$ is both integrable, and square-integrable ...
- 5,487
1 vote
0 answers
168 views
Parabolic regularity for weak solution with $L^2$ data
I want to study the regularity of weak solutions $u\in C([0,T];L^2(\Omega))\cap H^1((0,T);L^2(\Omega))\cap L^2(0,T;H^1(\Omega))$ of the heat equation with Neumann boundary conditions: $$\begin{cases}\...
- 2,029
3 votes
1 answer
463 views
$L^{\infty}$ estimate for heat equation with $L^2$ initial data
Is there a way to show in a relatively simple manner that for a Lipschitz bounded, connected, open domain $\Omega\subset\mathbb{R}^N$ for any $f\in L^2(\Omega)$ the solution of the problem: $$\begin{...
- 2,029
0 votes
0 answers
74 views
Reference needed for powers of semi-group generators
Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$. For example, if the ...
- 190
3 votes
1 answer
250 views
Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard
In Appendix A of the paper Stabilization of wave equations on the torus with rough dampings https://msp.org/paa/2020/2-3/p04.xhtml or https://arxiv.org/abs/1801.00983 by Nicoulas Burq and Patrick ...
4 votes
1 answer
248 views
Reference request: Uniformly elliptic partial differential operator generates positivity preserving semigroup
I am looking for a reference of the following result: Let $\Omega\subset \mathbb{R}^n$ be be a bounded domain with smooth boundary. Let $$A = \sum_{i,j=1}^n \partial_i ( a_{ij} \partial_j) + \sum_{i=1}...
- 300
1 vote
1 answer
134 views
Bounded $C_0$-semigroups on barrelled spaces are equicontinuous
I have the following question: Let $X$ be a barrelled locally convex space (every absolutely convex, absorbing and closed set is a neighborhood of zero) and let $(T(t))_{t\geq0}$ be a $C_0$-semigroup, ...
- 129
3 votes
0 answers
309 views
A few questions on Feller processes
Update. Most of my questions have been answered in the comments. I am adding these answers to the post. There are at least three definitions of Feller semigroup and the corresponding processes: $C_0 \...
- 690
1 vote
0 answers
90 views
A question about semigroups in a Heisenberg group
I'm trying to understand if the regularity of solutions in Heisenberg groups works like in the Euclidean case. So far I haven't found any results, so I'm trying to check if the Regularity Theorems ...
- 687
3 votes
0 answers
155 views
Algebra core for generator of Dirichlet form
This is a question about the existence of a core $C$ for the generator $A$ of a regular Dirichlet form $\mathcal{E}$ having a carré du champ $\Gamma$, so that $C$ is an algebra with respect to ...
- 153
4 votes
0 answers
94 views
Reference/Help request for formula $[A,e^{-itB}]$ found in physics thread
I'm wondering if anyone has a rigorous reference or a proof of the formula (2) found in the main answer of this thread on the physics stack exchange. I want to use it but in the case where $A, B$ are ...
- 71
1 vote
0 answers
85 views
Existence for a nonlinear evolution equation with a monotone operator that is not maximal
We consider the nonlinear evolution equation $$ \dot{u}(t) + Bu(t) = 0, \quad u(0)=0 $$ with $$ A: \mathcal{C}(\Omega)\to \mathcal{M}(\Omega),\; p \mapsto \arg\min_{\mu\in\partial\chi_{\{||\...
- 181
1 vote
0 answers
121 views
Commutator of self-adjoint operators and $C^1$-type formula
Let $\mathcal{H}$ be a (complex) Hilbert space. Let $H$ be a self-adjoint operator on $\mathcal{H}$ with dense domain $\mathcal{D}(H) \subset \mathcal{H}$, generating the unitary one-parameter ...
- 71
3 votes
2 answers
234 views
Lumer-Phillips-type theorem for non-autonomous evolutions
The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
- 300
0 votes
0 answers
181 views
Convergence of Solutions of Integral Equations with Weakly Converging Forcing Terms
Let $\Omega$ be a bounded interval of $\mathbb{R}$ and let $y\in L^\infty(\Omega \times (0,T))$ be a mild solution of the integral equation $$ y(\cdot,t)=S(t) y_0+\int_0^t S(t-s) \left[u(\cdot,s)y(...
- 55
4 votes
0 answers
431 views
Regularity up to the boundary of solutions of the heat equation
Given the heat problem: $$\begin{cases} \frac{d}{dt}u(x,t)=\Delta u(x,t) & \forall (x,t)\in \Omega\times(0,T) \\ u(x,0)=u_0(x) & \forall x\in\Omega \\ u(x,t)=0 & \forall x\in\partial\...
- 91
4 votes
1 answer
206 views
approximation of a Feller semi-group with the infinitesimal generator
Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator. If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$. Is this formula always ...
- 315
1 vote
0 answers
78 views
Solution to $u_t = A(t)u + f(t)$ on bounded domain
I am dealing with the problem \begin{align}u_t &= \nabla \cdot (a(x,t) \nabla u) + f(x,t) &\text{ on } \Omega \times (0,T)\\ \partial_{\nu} u &= 0 &\text{ on } \partial \Omega \...
- 11
3 votes
1 answer
206 views
On the Fractional Laplace-Beltrami operator
I would appreciate it if a reference could be given for the following claim. Let $g$ be a Riemannian metric on $\mathbb R^n$, $n\geq 2$ that is equal to the Euclidean metric outside some compact set. ...
- 4,189
3 votes
1 answer
234 views
Looking for an electronic copy of Kato's old paper
I would like to know if anyone has an electronic copy of the following paper: MR0279626 (43 #5347) Kato, Tosio Linear evolution equations of "hyperbolic'' type. J. Fac. Sci. Univ. Tokyo Sect. I ...
- 519
1 vote
0 answers
99 views
Continuity in the uniform operator topology of a map
I have a question concerning the continuity for $t>0$ in the uniform operator topology $L(X)$ of the following map: $$t\mapsto A^\alpha R(t)$$ where A is the infinitesimal generator of an analytic ...
- 31
2 votes
0 answers
93 views
Are there results that relate the restriction of the heat semigroup in $\mathbb{R}^n$ and the semigroup in a domain?
I'm thinking about the following situation:0 suppose that $$ S_{\Omega}(t)f = \int_{\Omega} K_{\Omega}(x,y,t)f(y)dy $$ where $K_\Omega(x,y ,t)$ is the Dirichlet heat kernel in the domain $\Omega$. It ...
- 687
3 votes
0 answers
171 views
Extrapolated Integral operator (compactness)
I am studying the compactness of some convolution operators. Let the convolution with extrapolation $$ \Gamma: X\longrightarrow X; x\mapsto\int_0^t T_{-1}(t-s)B(s)x\mathrm{d}s. $$ Here $T(\cdot)$ is a ...
- 299
0 votes
0 answers
144 views
Characterization of the adjoint of a $C_0$-Semigoup infinitesimal generator
I am looking for characterizations of the adjoint operator of the infinitesimal generator of a $C_0$-semigoup in Hilbert spaces. All I could find in the literature is that if $(A, D(A))$ is the ...
- 101
2 votes
1 answer
435 views
Periodic solution for linear parabolic equation - existence, regularity
I am interested in proving the existence and regularity of solution to the following problem: $$\begin{cases} \dfrac{\partial y}{\partial t}(t,x)-\Delta y(t,x)+c(t,x)y(t,x)=f(t,x), & (t,x)\in (0,T)...
- 2,029
4 votes
1 answer
509 views
Strong positivity of Neumann Laplacian
There are many places in the literature where the positivity of some semigroups is treated. However I did not know anyone which states and proves the strong positivity even for the basic semigroups ...
- 2,029