All Questions
1,304 questions
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352
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Heat semigroup ultracontractive?
Let $g(x,t)= \frac{1}{(4 \pi t)^{\frac{n}{2}}}e^{\frac{-|x|^2}{4t}}$ be the heat kernel on $\mathbb{R}^{n}.$ Is the standard definition now to say that this heat-semigroup $T(t)(f):=g *f(.,t)$ is ...
- 11
1
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0
answers
158
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On the differentiability of a certain map from $ (0,\infty) $ to $ \Bbb{R} $
This problem arose from my study of energy-conservation for non-linear Schrödinger equations. Suppose that we have the following data:
$ u \in C^{1} \! \left( (0,\infty),{L^{2}}(\Bbb{R}^{n}) \right) $...
- 1,396
1
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0
answers
86
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Asymptotics of a elliptic pde when exponent gets large
I am interested in the following pde
$$ -\Delta w_p + \left( \frac{1}{p-2} +1 \right) \frac{ | \nabla w_p|^2}{w_p} + \epsilon(p) \left( \frac{1}{w_p} \right)^{(p-2)} = (p-2) w_p $$ in the unit ball $...
- 1,385
1
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0
answers
154
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One-parameter group of unitary operators and Core
Question : For what condition on $V$ (we can take it smooth, bounded, whatever necessary), the one-parameter unitary group $U(t)$ associated to the seladjoint operator $A=-\Delta+V$ on $\mathbb{R}^n$ ...
- 11
1
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205
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A Question about compactness of an embedding into $L^p$ spaces
Assume $ \Omega \subset \mathbb{R}^N$ is a smooth bounded domain. There is well known Hardy inequality that says
For any $ u \in W_0^{1,2}(\Omega) $, $N\geq3$ we have
$$ \Lambda \int_{\Omega} \frac{u^...
- 371
1
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0
answers
95
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Construct a PDE solution from a net of approximations
Consider $P$ a linear partial differential operator in $\Bbb R ^n$. Consider some boundary condition given in the generic form $C(u) = 0$, that guarantees a unique solution (if any) of $Pu = 0$.
Let $...
- 5,407
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answers
62
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Reference request - Compact embedding of intermediate space
Given two Banach spaces $X_0$ and $X_1$ with norms $\|\cdot\|_0$ and $\|\cdot\|_1$, respectively, such that $X_0\subset X_1$ and $X_0\hookrightarrow X_1$, i.e., $X_0$ is continuous embedded in $X_1$.
...
- 679
1
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0
answers
147
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Energy inequalities for Sobolev spaces of negative integer
I asked this question in mathematics stackexchange and couldn't get an answer.
Let $\phi\in H^{s}$ such that the following energy inequality is true:
$$\|\phi(t,\cdot)\|_s \le\int^t_0 C \| P\phi(t,\...
- 101
1
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0
answers
82
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boundedness of a sequence $ \in L^{\infty}(I,H^1(M))\cap Lip(I,L^2(M))$ implies that its temporal derivative is bounded as well [closed]
Hi I have the next claim which I would like to find a proof of it.
I have a sequence of functions $u_\epsilon(t,x) \in H^1(M)$ where $M$ is a compact manifold, and $u_\epsilon \in L^\infty(I,H^1(M))\...
- 1,594
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0
answers
70
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Equivalence of two definitions of weak solution (subtlety with null sets)
Consider
$$y_t - \Delta y = f$$
$$y(0) = y_0$$
with zero boundary condition. Let $a(t,.,.)$ be the bilinear form associated to $-\Delta$. We have two definitions of weak solutions:
We have $y \in L^...
- 266
1
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0
answers
191
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$L^2$ bound on solution of PDE in terms of $L^2$ norm of initial value
Let $u \in H^1((0,T)\times S)$ be the unique solution of
$$u_{tt} + \Delta u =0$$
$$u|_{t=0}= u_0$$
$$u|_{t=T}=0$$
where $u_0 \in H^{\frac 12}(S)$ and $S$ is some Euclidean hypersurface without ...
- 21
1
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0
answers
129
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persistence of regularity for nonlinear Klein-Gordon equation
I have been reading the paper on nonlinear Klein-Gordon equation(NLKG) for initial data in modulation space: For detail please see the paper "Klein-Gordon Equations on Modulation Spaces (2014)" (...
- 1,051
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0
answers
116
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Strong solution to parabolic equation without differentiability assumption on coefficient?
Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain
$$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$
$$u|_{\partial\Omega} = 0$$
where $a$ is real-valued and satisfies
$C_1 \leq a(r) \leq C_2$ ...
- 251
1
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0
answers
213
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Harmonic extension of $L^\infty$ function is in $L^\infty$?
Let $u \in H^{\frac 12}(\Omega)$ with $\int_\Omega u = 0$ and consider the solution $v \in H^1(C)$ where $C=\Omega \times (0,\infty)$ of
$$-\Delta v(x,y) = 0$$
$$\partial_\nu v = 0$$
$$v(x,0) = u(x)$$...
- 21
1
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0
answers
215
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Solution to a PDE with constant data - what is the fault in my proof? [closed]
Let $C=\Omega \times (0,\infty)$. We want to find a solution $v \in H^1(C)$ such that given $u \in H^{\frac 12}(\Omega)$,
$$\int_0^\infty\int_\Omega \nabla v \nabla \varphi + v_y\varphi_y = 0\quad\...
- 49
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0
answers
108
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A bilinear estimate in Lp space
Let $\varphi(D)$ be a Fourier multiplier with symbol $\varphi(\xi) = \xi/(1+|\xi|^2)$. It's easy to prove that
\begin{equation}
\|\varphi(D)u^2\|_{H^s(R)}\lesssim \|u\|^2_{H^s(R)} \quad (*)
\end{...
- 425
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0
answers
115
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Free Endpoint of Minimization Problem
Consider the following minimization problem $$\inf \left\{ \int\limits_{-\infty}^0 \left[ (\psi')^2 + m(y)(\psi - F)^2 \right]\; : \; \psi \in H^1(\left(-\infty,0\right]) \right\}$$ where $m(y) > 0$...
- 516
1
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0
answers
99
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decomposition of tempered distributions by entire analytic functions
Let $\phi$ be a $C^{\infty}$ function on $\mathbb R^{n}$ with
$$ \operatorname{supp} \phi \subset \{\xi \in \mathbb R^{n}: |\xi|\leq 2, \phi(\xi)=1~~\text{if}~|\xi|\leq 1\}$$
Let $j\in \mathbb N$ ...
- 1,051
1
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0
answers
136
views
A linear operator equation (PDE) with non-monotone term
I'm interested in the existence and/or uniqueness to the following problem. Let $V$ and $H$ be Hilbert spaces and $V \subset H \subset V^*$ form a Gelfand triple.
There is a linear operator $L:{D}(L) ...
- 11
1
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0
answers
182
views
Laplacian mapping on various function spaces
I have a question related to a certain elliptic operator on $R^N$ but I think i can clarify my confusion if I just consider the Laplacian $\Delta$ on the unit ball in $R^N$.
If $ 1 <p< \infty$...
- 539
1
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0
answers
218
views
Compact embedding
Let $\Omega$ be a domain in $\mathbb{R}^d$ (not necessarily bounded, no regularity assumption) and $K \subset \Omega$ a compact.
Is it true that the embedding $H^1_0(\Omega) \rightarrow L^2_K(\Omega)$...
- 185
1
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0
answers
123
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$L^p$ norm of solution to porous medium equation decreases in time: how to make formal calculation rigorous?
Let $u \in C^0([0,\infty);L^1(M)) \cap W^{1,1}_{\text{loc}}((0,\infty);L^1(M))$ with $u(t) \in H^1(M)$ for a.e. $t$ be the solution of the porous medium equation $\dot u = \Delta (u^m)$ on a compact (...
- 13
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0
answers
134
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Comparison principle using truncation for porous medium equation
For a porous medium equation (eg. $u_t - \Delta \Phi(u) = f$), is it possible to obtain a comparison principle for very weak solutions (eg. if $u_0 \geq 0$ and $f \geq 0$ then $u \geq 0$ a.e.) using ...
- 43
1
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1
answer
88
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Embedding to $L^\alpha(0,T;L^\beta(\Omega))$
Good day!
Let $V = H^1(\Omega)$, $\Omega \subset \mathbb R^3$.
Consider the space
$W = \{ y \in L^2(0,T;V) \colon dy/dt \in L^2(0,T;V') \}$.
It is well-known that $W \subset C([0,T];H)$ where $H = ...
- 243
1
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0
answers
134
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on high order Laplacian
Roughly speaking, we have good understanding of the solution to heat equation $u_t-\Delta u=0$, on bounded or unbounded domain. For example, we know the decay rate, we know it generates analytic ...
- 39
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0
answers
198
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Passing to the limit in a PDE (subsequence problems)
For $w \in L^2(0,T;H^1)$, consider the PDE
$$\int u'(t)v(t) + \int g(w(t))\nabla u(t) \nabla v(t) = \int f(t) v(t)\quad \forall v \in L^2(0,T;H^1)$$
where $u \in H^1(0,T;L^2)\cap L^2(0,T;H^1)$, and $f$...
- 155
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0
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164
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How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
- 1,051
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0
answers
103
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Regularity of weak solutions for a quasilinear problem
Theorem 6.2.7 in the book Nonlinear Analisis of Gasisnski and Papageorgiou states: If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\Delta_pu \in L^\infty(\Omega)$ then we have $u\in C^{1,\beta}_0(\...
- 11
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0
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168
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Estimates on gradients of diffusion semigroups
Consider the Dirichlet or Neumann Laplacian on a manifold with boundary. Suppose we have some estimate of the form
$$||e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for some $p, q$. For a specific ...
- 11
1
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1
answer
214
views
well-posedness of heat equation with Neumann BC and periodic data
On a domain $\Omega$ with $f \in L^2(0,T;H^{-1})$ such that $f(0) = f(T)$, consider
$$u_t - \Delta u = f\quad\text{on $\Omega$}$$
$$\frac{\partial u}{\partial \nu} = 0\quad\text{on $\partial\Omega$}$$
...
- 85
1
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2
answers
353
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How to show this integral on boundary of Lipschitz domain is finite?
Sorry for asking a basic question but this did not get answered on M.SE.
Let $\Omega \subset \mathbb{R}^n$ be a Lipschitz domain. How do I show rigorously that
$$\int_{\partial\Omega} \frac{1}{|y|^{...
1
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0
answers
122
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Checking initial condition of PDE is satisfied in Galerkin method
I asked this question here: https://math.stackexchange.com/questions/416885/checking-initial-condition-of-pde-is-satisfied-in-galerkin-method
But I did not receive the solution so I post it here.
...
- 21
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0
answers
103
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Generalized bilinear estimates
Hello. Let $ X^{s,b} $ be the Bourgain space generated by $ \tau - \xi^3 $. It is proved that, for $ s\in (-\frac{1}{2}, 0] $, we have
$$
\|(u^2)_x\| _{X^{s,b'-1}} \leq c \|u\|_{X^{s,b}} \|u\|_{X^{-...
- 425
1
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0
answers
76
views
h-oscillating function
I need help understanding the following condition:
$u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the flat torus, is ...
- 11
1
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0
answers
149
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(localized) L^2 norm of quasimode for Laplacian
Lately I've been thinking about the following distribution on the flat torus $\mathbb{T}^2$:
$u_k=\frac{1}{\sqrt{2\lfloor k^{0.99}\rfloor+1}}\sum_{|l|\leq k^{0.99}}e^{ikx}e^{ily}=\frac{1}{\sqrt{2\...
- 11
1
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0
answers
294
views
Galerkin method for existence for PDE with nonsymmetric bilinear form
Suppose we have a PDE
$$\langle u', v \rangle + a(u,v) = 0$$
where $a:V\times V \to \mathbb{R}$ is a bounded symmetric bilinear form, then if $u_0 \in V$ then $u \in L^2(0,T; V)$ with $u' \in L^2(0,T;...
- 113
1
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0
answers
316
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"Integration by parts" formula for functionals
We know that for a Hilbert triple $V \subset H \subset V^{'}$, if we have $u, v \in L^2(0,T;V)$ with $u',v' \in L^2(0,T;V')$
then
$$\frac{d}{dt}(u(t), v(t))_H = u'(t)(v(t)) + v'(t)(u(t))$$
where the $...
- 29
1
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0
answers
205
views
Looking for higher order Sobolev inequality
Hello,
On a compact (without boundary) Riemannian manifold (eg. some surface in $\mathbb{R}^n$), I'm looking for a result like
$$\lVert \nabla u\rVert_{L^2}^2 \leq \epsilon\lVert u\rVert_{H^1}^2+\...
- 29
1
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0
answers
125
views
base change for distributions
For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
- 2,649
1
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0
answers
202
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Weak solution of a certain pde with integral term
Let us consider the following pde on the domain $(0,T)\times(0,1)$
$
\dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0
$
with initial data $p(0,x)=p_{0}(x)$ and boundary data $...
- 11
0
votes
1
answer
507
views
Possible research directions in analysis? [closed]
I am an undergraduate student who loves basic mathematics in the analysis branch, but I have learned that some directions, for example, harmonic analysis, are already well developed and difficult to ...
- 101
0
votes
2
answers
197
views
Concerning the decay of the ground state of certain Schrodinger operators
Consider the Schrodinger operator in $n$ dimensions with a potential $V$, which grows rather quickly as $\mid x\mid$ tends to infinity, but with negative potential in a bounded region, for example, a ...
- 219
0
votes
1
answer
185
views
Can we approximate a Hölder pdf by higher-order Hölder pdf's?
$\newcommand{\RR}{\mathbb R}\newcommand{\NN}{\mathbb N}$
Let $\alpha \in (0, 1)$ and $j \in \NN$. We denote by $H^{j + \alpha} := H^{j + \alpha} ({\RR}^d)$ the space of real-valued functions $f$ on $\...
- 825
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votes
1
answer
414
views
What functions are equal to their symmetric decreasing rearrangement?
I am trying to understand the set
$$\mathcal{A}=\{f:f\text{ non-negative, measurable}, \text{vanishes at infinity and } f(x)=f^*(x)\}$$
where $f:\mathbb{R}^n\to \mathbb{R}$ is vanishing at infinity if ...
- 537
0
votes
1
answer
239
views
A proof for the existence of smooth solution of PDE in form $\Delta u=f(x, u)$ in Michael E. Taylor's book Partial Differential Equations III
This part is from page 107 in Michael E. Taylor's book Partial Differential Equations III.
In this part, we want a proof for the existence of smooth solution of the PDE
$\Delta u=f(x, u)$ on $U$ with ...
- 809
0
votes
1
answer
311
views
Bilinear Strichartz estimates for the Schrodinger equation
Let $d\geq 2$, $I\subset\mathbb{R}$ a time interval and $t_0\in I$. Let $u,v$ be solutions to the free Schrodinger equation on $\mathbb{R}^d$, i.e. $(i\partial_t+\Delta)u=(i\partial_t+\Delta)v=0.$ Let ...
- 943
0
votes
2
answers
388
views
Derivative of fractional Laplacian is the fractional Laplacian of the derivative
Is it true that $$\partial_x ((-\Delta)^s u(x)) = ((-\Delta)^s \partial_x
u(x))?$$
user104647
0
votes
1
answer
381
views
Converse of Lax-Milgram theorem [closed]
Suppose that $a(\cdot,\cdot):V \times V \rightarrow \mathbb{R}$ is a symmetric, continuous bilinear form defined on the Hilbert space V.
Assume that, for any continuous linear functional on $l \in V’...
- 23
0
votes
1
answer
152
views
Solution of Poisson equation vanishing at the boundary of any order
Let $f$ be a compactly supported function in $\Omega \subset \mathbb{R}^3$ and
$\Delta u=f$ in $\Omega$
such that $D^{\alpha}u=0$ on $\partial \Omega$ for every multi-index $\alpha$ with $|\alpha| \...
0
votes
1
answer
562
views
$H_0^1(\Omega)$ in the study of the Navier-Stokes Equations
This is cross-posted on MSE: https://math.stackexchange.com/q/1584519/9464
Let $\mathcal{V}$ be the space (without topology)
$$\displaystyle \mathcal{V}=\{u\in C_0^\infty(\Omega)\mid \nabla\cdot u=0\...
user14319