Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
0
votes
0answers
35 views
A parabolic PDE with Lipschitz nonlinearity, how to obtain well-posedness?
Let $\Omega$ be a bounded smooth domain in $\mathbb{R}^n$ (or more generally a compact manifold). I'm interested in well-posedness (existence most importantly) of equations of the form
$$u_t(t) - ...
0
votes
0answers
68 views
Existence and uniqueness on this semi-linear parabolic PDE
I want to know whether the existence and uniqueness of a classical solution can be found about this question:
Find a classical solution $u : [0,T]\times [0,\infty] \rightarrow {\mathbb R}$, such ...
0
votes
0answers
23 views
When is Dirichlet solution from disk to Riemannian manifold Holder continuous near the boundary?
$D$ is the two-dimensional unit disk. $X$ is a compact Riemannian manifold.
Let$\phi \in W^{1,2}(D,X)$ and define
$$
W^{1,2}_{\phi}=\{v \in W^{1,2}(D,X):Trace(v)=Trace(\phi)\}
$$
Let $$
...
0
votes
1answer
83 views
Can we expect $\left\||f|^{2}f-|g|^{2}g\right\|\leq C ||f-g||$ in the Banch algebra $A(\mathbb T)$ ?
Let $f\in L^{1}(\mathbb T)$ and define the Fourier coefficient of $f$ : $\hat{f}(n)=\frac{1}{2\pi} \int _{-\pi}^{\pi} f(t) e^{-int} dt; (n\in \mathbb Z)$.Consider the space,
$$A(\mathbb T):= \{f\in ...
0
votes
1answer
32 views
uniqueness for Poisson equation in R^d with mildly regular data
I'm interested in Poisson's equation $-\Delta u=f$ set in the whole space $R^d$ (let's say $d\geq 3$ for simplicity) when $f$ has very little integrability, specifically $f\in L^{1+\varepsilon}$ for ...
4
votes
1answer
107 views
Simultaneous Orthogonal basis for $L^2(\mathbb{R}^n)$ and $H^1(\mathbb{R}^n)$
Given a smooth bounded set $U\subset \mathbb{R}^n$, there is a simultaneous orthogonal basis for $L^2(U)$ and $H^1_0(U)$ by the existence of eigenvectors to the Laplacian in a bounded domain, which ...
12
votes
3answers
569 views
Sobolev spaces and geometry
This is a very naive question, is there a way to geometrically understand Sobolev spaces without going through analysis and PDE's? To my knowledge, Sobolev spaces where created precisely to study ...
0
votes
0answers
72 views
How to use, $(|u|^{2}u - |v|^{2}v)(s)= (u-v)|u|^{2}(s)+ v(|u|^{2}-|v|^{2}) (s)$; to prove contraction in a Banach space $C([0,T]; M^{p,1})$?
(May be this is very basic question for MO)
(For details or this question you may see the paper page no. 9, MR2506839, Local well-posedness of nonlinear dispersive equations on modulation spaces; ...
0
votes
1answer
95 views
Integrability of the Poisson integral
Maybe this is rather obvious, but I'm stuck. Let's consider the Laplace equation in the upper half plane with boundary condition $g$, $i.e.$
$$
\Delta u(x,y)=0, u(x,0)=g(x).
$$
Then the solution is ...
1
vote
1answer
165 views
Are smooth functions dense in the space $\{u \in H^1(Q) \text{ with } \Delta_\Gamma u \in L^2(Q)\}$?
Define $$Q = \bigcup_{t \in (0,T)}\Gamma \times \{t\}$$ where $\Gamma$ is a compact (without boundary) hypersurface. Assume whatever smoothness is required.
Define $L^2(Q) := L^2(0,T;L^2(\Gamma))$ ...
1
vote
0answers
67 views
Regularity of weak solutions for a quasilinear problem
Theorem 6.2.7 in the book Nonlinear Analisis of Gasisnski and Papageorgiou states: If $u\in W^{1,p}(\Omega)\cap L^\infty(\Omega)$ and $\Delta_pu \in L^\infty(\Omega)$ then we have $u\in ...
3
votes
1answer
81 views
A question about solutions to Floer's equation which are asymptotic to a stationary point
Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation
$$ u(t,s): S^1 \times \mathbb{R} \to M$$
$$(du+X_H\otimes ...
0
votes
0answers
80 views
Localized convolution operators
Take our favorite differential operator $\Delta$ on $\mathbb{R}^3$. Module a constant, we can represent its inverse (in some sense) by a convolution operator, namely
$$\Delta^{-1}f=f*\frac{1}{|x|}.$$
...
0
votes
0answers
43 views
Mean value formula for high order elliptic operators
Weyl's lemma says any distributionally harmonic function $u\in L^1_{\text{loc}}$ is harmonic. One way to prove it is as follows.
Take radially symmetric compactly supported and smooth approximation ...
0
votes
0answers
48 views
Equating two systems of PDEs [migrated]
I'm trying to relate two sets of a pair of PDEs, but massively struggling! They should be equivalent up to a linear transformation. Any help would be wonderful!
First set:
$$\frac{\partial ...
0
votes
0answers
57 views
Invertibility of Paneitz operator on a compact manifold without boundary
Let $(M,g)$ be a Riemannian manifold of any even dimension that we assume compact and without boundary and $P_g$ the Paneitz operator in the metric $g$. The question is the following:
how to define a ...
3
votes
1answer
185 views
Bounding $\lVert{u}\rVert_{C^0([0,T];V)} \leq C\left(\lVert{u}\rVert_{L^2(0,T;V)} + \lVert{u'}\rVert_{L^2(0,T;H)}\right)$?
Define $W(X,Y) = \{ u \in L^2(0,T;X) \mid u' \in L^2(0,T;Y)\}$
where $u'$ is the usual weak derivative.
Let $V \subset H \subset V^*$ be a Hilbert triple. If $u \in W(V,V)$ (so $$u \in L^2(0,T;V) ...
3
votes
1answer
140 views
parabolic PDE with almost-monotone elliptic operator, existence results?
Are there any existence results for parabolic PDE of the type $$u_t - Au = f$$ in some Gelfand triple setting ($V \subset H \subset V^*$) with $A$ an operator that it is not quite monotone but close: ...
-2
votes
1answer
121 views
$L^{1}(\mathbb R) \cap L^{2}(\mathbb R) \cap C_{0}(\mathbb R)\subset H_{1}(\mathbb R)$?
Put, $C_{0} (\mathbb R)=\{f:\mathbb R \to \mathbb C: f \text { is continuous on} \ \mathbb R \ \text {and } \lim_{|x|\to \pm \infty}f(x)=0 \}$(= Continuous functions on $\mathbb R$ vanishing at ...
0
votes
1answer
70 views
harmonic maps from cone to $S^2$ locally lipschitz?
Are the harmonic maps from a 2-dimensional cone to $S^2$ locally lipschitz or Holder continuous?
0
votes
0answers
80 views
Solvable PDEs and their Green's functions
I have a class of PDEs of the form
$$
-\Box\phi(x)+\lambda\phi_0^2(x)\phi(x)=0
$$
with $\phi_0^2(x)=\sum_{n=-\infty}^\infty b_ne^{ip_n\cdot x}$. I know some exact solutions for them (see here and ...
2
votes
0answers
65 views
Is a certain set of periodic solutions of the 2D Navier-Stokes equations closed generically?
I would be interested to know if a certain set of periodic solutions for
the two-dimensional Navier-Stokes equations is closed generically.
Many similar (yet not identical) set-ups can be found in the ...
0
votes
0answers
82 views
Does the Laplace-Beltrami/surface gradient commute with orthogonal projection? (related to Galerkin method)
Let $\Gamma$ be a $C^k$ $(n-1)$-dimensional hypersurface embedded in $\mathbb{R}^n$. Let $H=L^2(\Gamma)$ and $V=H^1(\Gamma)$.
Suppose that $\{v_j\}$ is a basis for $H$ and $V$ (not necessarily ...
2
votes
1answer
250 views
How fast does the Heat equation with boundary condition $\frac{\partial u}{\partial \vec{n}}=u^2$ decay?
Consider the heat equation $\frac{\partial u}{\partial t}=\frac{1}{2}\Delta u$ in a bounded domain (say the interval [0,$\pi$]) with boundary condition $$\frac{\partial u}{\partial \vec{n}}=u^2$$
with ...
1
vote
1answer
42 views
Equivalence of distributional and viscosity solution in parabolic case?
In the answer to http://math.stackexchange.com/questions/166286/viscosity-solution-vs-weak-solution
H. Ishii, "On the equivalence of two notions of weak solutions, viscosity solutions and ...
2
votes
0answers
95 views
geometric irregularities in pde's
The following question is intended for a person more acquainted with the works of Yves Laurent.
see: http://archive.numdam.org/article/ASENS_1987_4_20_3_391_0.pdf (French)
...
0
votes
0answers
35 views
Well-posedness of a Stefan problem with Faedo-Galerkin approach
Given a domain $\Omega$ which is divided by $\Omega_1(t)$ and $\Omega_2(t)$ and the interface $\Gamma(t)$, does anyone have a reference to where a Stefan problem of the type
$$\frac{d}{dt}H(u) - ...
2
votes
0answers
100 views
How to show the identity $\int_0^T \int_{\Gamma(t)}f(s,t)\;dsdt = \int_S f(\sigma)(1+(\mathbf w \cdot \mathbf n)^2)^{-\frac{1}{2}}\;d\sigma$?
I am reading this paper.
Let $\Gamma(t)$ be a smooth closed connected oriented hypersurface for each $t \in [0,T]$. Define the set $$S = \bigcup_{t \in (0,T)}\Gamma(t) \times \{t\}.$$
On page 5 of ...
2
votes
0answers
124 views
Estimates on gradients of diffusion semigroups
Consider the Dirichlet or Neumann Laplacian on a manifold with boundary. Suppose we have some estimate of the form
$$||e^{t\Delta} f||_{L^p} \leq C(t)||f||_{L^q}$$ for some $p, q$. For a specific ...
0
votes
1answer
83 views
semiclassical principal symbol
What is the semiclassical principal symbol $\sigma_h$ of the operator $h^2\Delta-1$ (here $\Delta=-\sum_j\partial^{2}_{x_j}$)? $h^2\Delta-1$ is a second order semiclassical partial differential ...
1
vote
0answers
88 views
Almost a Green formula
Let $\Omega$ be the half-space $\mathbb{R}^{n-1}\times \{ x_n>0 \}$, let $v \in L^2(\Omega)$ and $\phi\in \mathcal{C}^{\infty}(\overline{\Omega})$ with compact support in $\overline{\Omega}$. What ...
1
vote
1answer
117 views
Finite propagation speed of second order operators
I was reading about the finite propagation speed of the wave equation on a Riemannian manifold. I was wondering if instead of the Laplace-Beltrami operator $\Delta$ we consider the equation ...
0
votes
2answers
127 views
A basic question about JL Lions' transformation of a Stefan problem
In J.L Lions' book "Quelques méthodes de résolution des problèmes aux limites non linéaires" (page 196), the author considers a two-phase problem with moving boundary separating the interface. The ...
1
vote
1answer
181 views
Reference request: Boundary behavior and quantitative lower bound for the principal eigenfunction of an elliptic PDE in a ball $B(r)$
Consider the elliptic eigenvalue problem
$$
\begin{cases}
\int_{B(r)} A(x) \nabla u \cdot \nabla \phi \, dx &= \ \ \frac{\lambda_1}{r^2}\int_{B(r)} u \phi \, dx \\
\qquad \qquad \qquad \quad ...
0
votes
1answer
80 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
$$u_t - \Delta u = f\quad\text{on $\Omega$}$$
$$\frac{\partial u}{\partial \nu} = 0\quad\text{on $\partial\Omega$}$$
...
1
vote
0answers
83 views
Weak periodic solution of parabolic PDE
Take
$$
u_t(t) + A(t)u(t) = f(t),
$$
$$
u(0) = u(T),
$$
where $A$ is an linear elliptic operator and the first equation is an equality in $L^2(0,T;V^*)$ for $V \subset H \subset V^*$ Hilbert triple. ...
4
votes
0answers
71 views
Linearizing and solving a nonlinear PDE numerically
Im trying to solve the following (transport & diffusion) nonlinear PDE numerically (via finite volume on a cuboid region. Some Material gets cooled down, s.t. in some areas the material becomes ...
-1
votes
1answer
155 views
Why can't I get global existence to linear PDE in this way? [closed]
For any $n > 0$, standard theory implies there is a unique $u_n \in L^2(0,n;V)$ with $u_n' \in L^2(0,n;V^*)$ such that
$$u_n' + Au_n = f\quad\text{as an equality in $L^2(0,n;V^*)$}$$
$$u_n(0) = ...
1
vote
1answer
147 views
Inequality in the Sobolev space $H^1$
I've found the following inequality
$$\int_{B_r}\vert u\vert^q\leq C \bigg(\int_{B_r}\vert\nabla u\vert^2\bigg)^{a}\bigg(\int_{B_r}\vert u\vert ...
1
vote
0answers
112 views
Half-wave group $e^{it\sqrt{-\Delta_g}}$ for large $t$
Consider the Laplace-Beltrami operator $\Delta_g$ on compact Riemannian manifold $(M,g)$, then $e^{it\sqrt{-\Delta_g}}f$ is the solution of the following Cauchy problem.
$$
...
2
votes
2answers
143 views
Perturbations of positive-definite self-adjoint operators
I was reading Kato's book on Perturbations of Linear Operators and have the following questions:
If we have a self-adjoint operator, what kinds of perturbations (other than relatively bounded ones) ...
3
votes
1answer
136 views
(Ref req) Schrödinger heat kernel is a weak solution of parabolic Schrödinger equation
If we have nonnegative $V \in L^1_{\textrm{loc}}(\mathbb{R}^{n})$, then the operator $H = -\Delta + V$ can be defined on $L^{2}(\mathbb{R}^{n})$ via quadratic form methods. This is done by, for ...
0
votes
0answers
67 views
Dirichlet integral of harmonic functions on manifold controlled by radius?
M is an n($\ge2$)-dim Riemannian manifold with Ricci curvature bounded below. $\Omega$ is a domain in M. $$\Delta u=0, x\in \Omega;u=f ,x\in \partial \Omega$$ f is Lipschitz.
Is there a constant c ...
0
votes
0answers
62 views
Harmonic/functional analysis question: Uniform bounds for $(1 - \varepsilon\Delta)^{-1}$ as $\varepsilon\to 0$?
Is there a space $X$, compactly embedded in $H^{-1}(R^2)$, such that the operators $(1 - \varepsilon \Delta)^{-1}$ are bounded from $L^1(R^2)$ into $X$, with operator norms that are in turn bounded by ...
5
votes
1answer
306 views
Do exist infinitely differentiable, compactly supported non zero solutions of the free Schrodinger equation?
I would like to get an answer for the following problem (and possibly be pointed to the relevant literature): given the one dimensional free Schrodinger equation $ i \, f_t + f_{xx}/2 = 0$ for the ...
0
votes
1answer
127 views
A Poincaré-type inequality with logarithmic function
For any function $f(x)$ we denote $\bar{f}:=\frac{1}{\Omega}\int_\Omega f(x)\,dx$.
Let $\Omega\subset \mathbb{R}^n$ be a bounded smooth domain and $u(x)> 0$ be a smooth function defined on ...
0
votes
1answer
162 views
Showing $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
Let $\Omega\subset\mathbb R^n$ be a bounded Lipschitz domain.
How does one prove that the inclusion $H^1(\partial\Omega) \subset H^{\frac 12}(\partial\Omega)$ is continuous?
I define $H^{\frac 1 ...
0
votes
2answers
68 views
References for well-posedness of weak solutions to Stefan problem
Can anyone recommend me any papers/texts that deal with the existence off weak solutions of the one-phase (or other) Stefan problem, or in general any sort of free boundary problem (for a beginner)?
...
1
vote
1answer
51 views
Does replacing 'local' with 'global' in the definition of viscosity sub(super)solution change the definition?
Let $F(x,u,Du,D^2u)=0$ for $x\in\Omega$ be our equation of interest. One way of defining a viscosity subsolution is this:
We say $u$ is a viscosity subsolution iff for any test function $\phi\in ...
1
vote
2answers
130 views
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} ...
