Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.

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2answers
416 views

Reference request: learn measure theory for PDEs

I am requesting some references to learn appropriate measure theory for PDEs. Specifically, I would like to learn all the measure theory necessary to understand well-posedness of PDEs with measure ...
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2answers
148 views

Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data

I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial ...
2
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2answers
124 views

Is the left regularizer for elliptic BVP a left inverse for the principal part?

Take a differential operator with elliptic symbol, consider just the principal part of the operator. Can one invert this principal part with some parametrix type construction (at least construct a ...
1
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0answers
186 views

local existence for a singular quasilinear parabolic equation

I'm considering the following type of PDE: $u_{t}=u_{xx}+u_{x}+u_{x}^2+u_{x}^3+\frac{u_{x}}{x(1-x)}+\left(\frac{u_{x}}{x(1-x)}\right)^3$ with periodic boundary conditions $u_{x}(0,t)=u_{x}(1,t)=0$, ...
1
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0answers
81 views

Kernel of perturbation of biharmonic operator

Suppose we have a linear fourth order operator defined on $\mathbb{R}^{2n}$ with $n\geq2$ of the form: $$\mathcal{L}(f)=\Delta^{2}f+\sum_{i,j=1}^{2n}P_{ij}(x)\partial_{i}\partial_{j}f$$ with ...
3
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0answers
108 views

Critical elliptic equation; kernel of linearized operator

I am interested in the critical equation $- \Delta w(x) = w(x)^p $ in $ R^N$ where $p=\frac{N+2}{N-2}$. After translation the solutions of this equation are all radial with maximum at the origin ...
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1answer
203 views

Density of certain functions in $C_c^\infty(0,T;V)$ in the space $W(0,T) \approx H^1(0,T;V)$?

EDIT: I need to think more about the question I want to ask given comments in the answer below. Please close the thread if required. I leave it undeleted because answer is useful. Let $V \subset H ...
4
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0answers
128 views

Asymptotic higher order derivative estimates for solutions of semi-linear parabolic PDEs

Question: Consider a semi-linear parabolic equation on a bounded domain $\Omega \subset \mathbb{R}^m$: $$ \frac{\partial f_t}{\partial t} = -\Delta f_t + Q(f_t, df_t), $$ with smooth initial data ...
3
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1answer
238 views

Two equivalent definitions of weak solution to parabolic PDE; don't understand proof

(Crossposted from MSE due to no replies) I'm trying to understand the proof that (c) implies (a) here in the following proposition (here, $\mathcal{V} = L^2(0,T;V)$). See the very last line in the ...
3
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1answer
220 views

Heat Equation on $[0,T] \times \mathbb{R}^n$

I'm currently looking for a complete proof of a classical result (very useful for viscosity methods) and surprisingly all the references I can get study the heat equation on bounded domain. Do you ...
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1answer
85 views

approximating functions pointwise [closed]

If we have in certain norm 1). $g_j(x) \rightarrow h(x), j\rightarrow\infty$ and 2). $ f_{ij}(x)\rightarrow g_j(x), i\rightarrow \infty, \forall j$ , then we can choose a subsequence ...
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3answers
414 views

PDEs involving measures; where to begin?

If I want to learn about existence of weak solutions to PDEs of the form $$u_t + Au = f$$ or $$Au = f$$ where $A$ is elliptic and $f$ is a measure, where do I start? I know the Galerkin method for ...
2
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0answers
95 views

Helmhotz decomposition and Regularity in Stokes equation

It is known that every function $f\in L^{q}(\Omega )^{n}$ can be uniquely decomposed as \begin{eqnarray*} \ f=f_{0}+\nabla Q, \text{ (Helmhotz decomposition)} \ \end{eqnarray*} with $f_{0}\in ...
11
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4answers
706 views

Einstein field equations in perspectives from PDE and functional analysis

The Einstein field equations have been subject of research in theoretical physics, and differential geometry, apparently with methods from classical analysis and geometry. In particular, solutions in ...
3
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0answers
146 views

How to use Galerkin method to obtain existence with spaces $V \subset H$ not compactly embedded

With $V \subset H \subset V'$ a Hilbert triple (separable spaces as well), let's consider $$u' + Au = f$$ in $L^2(0,T;V')$, where $A:V \to V'$ is bounded and linear. If $V \subset H$ is not compact, ...
3
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0answers
197 views

Tangential boundary regularity for optimal transport maps

I'm interested in (and a bit confused by) the following theorem of Caffarelli, proven in section $4$ of his paper Boundary regularity of maps with convex potentials II: Assume $u$ is a convex ...
2
votes
2answers
264 views

worst regularity of f ensuring u is locally Lipschitz for $\Delta u = f$

Assume $M$ is a Riemannian manifold, and $\Omega $ is a bounded domain. Consider the Poisson equation: $$\Delta u = f \qquad \text{with }u \in {W^{1,2}}$$ What is the worst regularity of $f$ which ...
3
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1answer
230 views

A closed extension of the Laplace operator with respect to the supremum norm

Let $X$ be a bounded connected open subset of the $n$-dimensional real euclidean space. Consider the Laplace operator defined on the space of infinitely differentiable functions with compact support ...
12
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0answers
403 views

Classes of (non-continuous) functions with the fixed point property

Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$. ...
2
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0answers
138 views

A microlocal representation for quantum operator dynamics

In Maciej Zworski's book $\textit{Semiclassical Analysis}$, an important step in proving $L^p$ bounds on quasimodes is deriving a microlocal oscillatory integral representation formula for families of ...
3
votes
2answers
316 views

Non symmetric coefficient matrix for elliptic PDE

Let $\Omega \subset \mathbb{R}^n$ be a domain and consider the PDE in divergence form $$ D_i(a_{i,j}D_ju)=0 \tag{1}$$ where $a_{i,j}(x)$ are measurable and satisfly the uniform ellipticity ...
0
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0answers
60 views

partial maximum principle for elliptic differential operators

Let $(M,g)$ be a closed, smooth Riemannian manifold. Let $P$ be a self-adjoint, elliptic differential operator defined on $C^\infty(M)$ with smooth coefficients. Suppose as well that the lowest ...
5
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1answer
520 views

Possible mistake in De Giorgi's paper on Holder's regularity

$\mu_{n-1}$ is the $n-1$ dimensional measure and $\operatorname{meas}$ is the $n$-dimensional one. $I(\varrho)$ is the ball of radius $\varrho$ around a fixed point $y$ in the domain $\Omega\subset ...
4
votes
1answer
278 views

Caccioppoli-Leray Inequality for De Giorgi's theorem proof

I am studying De Giorgi's proof of Holder continuity of solutions of elliptic equations with bounded measurable coefficients. This is the translation of the original paper De Giorgi paper At page ...
3
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1answer
124 views

On-diagonal to off-diagonal heat kernel lower bounds, Davies' argument

Theorem 3.3.4 in Davies' Heat Kernels and Spectral Theory begins with ``on-diagonal'' lower bounds for the heat kernel $K$ of $H$, (i.e. $K = e^{-Ht}$), where $H$ is a uniformly elliptic operator ...
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0answers
79 views

Parabolic PDE; uniform bound on approximations $u'_n$ in $L^2(0,T;V^*)$ without using orthogonal basis?

Let $V \subset H \subset V^*$ be a Gelfand triple, all Hilbert and separable spaces. I consider the PDE with weak form: find $u \in L^2(0,T;V)$ with $u' \in L^2(0,T;V')$ such that $$\langle u'(t), ...
1
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1answer
166 views

Galerkin approximations for parabolic PDE weak solution, getting a uniform bound

(As usual $V \subset H$ are separable Hilbert spaces) In a book I read this about existence of the solutions to parabolic PDEs: the approximate solution $u_n(t)$ solves the equation $$(u_n', ...
2
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1answer
98 views

Is $R^n$ stochastically complete for the heat kernel of a Schrödinger operator?

Suppose $V:\mathbb{R}^{n} \to \mathbb{R}$ is just a positive polynomial and $K_{t}(x,y)$ is the heat kernel of $H = -\Delta + V$. Then does it follow $$\int_{\mathbb{R}^{n}} K_{t}(x, \cdot)\,dy = ...
2
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0answers
73 views

Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces

Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
2
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0answers
134 views

Idea behind distributional solutions

I have a problem understanding the meaning of a distributional solution. Let me tell you the context the problem appeared: I read thorugh some papers by DiPerna and Lions concerning the Cauchy Problem ...
2
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1answer
265 views

Best approach to solve this PDE

I need to solve this Partial Differential Equation for $\lambda(x,y)$, $$\frac{\partial \lambda}{\partial x} + h(x,y)\frac{\partial \lambda}{\partial y} - \lambda \frac{\partial h}{\partial y} = 0$$ ...
2
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0answers
117 views

linear operator associated with semilinear elliptic pde

I am reading a paper where at some point they analyse the following linear operator: $$L_\lambda(\phi)= - \Delta \phi - C_\lambda(x) \phi$$ where $ C_\lambda(x)>0$ (smooth) in $ \Omega$ a bounded ...
1
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1answer
115 views

metric scaling for an inequality

I read a lemma 1.12 in Tobias H.Colding's paper "Ricci curvature and volume convergence"."Suppose that $Ri{c_{{M^n}}} \ge \left( {n - 1} \right)\Lambda {R^{ - 2}}$,p and $q \in M$ with d(p,q)>8R,and ...
1
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0answers
419 views

A variation of Poisson's equation in cylindrical coordinates

Our team of undergraduate physicists are familiar with finding numerical approximations to the following Poisson-like PDE central to our plasma research in a torus: $\nabla^2 V = \frac{f(V)}{R^2}$ ...
2
votes
0answers
98 views

Regularity of solution of nonlinear equation

Hi! Let $L$ be a linear elliptic operator of order $4$ with smooth and bounded coefficients on the ball $B_1$ of $R^{2n}$ and let $N\in C_{loc}^{0,\alpha}(R^{3})$. Let $f\in C^{0,\alpha}(B_1)$ ...
1
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1answer
156 views

Reference for: $C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$

If it is true, where may I find a reference/proof for: $C_c^\infty(0,T;H)$ dense in $L^2(0,T;H)$ where $H$ is a Hilbert space. Thanks
1
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0answers
66 views

Checking initial condition of PDE is satisfied in Galerkin method

I asked this question here: http://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. The ...
11
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1answer
402 views

Do circular pipes maximize flow rate?

Suppose that $U \subset \mathbb{R}^2$ is nonempty, open, connected and bounded. Consider a Poisseuille flow in the pipe $U \times \mathbb{R}$. That is: a time-independent incompressible flow of the ...
3
votes
1answer
305 views

First integrals of a 3D incompressible flow

Let $\Omega$ be an unbounded periodic smooth domain of $\mathbb{R}^3$. We are Given an incompressible vector field $q:\Omega\subset\mathbb{R}^3\rightarrow \mathbb{R}^3$ (i.e. $\nabla\cdot q\equiv 0$ ...
2
votes
0answers
519 views

Nonlinear PDE and Green functions

This is somewhat of a curiosity that can hide somewhat deeper. For a Green function of a nonlinear PDE I mean something like $$ \partial^2\phi+V(\phi)=\delta^D(x). $$ I do not know if a real ...
0
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0answers
94 views

Reference Search for a Functional Minimization Problem

Let $u(x) \ge 0$ be a non-negative, piecewise-differentiable function on the real line. Moreover, let $u(x)$ be integrable with fixed positive mass, that is $$M \equiv\int_{x=-\infty}^\infty u(x) ~ ...
7
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2answers
343 views

Fredholm alternative result for general elliptic system?

Now I have known that Fredholm alternative result is valid for the strong elliptic system. But I'm not sure that is it still valid for the general elliptic system, in which the second-order heading ...
2
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0answers
129 views

Examples of non-uniqueness in reaction-diffusion equations

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
4
votes
1answer
232 views

Please recommend some literature on the systematical theory of the elliptic systems!

Now I'm interested in the theory of elliptic systems, for example, both the linear and nonlinear case, the exsitence and regularity results, and is there a Fredholm alternative result for the linear ...
2
votes
0answers
116 views

Existence of solutions to a reaction-diffusion problem.

Consider the problem of finding a bounded classical solution $u:\mathbb{R}\times [0,T]\to\mathbb{R}$ (such that $u$ is continuous and $u_t$, $u_x$ and $u_{xx}$ exist and are continuous on ...
1
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0answers
82 views

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}} ...
1
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1answer
269 views

Regularity of the right hand side (the source term) in Evans-Krylov theory

A well-known theorem of Evans and Krylov states that in an equation of the form $F(D^2 u)=g$, provided that the right hand side and $u$ both have Lipschitz gradient, and that $F$ is concave or convex ...
2
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0answers
170 views

Similarity solutions of the imaginary time Benjamin--Ono equation

This problem arose in the course of a theoretical physics project. We seek (complex) solutions of the imaginary time Benjamin--Ono equation $$u_t-iu u_x-iu_{H,xx}=0$$ where $u_H(x,t)$ denotes the ...
3
votes
2answers
195 views

Elliptic Harnack inequality for 1D Schrodinger operator?

For a nonnegative polynomial $V: \mathbb{R} \to \mathbb{R}$, write $H = -\Delta + V$. I am wondering if there is an elliptic Harnack inequality for H. That is: There exist $C_{H} > 0$ and ...
3
votes
2answers
153 views

Boundedness of Solutions to $\Delta u = f u$ on $R^2$

Consider the Laplacian $\Delta = d/dx^2 + d/dy^2$ on $\mathbb{R}^2$. This is true: Let $f$ be a nonnegative function, not identically zero. Then any positive solution of $\Delta u = f u$ is ...