A note on convergence of the solutions of Benjamin–Bona–Mahony type equations
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Cited by 4 publications
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We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_{\varepsilon ,\delta } +\mathrm {div} {\mathfrak f}_{\varepsilon ,\delta }(\mathbf{x}, u_{\varepsilon ,\delta })=\varepsilon \Delta u_{\varepsilon ,\delta }+\delta (\varepsilon ) \partial _t \Delta u_{\varepsilon ,\delta }, \ \ \mathbf{x} \in M, \ \ t\ge 0\\ u|_{t=0}=u_0(\mathbf{x}). \end{array}\right. } \end{aligned}$$ ∂ t u ε , δ + div f ε , δ ( x , u ε , δ ) = ε Δ u ε , δ + δ ( ε ) ∂ t Δ u ε , δ , x ∈ M , t ≥ 0 u | t = 0 = u 0 ( x ) . Here, $${{\mathfrak {f}}}_{\varepsilon ,\delta }$$ f ε , δ and $$u_0$$ u 0 are smooth functions while $$\varepsilon $$ ε and $$\delta =\delta (\varepsilon )$$ δ = δ ( ε ) are fixed constants. Assuming $${{\mathfrak {f}}}_{\varepsilon ,\delta } \rightarrow {{\mathfrak {f}}}\in L^p( {\mathbb {R}}^d\times {\mathbb {R}};{\mathbb {R}}^d)$$ f ε , δ → f ∈ L p ( R d × R ; R d ) for some $$1<p<\infty $$ 1 < p < ∞ , strongly as $$\varepsilon \rightarrow 0$$ ε → 0 , we prove that, under an appropriate relationship between $$\varepsilon $$ ε and $$\delta (\varepsilon )$$ δ ( ε ) depending on the regularity of the flux $${{\mathfrak {f}}}$$ f , the sequence of solutions $$(u_{\varepsilon ,\delta })$$ ( u ε , δ ) strongly converges in $$L^1_{loc}({\mathbb {R}}^+\times {\mathbb {R}}^d)$$ L loc 1 ( R + × R d ) toward a solution to the conservation law $$\begin{aligned} \partial _t u +\mathrm {div} {{\mathfrak {f}}}(\mathbf{x}, u)=0. \end{aligned}$$ ∂ t u + div f ( x , u ) = 0 . The main tools employed in the proof are the Leray–Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.
We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the dynamics capillarity equation $$\begin{aligned} {\left\{ \begin{array}{ll} \partial _t u_{\varepsilon ,\delta } +\mathrm {div} {\mathfrak f}_{\varepsilon ,\delta }(\mathbf{x}, u_{\varepsilon ,\delta })=\varepsilon \Delta u_{\varepsilon ,\delta }+\delta (\varepsilon ) \partial _t \Delta u_{\varepsilon ,\delta }, \ \ \mathbf{x} \in M, \ \ t\ge 0\\ u|_{t=0}=u_0(\mathbf{x}). \end{array}\right. } \end{aligned}$$ ∂ t u ε , δ + div f ε , δ ( x , u ε , δ ) = ε Δ u ε , δ + δ ( ε ) ∂ t Δ u ε , δ , x ∈ M , t ≥ 0 u | t = 0 = u 0 ( x ) . Here, $${{\mathfrak {f}}}_{\varepsilon ,\delta }$$ f ε , δ and $$u_0$$ u 0 are smooth functions while $$\varepsilon $$ ε and $$\delta =\delta (\varepsilon )$$ δ = δ ( ε ) are fixed constants. Assuming $${{\mathfrak {f}}}_{\varepsilon ,\delta } \rightarrow {{\mathfrak {f}}}\in L^p( {\mathbb {R}}^d\times {\mathbb {R}};{\mathbb {R}}^d)$$ f ε , δ → f ∈ L p ( R d × R ; R d ) for some $$1<p<\infty $$ 1 < p < ∞ , strongly as $$\varepsilon \rightarrow 0$$ ε → 0 , we prove that, under an appropriate relationship between $$\varepsilon $$ ε and $$\delta (\varepsilon )$$ δ ( ε ) depending on the regularity of the flux $${{\mathfrak {f}}}$$ f , the sequence of solutions $$(u_{\varepsilon ,\delta })$$ ( u ε , δ ) strongly converges in $$L^1_{loc}({\mathbb {R}}^+\times {\mathbb {R}}^d)$$ L loc 1 ( R + × R d ) toward a solution to the conservation law $$\begin{aligned} \partial _t u +\mathrm {div} {{\mathfrak {f}}}(\mathbf{x}, u)=0. \end{aligned}$$ ∂ t u + div f ( x , u ) = 0 . The main tools employed in the proof are the Leray–Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second. These results have the potential to generate a stable semigroup of solutions to the underlying scalar conservation laws different from the Kruzhkov entropy solutions concept.
A note on the convergence of the solution of the Novikov equation
We consider the Novikov and Camass-Holm equations, which contain nonlinear dispersive effects. We prove that as the diffusion parameter tends to zero, the solution of the dispersive equation converges to the unique entropy solution of a scalar conservation law. The proof relies on deriving suitable a priori estimates together with an application of the compensated compactness method in the L p setting.
We prove existence and uniqueness of a solution to the Cauchy problem corresponding to the equationHere, f ε,δ and u 0 are smooth functions while ε and δ = δ(ε) are fixed constants. Assuming f ε,δ → f ∈ L p (R d ×R; R d ) for some 1 < p < ∞, strongly as ε → 0, we prove that, under an appropriate relationship between ε and δ(ε) depending on the regularity of the flux f, the sequence of solutions (u ε,δ ) strongly converges in L 1 loc (R + × R d ) towards a solution to the conservation law ∂tu + divf(x, u) = 0.The main tools employed in the proof are the Leray-Schauder fixed point theorem for the first part and reduction to the kinetic formulation combined with recent results in the velocity averaging theory for the second.
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